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Optimal ballistically captured Earth–Moon transfers
Acta Astronautica 76 (2012) 1–12
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Optimal ballistically captured Earth–Moon transfers Paul Ricord Griesemer a,n, Cesar Ocampo a, D.S. Cooley b a b
Department of Aerospace Engineering and Engineering Mechanics, 1 University Station C0600, The University of Texas, Austin, TX 78712, USA Navigation and Mission Design Branch, NASA Goddard Space Flight Center, Mailstop 595.0, Greenbelt, MD 20771, USA
a r t i c l e i n f o
abstract
Article history: Received 29 January 2011 Received in revised form 5 January 2012 Accepted 7 January 2012 Available online 28 February 2012
The optimality of a low-energy Earth–Moon transfer terminating in ballistic capture is examined for the first time using primer vector theory. An optimal control problem is formed with the following free variables: the location, time, and magnitude of the transfer insertion burn, and the transfer time. A constraint is placed on the initial state of the spacecraft to bind it to a given initial orbit around a first body, and on the final state of the spacecraft to limit its Keplerian energy with respect to a second body. Optimal transfers in the system are shown to meet certain conditions placed on the primer vector and its time derivative. A two point boundary value problem containing these necessary conditions is created for use in targeting optimal transfers. The two point boundary value problem is then applied to the ballistic lunar capture problem, and an optimal trajectory is shown. Additionally, the problem is then modified to fix the time of transfer, allowing for optimal multi-impulse transfers. The tradeoff between transfer time and fuel cost is shown for Earth–Moon ballistic lunar capture transfers. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Optimal transfer Primer vector Ballistic capture
1. Introduction With Belbruno’s introduction of the Weak Stability Boundary (WSB) [1], and Belbruno and Miller’s [2] subsequent application of the concept to ballistic lunar capture transfers (BLCTs), low energy transfers have been shown to be a useful tool in transferring between orbits around different gravitational bodies. Although they have demonstrated savings in fuel consumption when compared to other classes of transfers the question of finding optimal WSB transfers raises issues not previously explored. This work applies optimal control theory to the problem of transferring a spacecraft from an orbit around a gravitational point mass to a ballistically captured state around a second gravitational point mass. The method relies on new optimality conditions for a ballistically captured spacecraft that are derived in agreement with primer vector theory. The new conditions are
n
Corresponding author. Tel.: þ512 619 7154. E-mail address: [email protected] (P. Ricord Griesemer).
0094-5765/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2012.01.001
applied to the case of a ballistic lunar capture transfer, in which a spacecraft in Earth orbit is transferred to a captured lunar orbit via a long low energy transfer that travels beyond the Moon to the Earth–Sun WSB and then returns to the Moon from the far side. The new optimality conditions are used in a numerical targeting algorithm to verify the optimality of a given ballistic lunar capture transfer. One mathematical framework that has proven useful in the study of BLCTs is dynamical systems theory. The notion of heteroclinic connections between periodic orbits in dynamical systems dates back at least to 1937 [3]. More recent studies by Koon et al. [4,5], Gomez and Masdemont [6], and Canalias and Masdemont [7] consider the use of heteroclinic connections in the circular restricted three body problem for the design of interplanetary transfers and of tours of planet–moon systems. These connections are established by numerically identifying connections between stable and unstable manifolds of periodic orbits in the circular restricted three body problem (CRTBP) [5], and methods for their computation have been provided by the authors.
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P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
Nomenclature
E
c WSB J
weak stability boundary performance index of the optimal control problem t time Dv change of velocity vector relating to an impulsive maneuver d(t) impulse function dn(t tc) delta sequence ath thrust magnitude _ m mass flow rate of the engine propellant c characteristic velocity of the engine propellant m mass of the spacecraft t time-like variable indicating the position of the spacecraft on a given orbit u vector of controls C control constraint r radius vector v velocity vector g acceleration vector of the general force field H boundary conditions at the initial time m* pre-defined initial mass of the spacecraft E* desired Keplerian energy state
When analyzed in the context of dynamical systems theory, BLCTs are shown to follow manifolds of libration point orbits in the Earth–Sun and Earth–Moon CRTBPs. In the first part of the transfer, the spacecraft trajectory shadows the stable and unstable manifolds of a Sun–Earth libration point orbit. In the second part of the transfers, the trajectory shadows the stable and unstable manifolds of an Earth–Moon libration point orbit. Though BLCTs do not meet the formal definition of low energy transfers given above, the term is applied to BLCTs to signify the reduction in relative energy of the spacecraft on its approach to the Moon. The libration point orbit manifolds were exploited for trajectory design by Howell et al. in the context of missions to libration point orbits [8]. In these mission designs, linear targeting algorithms were used to design missions from a specified orbit to a stable manifold of the desired trajectory. The spacecraft follows the manifold and ballistically enters into the libration point orbit. The method has been applied to libration point missions such as Genesis [9]. Several authors have applied similar theory to the analysis and targeting of low energy transfers to the Moon [10–12]. The problem becomes more complicated for the lunar transfers due to the need for the simultaneous calculation of manifolds of the Earth–Moon libration points and the Earth–Sun libration points. Algorithms exist that utilize the invariant manifolds associated with libration point orbits in both the Sun–Earth and Earth– Moon CRTBPs. For example, Koon et al. [10] targeted BLCTs by finding intersections of these invariant manifolds. Similarly, Yamato and Spencer [11] approximated
a G u
n
m ^ H
k p f a c
F(tf,t0) X a e i
o O
actual Keplerian energy state boundary condition at the final time slack variable endpoint function constant Lagrange multiplier associated with the final boundary condition vector of constant Lagrange multipliers associated with the initial boundary conditions Lagrange multiplier associated with the control constraint extended Hamiltonian function adjoint variable vector primer vector state equation vector optimization parameters in the numerical optimization problem vector of constraints in the numerical optimization problem state transition matrix vector of state variables semi-major axis eccentricity inclination argument of perigee right ascension of the ascending node
the invariant manifolds in a perturbed CRTBP, yielding transit orbits and lunar capture. These transfers are typically found with manifolds of halo orbits around the libration points. The problem of minimizing the fuel consumption associated with impulsive orbit-to-orbit transfers has received a great amount of attention. Formalized by Lawden in 1963 [12], the necessary conditions for an optimal impulsive transfer are imposed on the primer vector, the reflection of the vector of adjoint variables in the optimal control problem that correspond to the velocity components of the spacecraft state vector. Lawden’s necessary conditions specify the magnitude and orientation of the primer vector at the times of impulse. Specifically, the primer vector must align with the direction of the impulse and have a uniform magnitude at each impulse. The magnitude has been traditionally chosen to be unity at each impulse, but the designated value is arbitrary. Lion and Handelsman [13] extended Lawden’s work to determine when an additional impulse would improve on the fuel cost of the transfer. Jezewski [14] later gives a summary of primer vector theory, and introduces an estimation of the magnitude of an intermediate impulse that would optimize the transfer. Primer vector theory has been used to evaluate the optimality of several different types of transfers. Optimal transfers in an inverse square gravitational field have been analyzed for various types of missions. Rendezvous in the vicinity of a circular orbit has been explored by Prussing [15], direct ascent rendezvous by Gross and Prussing [16], and rendezvous between circular orbits by Prussing and Chiu [17]. Jezewski [14] has studied the
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
problem of multiple impulse transfers and transfers with inequality constraints. The primer vector in the three body problem was studied by Hiday-Johnston and Howell [18] to evaluate the optimality of transfers between libration point orbits, by D’Amario and Edelbaum [19] in the transfer from a collinear libration point to a lunar orbit, and by Ocampo [20] in the problem of insertion into distant retrograde orbits. The use of primer vector theory in the four body problem, the dynamics of which is required for Lunar WSB transfers, has been less comprehensive. In one example, Pu and Edelbaum [21] created a numerical method to compute 2-impulse and 3-impulse optimal transfers in the four body problem. Once the trajectories were numerically optimized, the primer vector history of the transfers was evaluated to verify their optimality. A similar method is employed here. The goals of the current research are twofold. First, we apply primer vector theory to derive new necessary conditions for a type of transfer that had not yet been considered in the context of primer vector theory, namely ballistic capture transfers. Previous implementations of primer vector theory have been insufficient to deal with ballistic capture trajectories. Necessary conditions have been traditionally based on the magnitude and orientation of the primer vector at the times of impulse. In a ballistic capture scenario with only one impulse occurring during the entire transfer, necessary conditions relating to the impulse provide an insufficient number of conditions to determine an optimal transfer. New conditions must be derived specifically for a ballistic capture to produce the required number of necessary conditions to define local optimality. Ballistic capture trajectories can be used in a number of contexts and offer advantages over traditional Hohmann-based approaches. As an example, the BepiColumbo mission cites the engine fault tolerance of a ballistic capture arc as a motivation for using a ballistic capture transfer to Mercury [22]. As these types of transfers become better understood, an analysis of their optimality becomes necessary. The second goal is to demonstrate, through the examination of BLCTs, an application of the new necessary conditions. Particular transfers in the Sun–Earth–Moon system will be found using the targeting algorithm and then analyzed to determine whether the transfers can be improved with an intermediate impulse. Multiple impulse fixed-time ballistic capture transfers will then be constructed to determine the tradeoff between optimal fuel consumption and transfer time.
optimal control of this spacecraft is based on Hull’s [23] formulation of the optimal control problem. Where the derivation overlaps with the orbit-to-orbit problem, it has equivalent results to those of Lawden, Jezewski, and Lion and Handelsman [12–14]. 2.1. Cost function An optimal control will minimize the total impulse provided to the spacecraft, represented by the following performance index: Z t2 J¼ Dv1 dðtt 1 Þ dt, ð1Þ t 1
where nv1 is the magnitude of the initial impulses, t2 is defined to be the final time of the transfer, and d(t t1) is an impulse function in which the impulse occurs at the beginning of the transfer, t ¼t1. The impulse function is used as a way to represent a finite change in the velocity of the spacecraft due to an infinite acceleration. By containing the limit functions within the impulse function, the necessary conditions for optimal control can be derived without taking the limit of the performance index itself. The delta function used here is defined by the relationship:
dðttc Þ ¼ lim dn ðttc Þ, n-1
Consider the orbit-to-capture transfer problem for an impulsively maneuvering spacecraft in a general force field. A spacecraft orbiting a body is to be transferred to any captured orbit of another body via a single impulse. In this situation, capture is defined by an instantaneous negative Keplerian energy level with respect to the second body, assuring an instantaneous elliptical orbit around the body. The derivation of the necessary conditions for an
ð2Þ
where dn(t tc) is a delta sequence that may be defined by the following piecewise continuous function [25]: 8 1 0 for t o t c 2n > > < 1 1 ð3Þ dn ðttc Þ ¼ n for tc 2n ot o tc þ 2n > > 1 : 0 for t c þ ot 2n When used in the cost function, the delta function represents an impulsive maneuver. The velocity change and fuel consumption associated with the maneuver can be determined through the integration of the function. The integral of the delta function has the property [24] Z 1 f ðtÞdðtt c Þ dt ¼ f ðt c Þ ð4Þ 1
The resulting velocity change due to a maneuver at time c can be determined explicitly using Eq. (4): Z tcþ DVc ¼ Dvc dðttc Þlc dt t c
Z
1
Dvc dðttc Þlc dt
¼ 2. Primer vector theory for an impulsive low energy transfer
3
1
¼ Dvc lc
ð5Þ
In this expression, lc is the steering vector that orients the impulse. The mass consumed by a maneuver can be calculated from the mass state equation. If thrust acceleration, ath, is represented by the expression: ath ¼
_ mc m
ð6Þ
_ is the mass flow rate of the propellant and c is its where m characteristic velocity, then the following equation holds
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for an impulsive maneuver occurring at time t1: _ mc Dv1 dðtt1 Þ ¼ m
ð7Þ
or _ ¼ m
mDv1 dðtt 1 Þ c
ð8Þ
2.2. Optimal control problem The controls in the orbit-to-capture problem, shown in Eq. (9), are the magnitude and direction of the transfer insertion maneuver at time t1 and the location of the spacecraft on a specified orbit around a gravitational body, represented by the time-like parameter t1. Following the methodology of Ocampo [20], t1 is introduced into the problem to define a plane in phase space that contains the initial orbit. This variable is defined such that r(t1) and v(t1) lie on the initial orbit for any t1. In this way, t1 is similar to the true anomaly of an orbit, or any other variable that defines a spacecraft’s location on a particular orbit. Both t1 and t2 are free parameters, but are not included among the controls according to the methodology of Hull: u ¼ ð Dv1
l1
T
t1 ÞT
ð9Þ
A control constraint exists on the thrust direction vector l1 so that its magnitude is unity: T
C ¼ ðl1 l1 1Þ ¼ 0
ð10Þ
The differential constraint for the problem is the state equation: ! v r_ ¼ , ð11Þ x_ ¼ g þ dðtt 1 ÞWv1 l1 v_ where r and v are the position vector and the velocity vector, respectively. The gravitational acceleration vector g represents a general force field, and although the low energy lunar transfer is typically calculated with a four body system, no assumptions about the composition of the gravity field will be made in the derivation. The acceleration vector for the spacecraft consists of the gravitational acceleration and the acceleration due to the impulsive maneuver. The initial conditions constrain the initial state of the spacecraft to lie on a pre-established orbit. The constraint is written as ! r1 rðt1 Þ h¼ ¼0 ð12Þ v1 vðt1 Þ The final conditions are generalized to require the spacecraft to arrive at the second body with a specific energy state with respect to the second body less than or equal to a stated value. For a spacecraft to be captured in a two body system, the stated energy level would be less than zero. In a multi-body system, however, a stronger constraint may be desired to increase the likelihood of the spacecraft orbiting the second body before it escapes. In the derivation, the desired energy level will be left unspecified.
An inequality constraint is used to allow for the possibility of an optimal transfer arriving at the secondary body at a more strongly captured state than specified. The resulting final condition is
c ¼ En
vs 2 ms r 0, 2 rs
ð13Þ
where rs and vs are the radius and the velocity with respect to the secondary body, respectively, ms is the mass parameter of the secondary body, and E* is the desired energy state. Following the methodology of Hull [23], a slack variable is introduced to convert the inequality constraint in Eq. (13) into an equality constraint. A consequence of the introduction of a slack variable into the constraint is an additional parameter in the problem. The final condition becomes
c¼
vs 2 m s En þ a2 ¼ 0 2 rs
ð14Þ
where a is the slack variable. The boundary constraint function is T
G0 ¼ uc þ n h
ð15Þ
where u and x are a scalar and a vector, respectively, of constant Lagrange multipliers adjoined to the endpoint constraints. An extended Hamiltonian is formed using a Lagrange multiplier m to adjoin the control constraint in Eq. (10) to the Hamiltonian: ^ ¼ kr T v þ kv T ðg þ Dv1 dðtt 1 Þl1 Þ þ Dv1 dðtt 1 Þ þ mC H
ð16Þ
The extended Hamiltonian is constant if the gravitational acceleration vector is not a function of time. It is not necessarily an integral of the problem for the general force field. The performance index and constraints described above form a time-free optimal control problem. The necessary conditions from the first differential are given by the Euler–Lagrange equations: x_ ¼ f
ð17Þ
k_ ¼ H^ x
T
^ uT 0¼H
ð18Þ ð19Þ
and the natural boundary conditions: ^ 2 ¼ G0 H t2 ^ 1 ¼ G0 H t1
k2 ¼ G0x2 T k1 ¼ G0x1 T
ð20Þ
The conditions of optimality pertaining to the impulse in the orbit-to-capture problem are equivalent to the orbit-to-orbit necessary conditions derived in the work of Lawden, Jezewski, and others. As previously stated, the primer vector must be aligned with the impulse at the time of the impulse, the time derivative of its magnitude must be zero, and its magnitude must be unity: p1 ¼
DV1 Dv1
ð21Þ
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
p_ 1 ¼ 0,
ð22Þ
where p ¼ kv p_ ¼ kr
ð23Þ
At time t2, there are two complications that call for necessary conditions that differ from the conventional derivation. First, the inequality endpoint constraint does not fit neatly into the optimal control theory formulation of Hull. After extending Hull’s formulation to account for an endpoint inequality constraint, the inclusion of the slack variable produces two separate cases, one when the solution is on the boundary of the constraint, and one when it is off of the boundary. Secondly, the traditional impulsive orbit-to-orbit problem contains at least two impulses. A first impulse initiates the transfer, and a second inserts the spacecraft from a transfer trajectory into the target orbit. This paradigm is insufficient when dealing with ballistic capture problems. New conditions need to be derived from primer vector theory to account for the possibility of reaching the desired state without an impulse to conclude the transfer. Hull’s derivation allows for the endpoint function to depend on the initial and final time, the state variables, and constant Lagrange multipliers. The slack variable associated with the inequality constraint cannot be considered to belong to any class of these variables. It can be dealt with outside of Hull’s formulation by treating it as a separate class of variable and including it in a modified endpoint function, G0 . The modified endpoint function shown in Eq. (15) depends on the initial and final time, the state variables, constant Lagrange multipliers, and the slack variable, a. Hull derives the first order conditions of optimality by taking the first differential of the extended cost function, which includes the endpoint function. By adding the slack variable to the parameters of the problem, a term appears in the differential that depends on da: 0
dG ¼ 2ua da
ð24Þ
In order for a trajectory to be optimal, the expression in Eq. (24) must be equal to zero. This optimality condition can be satisfied in two cases. In the first case, a is equal to zero and the solution lies on the boundary of the inequality constraint. In the second case, the Lagrange multiplier is equal to zero. This condition applies to the situation where the optimality conditions are satisfied on a trajectory that terminates with an energy level less than the targeted energy level. All other conditions on G0 are equivalent to the conditions on G in Hull’s formulation. To determine necessary conditions appropriate for an orbit to capture trajectory at the terminal time, consider the endpoint conditions on from Eq. (20), along with the definition of the primer vector given in Eq. (23): ms p_ 2 ¼ G0T r2 ¼ u r 3 rs
at the final time is aligned with the velocity vector of the spacecraft with respect to the second gravitational body, and the rate of change of the primer vector is aligned with the radius vector of the spacecraft with respect to the gravitational body. Furthermore, because the constant Lagrange multiplier u appears in both expressions, its value is not arbitrary. The parameter u establishes a relationship between the magnitude of the primer vector and its time derivative. It is either zero in the off boundary case, or it can be solved for by post-multiplying the equation for the Lagrange multipliers associated with the velocity by the relative velocity vector: u¼
kTv2 vs
ð26Þ
v3s
kv2 and vs must be parallel to satisfy Eq. (25), so their inner product can be reduced to the product of their two magnitudes, resulting in a simplified expression for u: u¼ 7
lv2
ð27Þ
vs 2
The sign of Eq. (27) is left to be determined by analyzing the results of a particular problem. Together, Eqs.(24) and (25) define the necessary conditions at the final time for a ballistic capture trajectory. They apply to the magnitude and direction of both the primer vector and the time derivative of the primer vector, and represent trajectories that terminate with an energy level either equal to that targeted or less than the targeted value. Additionally, it can be shown that the magnitude of the primer vector must be less than unity to preclude an additional impulse. 2.3. Additional impulses A separate analysis can be made to determine whether an intermediate impulse has the potential to reduce the cost of the maneuver. Consider a trajectory that has an additional maneuver at time tk. The performance index of the trajectory would be changed to Z tþ 2 J¼ Dv1 dðtt1 Þ þ Dvk dðttk Þ dt, ð28Þ t 1
and the vector of controls would include the magnitude, time, and direction of the intermediate maneuver: u ¼ ð Dv1
l1
t1 Dvk lk tk ÞT
ð25Þ
Eq. (25) establishes new necessary conditions for the orbit-to-capture problem. It dictates that the primer vector
ð29Þ
The unit thrust vectors are subject to the constraint: ! T l1 l1 1 C¼ ¼ 0, ð30Þ T lk lk 1 and the extended Hamiltonian is ^ ¼ kT v þ kT ðg þ Dv1 dðtt 1 Þl1 þ Dvk dðtt k Þlk Þ H r v þ Dv1 dðtt 1 Þ þ Dvk dðtt k Þ þ lT C
s
p2 ¼ G0T v2 ¼ uvs vs
5
ð31Þ
Applying the optimality condition in Eq. (19) to Eq. (31) gives the following result when considering the magnitude of the primer vector at an intermediate
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impulse: ^ Dv ¼ dðtt k ÞkT lk þ dðtt k Þ ¼ 0 H v k
ð32Þ
At time tk Eq. (32) reduces to kvk ¼ 1
ð33Þ
The result obtained from applying the same condition at the initial time is kv1 ¼ 1 ð34Þ It can be seen from comparing Eq. (33) to Eq. (34) that a necessary condition for an intermediate impulse is p ¼ p ð35Þ k
1
Eq. (35) requires that an optimal trajectory with an intermediate impulse have primer vectors with equal magnitudes at each impulse. As stated previously, it is common to assign a unit magnitude to the primer vector at the initial impulse. Consequently, if the primer vector never again returns to unit magnitude between times t1 and t2, there is no potential for an improvement in the performance index via an intermediate impulse. 3. Numerical analysis The necessary conditions on the state and co-state vectors described above can be used in a targeting algorithm to verify the optimality of transfers or optimize near-optimal trajectories. The boundary conditions and the necessary conditions of the optimal control problem define a two point boundary value problem that can be numerically solved to yield a control that satisfies the conditions of optimality. The unknown and constraint vectors associated with the problem are shown in Eq. (36). The differential equations to be satisfied are the state and co-state equations shown in Eqs.(17) and (18): a11x1 ¼ ð t 1
^ c11x1 ¼ H1
p_ 1
t2
t1
^2 H
EEn þ a2
Dv1 2am
u
used for this analysis is to remove the initial and final times from the vector of free parameters in the optimal control problem, and thus fix the transfer time. With the addition of a second impulse, optimal transfers can be constructed that are initiated by a transfer insertion burn and have a second burn that places the spacecraft on a ballistic arc that transitions to a captured state at the final time. By constructing transfers using this methodology, the transfer time can be set anywhere between an upper bound of the transfer time of a single impulse low energy transfer and lower bound of the transfer time of a direct two impulse transfer. The necessary conditions for an optimal two impulse transfer to a ballistic arc terminating at a desired energy state mirror those presented in Section 2 for an optimal ballistic lunar transfer. The two modifications of the problem are the removal of the initial and final times from the vector of free parameters and the addition of an intermediate impulse. The two point boundary value problem that is used to target optimal trajectories shown in Eq. (36) can be modified in the following way to target optimal time-fixed trajectories. The initial and final times are removed from the parameter vector, and the time, magnitude, and direction of the intermediate impulse are added. The constraint vector is modified by eliminating the constraints on the initial and final Hamiltonians and adding the optimality conditions on the intermediate impulse (Eq. (35)). The resulting two point boundary value problem, shown in Eq. (37), is thirteen dimensional and can be solved by iterating from an initial guess using a numerical nonlinear equation solver: a13x1 ¼ ð tk
Dvk
f
p_ 1
Dv1
u
a ÞT ,
T m Dv n T 2 c13x1 ¼ pk jDvkk j pk p_ k EE þ a 2am kr2 u r3ss rs kv2 uvs vs p_ 1
ð37Þ
T
aÞ ,
kr2 u mr3s rs s
kv2 uvs vs
p_ 1
T
3.2. The restricted four body problem
ð36Þ This system, with an appropriate initial guess, can be solved by a numerical nonlinear equation solver such as NS-12 [25]. The initial state vector is formed from the prescribed initial orbit and the parameter t1. The initial co-state vector is determined from the parameter p_ 1 and by aligning the primer vector in the direction of the impulse and giving it a magnitude of unity. A solution to this system is not guaranteed to meet the necessary conditions of optimality, as there are no path constraints to ensure the magnitude of the primer vector remains below unity. An inspection of the primer vector history to verify this condition is required to ensure that the trajectory meets the necessary conditions once a solution to the boundary value problem is found. 3.1. Multiple impulses In this section, analysis is presented with the intent to provide a means of analyzing the trade space between transfer time and fuel cost in lunar transfers. The method
The two point boundary value problem shown above can be solved for any dynamical system. However, single impulse ballistic lunar transfer trajectories rely on the gravitational attraction of the Sun, Earth, and Moon [2]. In this paper, the governing dynamics of the spacecraft are generated by assuming these three primary bodies act as point masses attracting the spacecraft. In the numerical analysis, their orbits about each other are defined by the JPL DE405 ephemeris [26]. A spacecraft moving through their gravitational field is assumed to be infinitesimally small, so that it will not perturb the orbits of the primary masses. A non-rotating coordinate system is established that is centered on the Earth–Moon barycenter. Assuming Newtonian gravity, the gravitational acceleration of a spacecraft in this four-body system is defined in Eq. (38) [27], where the vectors are defined in Fig. 1. r€ ¼
me jr-re j3
ðrre Þ
mm jr-rm j3
ðrrm Þ
ms jr-rs j3
ðrrs Þ
ms r 3s
rs ð38Þ
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
convergence. The algorithm used to target optimal trajectories requires partial derivatives that relate the constraints, c, of the problem to the free parameters, a. For the constraints that are applied at the initial time, partial derivatives can be derived directly from the initial state. For the constraints at the final time, the state transition matrix can be used in conjunction with the chain rule to determine partial derivatives that relate parameters at the initial time to the final constraints. It should be noted, however, that the state transition matrix can only relate changes in the state that occur at fixed times. Therefore, when relating constraint values to the variation of the initial time, an additional step must be taken. The changes in the state at t ¼ t1 must be related to the change in the initial time by calculating a numerical derivative that relates to change in the state at a fixed time t1 to a change in the initial time. The time derivative of the state vector does not apply to this calculation, as discussed by Zimmer [28]. This derivative can be calculated with the forward difference method by simply integrating the state from a perturbed time back to t1. The partial derivatives used in the numerical routine are shown below:
Fig. 1. Restricted four body problem.
Table 1 Gravitational parameter values. Gravitational parameter
Value (km3/s2)
ms me mm
1.32715 1011 3.986004 105 4.9029 103
@Xðt 2 Þ @Xðt 1 Þ ¼ Fðt 2 ,t 1 Þ @t 1 @t 1
The gravitational parameters ms, me, and mm are defined in Table 1. Eq. (38) contains three direct gravitational accelerations followed by an indirect term. The indirect term accounts for the acceleration of the reference frame around the Sun–EarthEarth–Moon barycenter. 0
1 kTv @g @t1
B B @H^ B 2 Fðt 2 ,t 1 Þ @X1 @t 1 B @X2 B B @c3 Fðt ,t Þ @X1 2 1 @t1 B @X2 @c B 0 ¼B B @a B @c 1 B @X52 Fðt 2 ,t 1 Þ @X @t1 B B @c6 1 B @X Fðt 2 ,t 1 Þ @X @t1 B 2 @ 0
^ 1 @X1 @H @X1 @f
0
7
ð39Þ
where q[x(t1)]/qt1 is calculated using the forward difference method by varying the initial time by a small amount and numerically integrating back to t1.
0
0
0
0
^ 2 @X2 @H @X2 @t 2
^2 @H @X2
1 Fðt2 ,t1 Þ @X @f
^2 @H @X2
1 Fðt2 ,t1 Þ @X @p_ 1
^2 @H @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
0
@c3 @X2 @X2 @t 2
@c3 @X2
1 Fðt2 ,t1 Þ @X @f
@c3 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c3 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
0
@c5 @X2 @X2 @t 2
@c5 @X2
1 Fðt2 ,t1 Þ @X @f
@c5 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c5 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
mr3s rs
@c6 @X2 @X2 @t 2
@c6 @X2
1 Fðt2 ,t1 Þ @X @f
@c6 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c6 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
vs vs
0
0
0
0
0
1 p_ 1
3.3. State transition matrix derivatives It should be stressed that convergence is an issue when working with these chaotic trajectories in the four body problem. Jacobian matrices are poorly conditioned and numerical targeting algorithms that rely on linear approximations to the complex dynamical systems have small convergence envelopes. A reasonable first guess is required to converge upon an optimal solution. A particularly problematic case for convergence is the case of a lunar flyby on the outbound portion of the transfer, which yield Jacobian matrices that are more poorly conditioned. Although transfers with these flybys can be more efficient, they will not be considered here. As documented by Zimmer and Ocampo [28] the use of analytical derivatives in targeting algorithms that require gradients increases both the speed and likelihood of
0
T p_ 1
0 s
0
0
1
C C 0 C C C 2a C C C 2m C C C 0 C C C 0 C C A 0
ð40Þ
where ^ 2 T @g @H ¼ kv2 @r22 @X2
kTr2
gT2
vT2
ð41Þ
@ðEEn þ a2 Þ mm rT vT 0 0 ¼ rm 3 m m @X2 @c5 T 1 1 0 ¼ umm rm 3 I þ3 rm 5 rm rm @X2 @c6 ¼ @X2
0
@X2 ¼ ð vT @t 2 @X1 ¼ 0 @p_ 1
u vm I þ v1m vm vm T
k_ r
gT 0
I
T
0
T
k_ v Þ
I
ð42Þ 0
0
I
ð43Þ
ð44Þ
ð45Þ
ð46Þ
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P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
@X1 ¼ 0 @Dv1
I
0
0
ð47Þ
0
1 aðcos O sin f þ sin O cos f cos iÞ B C B aðsin O sin f þ cos O cos f cos iÞ C B C C a cos f sin i @X1 B B C ¼B C ð O cos f sin O sin f cos i Þ v cos B C @f B C B vðsin O cos f þ cos O sin f cos iÞ C @ A v sin f sin i
ð48Þ
Note that in these equations the initial orbit around the Earth is circular, where a is the semi-major axis of the orbit, f is the true anomaly of the orbit, i is the inclination, v is the magnitude of the velocity, and O is the right ascension of the ascending node. F(t2,t1) is the state transition matrix associated with the four body equations of motion given in Eq. (38). These partial derivatives are employed in the targeting code to iterate from an initial guess to an optimal solution. The guesses are propagated using a numerical predictor-corrector integration method [29]. 4. Results An optimal transfer is sought from the low Earth orbit and detailed in Table 2 to a captured state around the Moon using a single impulsive maneuver. A previously targeted nonoptimal transfer is used as an initial guess, and improved upon by targeting the necessary conditions developed above. Fig. 2 shows a converged solution in non-rotating coordinates centered at the Earth–Moon barycenter. Table 2 Parking orbit properties at t0. Orbit property
Value
epoch (JED) a (km) e i (deg.) o (deg.) O (deg.)
2,455,658.47677 7200 0 47.188 0 1.09948
In an effort to find a suitable initial guess for the targeting of an optimal transfer, parameters from a non-optimal ballistic lunar capture trajectory were calculated using a method detailed in previous work [30]. The parameters of the transfer are listed in Table 3, where the subscript (v) refers to the component of the vector in the direction of the velocity, the subscript (h) refers to the component in the direction of the angular momentum, and the subscript (r) refers to the direction that completes the right-handed coordinate system. The final state of the transfer lies on an elliptical two body orbit about the Moon. The parameters of this transfer are used as an initial guess in the two point boundary value problem described above. The initial values of the co-state associated with the velocity are assigned to align the primer vector with the impulse and give it a unit magnitude. The remaining initial values, those in the co-state associated with the radius, are assigned very small numbers. The solution to the two point boundary value problem in Eq. (36) yields a transfer that meets all of the necessary conditions of optimality. The targeted energy state is E* ¼ 0.02 MJ/kg in accordance with the experience gained in targeting states that result in a physical orbit of the Moon [30]. The time history of the magnitude of the primer vector for the converged solution to the two point boundary value problem is shown in Fig. 3. This primer vector history satisfies the necessary conditions detailed above: it begins with a magnitude of unity and a slope of zero, shown in detail in Fig. 4, and never again returns to a magnitude of unity. Additionally, this solution lies on the boundary of the inequality constraint, with slack variable, a, having a value of zero for this transfer and satisfying the condition in Eq. (24). Through the computation of the necessary conditions, the maneuver was improved from 3.050475 km/s to 3.050474 km/s. It should be noted that this modest reduction is with respect to a numerically optimized result. The role of this analysis is to verify the optimality of the previous findings. Finally, it is not evident in Fig. 3, but the primer vector at the final time is aligned with the velocity vector and its time derivative is aligned with the radius vector with respect to the moon. Not only are the vectors aligned but their magnitudes are related through the Lagrange multiplier u. Table 4 shows the angles of the related vectors relative to their counterparts. 4.1. Two impulse ballistic capture transfers The methodology of Section 3.1 is applied to determine the tradeoff between time of flight and fuel consumption for BLCTs. In this approach, transfer time is a fixed Table 3 Transfer properties. Transfer property
Value
DVv (km/s) DVr (km/s) DVh (km/s) Dt (days)
3.05048 0.0 0.0 117.32581 22.6881
f (deg.) Fig. 2. Single burn low energy transfer to lunar orbit.
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
9
Fig. 3. Primer vector magnitude of a single burn transfer.
Fig. 4. Detail of the primer vector magnitude near the initial time.
Table 4 Primer vector properties at tf. Primer vector angle with the velocity vector (deg.) Primer rate vector angle with the radius vector (deg.)
0.01 179.95
variable, and the two point boundary value problem is slightly modified to the form of Eq. (37). E* remains 0.02 MJ/kg. The necessary conditions on the primer vector for this problem are that the magnitude begins at unity and returns to a maximum of unity at the time of the impulse. A very good initial guess near the solution in both the state space and the co-state space must be provided to the nonlinear targeting algorithm. A numerically optimized transfer can be used as such a guess and can provide a solution within the convergence envelope. Based on the previous results of optimal orbit-to-orbit transfers, it would be expected that an optimal low energy transfer would terminate at or near a periapse with respect to
Fig. 5. Time-fixed reference trajectory in non-rotating coordinates.
the Moon. Therefore, it will be assumed that an optimal transfer to periselene is the desired target for generating an initial guess. A transfer of this type can be found by adding a second impulse at the apogee of a low energy orbit-to-orbit transfer and then targeting a minimum Dv transfer to the Earth–Moon L2 point as the transfer time is reduced. Fig. 5 shows a transfer that was formed to meet these conditions with a transfer time of 54 days. The numerical optimization of this transfer is continued with the constraint vector altered to target a periselene. Fig. 6 shows the results of the numerical optimization. This trajectory is the initial guess for the two point boundary value problem described in Eq. (37). The initial guess for the co-state parameters is formed by satisfying two of the necessary conditions. The initial primer vector is chosen to align with the velocity vector and has a unit magnitude. The primer rate vector is calculated based on the necessary condition that the primer vector be aligned with the second impulse.
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P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
1.1 1
primer magnitude
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2
Fig. 6. Fixed time initial guess in non-rotating coordinates.
3
3.5
4 4.5 time
5
5.5
6
6.5 x 106
Fig. 8. Primer vector history of the converged optimal solution.
3.8
5
3.7 delta V (km/s)
4.5 4 primer magnitude
2.5
3.5 3 2.5
3.6 3.5 3.4 3.3 3.2
2
3.1 0
1.5
10
20
1
30 40 50 transfer time (days)
60
70
80
70
80
Fig. 9. Optimal time-fixed transfer parameters.
0.5 0 2.5
3
3.5
4 4.5 time
5
5.5
6
6.5 x 106
Fig. 7. Primer vector history of the initial guess.
Fig. 7 shows the primer vector history of the initial guess, with the impulse applied at the time signified by the overlaid circle. The trajectory does not meet the conditions of optimality due to the segment near the end of the trajectory where the magnitude exceeds unity. The initial guess is iterated to an optimal solution using a numerical non-linear equation solving routine until the two point boundary value problem is solved. The primer vector history of the converged solution is shown in Fig. 8. In the process of solving the optimization problem, the magnitude of the cost was reduced from 3.30535 km/s to 3.288372 km/s. The method described above is repeated for different transfer times spanning the space between the Hohmann transfer time and the single impulse low energy transfer time. Fig. 9 shows the results of the trade-space study. It displays a linear increase in fuel cost as transfer time is shortened. Figs. 10 and 11 show the change in magnitude of the first and second maneuver, respectively, for the
3.06 3.05 delta V1 (km/s)
2
3.04 3.03 3.02 3.01 3 0
10
20
30 40 50 transfer time (days)
60
Fig. 10. Time-fixed transfer initial maneuver magnitude.
transfers included in Fig. 9. The cost of shortening the transfer time is shown to be entirely contained in the second impulse. The initial impulse actually decreases in magnitude as the transfer is shortened. This result can be interpreted by assuming the second impulse does the work of the Sun as the transfer time is shortened and the spacecraft does not travel far enough from the Earth to be influenced by the Sun’s dynamics.
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
consumption is explored for ballistic transfers. Optimal multiple impulse transfers are computed that show the cost of the initial impulse is reduced, but offset by the addition of a second impulse when shortening the travel time.
0.8 0.7 delta V2 (km/s)
11
0.6 0.5 0.4 0.3
Acknowledgments
0.2 0.1 0 0
10
20
30 40 50 transfer time (days)
60
70
80
Fig. 11. Time-fixed transfer intermediate maneuver magnitude.
5. Conclusions A method of targeting optimal low energy transfers, relying on multi-body dynamics, from an orbit about one body to another is presented. The targeting algorithm applies necessary conditions derived from primer vector theory to the orbit-to-capture transfer problem. With the help of precise derivatives obtained from the chain rule and the state transition matrix, optimal solutions can be converged using the non-linear equation solving techniques. The necessary conditions for an impulsive single impulse orbit-to-capture transfer offer a tool for evaluating low energy missions such as lunar tour missions of the outer planets or ballistic lunar capture trajectories. The use of primer vector theory to obtain necessary conditions for the problem offers an extension of previous work with primer vector theory. The resulting conditions can be seen as slight modifications of, but still consistent with, previous results. For example, for the off boundary case, the primer vector of an optimal transfer must align with the velocity vector of the spacecraft with respect to the secondary mass, and the primer rate vector must align with the radius vector. Previously, it has been shown that the primer vector in an optimal orbit-to-orbit transfer must align with the impulse vector at the final time, and the time derivative of its magnitude must be zero. The new result is equivalent to previous findings as long as the second orbit insertion burn occurs at perigee and the burn is in the direction of the spacecraft’s relative velocity, two conditions previously associated with optimal transfers. At the perigee point, the velocity vector is perpendicular to the radius vector. At this point on an optimal trajectory, both the radius vector and the primer vector have magnitudes that are unchanging. Using the new necessary conditions, a ballistic lunar capture transfer was optimized and then analyzed to determine whether a second maneuver could improve the result. The transfer was found to be a local optimal with no available savings from an additional burn. It was therefore shown that a single burn transfer from the Earth to the Moon can meet the necessary conditions of optimality from primer vector theory. The methods are applied to multiple impulse transfers in which the tradeoff between transfer time and fuel
This work was supported by NASA Contract NNG05GL28H, and was approved for publication by NASA personnel. References [1] E.A. Belbruno, Lunar capture orbits, a method of constructing Earth–Moon trajectories and the lunar GAS mission, in: Proceedings of AIAA/DGLR/JSASS International Propulsion Conference, AIAA, 1987, pp. 97–1054. [2] E.A. Belbruno, J. Miller, Sun-perturbed Earth-to-Moon transfers with ballistic capture, J. Guid. Control Dyn. 16 (1993) 770–775. [3] A.A. Andronov, E.A. Vitt, S.E. Khaiken, Theory of Oscillators, Pergamon Press, Oxford, 1966. [4] W.S. Koon, M.W. Lo, J.E. Marsden, S.D. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos 6 (2000) 427–469. [5] W.S. Koon, M.W. Lo, J.E. Marsden and S.D. Ross, Shoot the Moon, in: AIAA/AAS Astrodynamics Conference, AAS 00-166 ,2000. [6] G. Gomez, J.J. Masdemont, Some zero cost transfers between libration point orbits, Adv. Astron. Sci. 105 (2000) 1199–1215. [7] E. Canalias, J.J. Masdemont, Impulsive transfers between lissajous libration point orbits, J. Astron. Sci. 51 (2003) 361–390. [8] K.C. Howell, B.T. Barden, M.W. Lo, Application of dynamical systems theory to trajectory design for a libration point mission, J. Astron. Sci. 45 (1997) 161–178. [9] K.C. Howell, B.T. Barden, R.S. Wilson and M.W. Lo, Trajectory design using a dynamical systems approach with application to GENESIS, in: Proceedings of AIAA/AAS Astrodynamics Conference, 1997, pp. 1665–1684. [10] W.S. Koon, M.W. Lo, J.E. Marsden, S.D. Ross, Low energy transfer to the Moon, Celest. Mech. Dyn. Astron. 81 (2001) 63–73. [11] H. Yamato, D.B. Spencer, Transit-orbit search for planar restricted three-body problems with perturbations, J. Guid. Control Dyn. 27 (2004) 1035–1045. [12] D.F. Lawden, Optimal Trajectories for Spacecraft Navigation, Butterworths, London, 1963. [13] P.M. Lion, M. Handelsman, Primer vector on fixed-time impulsive trajectories, AIAA J. 6 (1968) 127–132. [14] D.J. Jezewski, Primer vector theory and applications, NASA TR R454, 1975. [15] J.E. Prussing, Optimal two- and three-impulse fixed-time rendezvous in the vicinity of a circular orbit, AIAA J. 8 (1970) 1221–1228. [16] L.R. Gross, J.E. Prussing, Optimal multiple-impulse direct ascent fixed-time rendezvous, AIAA J. 12 (1974) 885–889. [17] J.E. Prussing, J.H. Chiu, Optimal multiple-impulse time-fixed rendezvous between circular orbits, J. Guid. Control Dyn. 9 (1986) 17–22. [18] L.A. Hiday-Johnston, K.C. Howell, Transfers between libration-point orbits in the elliptic restricted problem, Celest. Mech. Dyn. Astron. 58 (1994) 317–337. [19] L.A. D’Amario, T.N. Edelbaum, Minimum impulse three-body trajectories, AIAA J. 12 (1974) 455–462. [20] C. Ocampo, Trajectory optimization for distant earth satellites and satellite constellations, Ph.D. Dissertation, University of Colorado at Boulder, 1996. [21] C.L. Pu, T.N. Edelbaum, Four-body trajectory optimization, AIAA J. 13 (1975) 333–336. [22] S. Campagnola, M. Lo, BepiColombo gravitational capture and the elliptic restricted three-body problem, PAMM 7 (2007) 1030905–1030906. [23] D.G. Hull, Optimal Control Theory for Applications, Springer-Verlag, New York, 2003. [24] P.A.M. Dirac, Quantum Mechanics, Oxford University Press, London, 1958.
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[25] NS12AD, Harwell Subroutine Library, Available from: /http://hsl.rl. ac.uk/hsl2007/distrib/hsl2007.htmlS. [26] E.M. Standish, JPL Planetary and Lunar Ephemerides, DE405/LE405, Jet Propulsion Laboratory Interoffice Memorandum 312.F-98-048, 1998. [27] R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, American Institute of Aeronautics and Astronautics Inc., Virginia, 1999.
[28] S. Zimmer, C. Ocampo, Analytical gradients for gravity assist trajectories using constant specific impulse engines, J. Guid. Control Dyn. 28 (2005) 753–760. [29] A.C. Hindmarsh, Large ordinary differential equation systems and software, IEEE Control Syst. Mag. 2 (1982) 24–30. [30] P.R. Griesemer, C. Ocampo, D.S. Cooley, Targeting ballistic lunar capture transfers using periodic orbits, J. Guid. Control Dyn. 34 (2011) 893–902.
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Optimal ballistically captured Earth–Moon transfers Paul Ricord Griesemer a,n, Cesar Ocampo a, D.S. Cooley b a b
Department of Aerospace Engineering and Engineering Mechanics, 1 University Station C0600, The University of Texas, Austin, TX 78712, USA Navigation and Mission Design Branch, NASA Goddard Space Flight Center, Mailstop 595.0, Greenbelt, MD 20771, USA
a r t i c l e i n f o
abstract
Article history: Received 29 January 2011 Received in revised form 5 January 2012 Accepted 7 January 2012 Available online 28 February 2012
The optimality of a low-energy Earth–Moon transfer terminating in ballistic capture is examined for the first time using primer vector theory. An optimal control problem is formed with the following free variables: the location, time, and magnitude of the transfer insertion burn, and the transfer time. A constraint is placed on the initial state of the spacecraft to bind it to a given initial orbit around a first body, and on the final state of the spacecraft to limit its Keplerian energy with respect to a second body. Optimal transfers in the system are shown to meet certain conditions placed on the primer vector and its time derivative. A two point boundary value problem containing these necessary conditions is created for use in targeting optimal transfers. The two point boundary value problem is then applied to the ballistic lunar capture problem, and an optimal trajectory is shown. Additionally, the problem is then modified to fix the time of transfer, allowing for optimal multi-impulse transfers. The tradeoff between transfer time and fuel cost is shown for Earth–Moon ballistic lunar capture transfers. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Optimal transfer Primer vector Ballistic capture
1. Introduction With Belbruno’s introduction of the Weak Stability Boundary (WSB) [1], and Belbruno and Miller’s [2] subsequent application of the concept to ballistic lunar capture transfers (BLCTs), low energy transfers have been shown to be a useful tool in transferring between orbits around different gravitational bodies. Although they have demonstrated savings in fuel consumption when compared to other classes of transfers the question of finding optimal WSB transfers raises issues not previously explored. This work applies optimal control theory to the problem of transferring a spacecraft from an orbit around a gravitational point mass to a ballistically captured state around a second gravitational point mass. The method relies on new optimality conditions for a ballistically captured spacecraft that are derived in agreement with primer vector theory. The new conditions are
n
Corresponding author. Tel.: þ512 619 7154. E-mail address: [email protected] (P. Ricord Griesemer).
0094-5765/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2012.01.001
applied to the case of a ballistic lunar capture transfer, in which a spacecraft in Earth orbit is transferred to a captured lunar orbit via a long low energy transfer that travels beyond the Moon to the Earth–Sun WSB and then returns to the Moon from the far side. The new optimality conditions are used in a numerical targeting algorithm to verify the optimality of a given ballistic lunar capture transfer. One mathematical framework that has proven useful in the study of BLCTs is dynamical systems theory. The notion of heteroclinic connections between periodic orbits in dynamical systems dates back at least to 1937 [3]. More recent studies by Koon et al. [4,5], Gomez and Masdemont [6], and Canalias and Masdemont [7] consider the use of heteroclinic connections in the circular restricted three body problem for the design of interplanetary transfers and of tours of planet–moon systems. These connections are established by numerically identifying connections between stable and unstable manifolds of periodic orbits in the circular restricted three body problem (CRTBP) [5], and methods for their computation have been provided by the authors.
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Nomenclature
E
c WSB J
weak stability boundary performance index of the optimal control problem t time Dv change of velocity vector relating to an impulsive maneuver d(t) impulse function dn(t tc) delta sequence ath thrust magnitude _ m mass flow rate of the engine propellant c characteristic velocity of the engine propellant m mass of the spacecraft t time-like variable indicating the position of the spacecraft on a given orbit u vector of controls C control constraint r radius vector v velocity vector g acceleration vector of the general force field H boundary conditions at the initial time m* pre-defined initial mass of the spacecraft E* desired Keplerian energy state
When analyzed in the context of dynamical systems theory, BLCTs are shown to follow manifolds of libration point orbits in the Earth–Sun and Earth–Moon CRTBPs. In the first part of the transfer, the spacecraft trajectory shadows the stable and unstable manifolds of a Sun–Earth libration point orbit. In the second part of the transfers, the trajectory shadows the stable and unstable manifolds of an Earth–Moon libration point orbit. Though BLCTs do not meet the formal definition of low energy transfers given above, the term is applied to BLCTs to signify the reduction in relative energy of the spacecraft on its approach to the Moon. The libration point orbit manifolds were exploited for trajectory design by Howell et al. in the context of missions to libration point orbits [8]. In these mission designs, linear targeting algorithms were used to design missions from a specified orbit to a stable manifold of the desired trajectory. The spacecraft follows the manifold and ballistically enters into the libration point orbit. The method has been applied to libration point missions such as Genesis [9]. Several authors have applied similar theory to the analysis and targeting of low energy transfers to the Moon [10–12]. The problem becomes more complicated for the lunar transfers due to the need for the simultaneous calculation of manifolds of the Earth–Moon libration points and the Earth–Sun libration points. Algorithms exist that utilize the invariant manifolds associated with libration point orbits in both the Sun–Earth and Earth– Moon CRTBPs. For example, Koon et al. [10] targeted BLCTs by finding intersections of these invariant manifolds. Similarly, Yamato and Spencer [11] approximated
a G u
n
m ^ H
k p f a c
F(tf,t0) X a e i
o O
actual Keplerian energy state boundary condition at the final time slack variable endpoint function constant Lagrange multiplier associated with the final boundary condition vector of constant Lagrange multipliers associated with the initial boundary conditions Lagrange multiplier associated with the control constraint extended Hamiltonian function adjoint variable vector primer vector state equation vector optimization parameters in the numerical optimization problem vector of constraints in the numerical optimization problem state transition matrix vector of state variables semi-major axis eccentricity inclination argument of perigee right ascension of the ascending node
the invariant manifolds in a perturbed CRTBP, yielding transit orbits and lunar capture. These transfers are typically found with manifolds of halo orbits around the libration points. The problem of minimizing the fuel consumption associated with impulsive orbit-to-orbit transfers has received a great amount of attention. Formalized by Lawden in 1963 [12], the necessary conditions for an optimal impulsive transfer are imposed on the primer vector, the reflection of the vector of adjoint variables in the optimal control problem that correspond to the velocity components of the spacecraft state vector. Lawden’s necessary conditions specify the magnitude and orientation of the primer vector at the times of impulse. Specifically, the primer vector must align with the direction of the impulse and have a uniform magnitude at each impulse. The magnitude has been traditionally chosen to be unity at each impulse, but the designated value is arbitrary. Lion and Handelsman [13] extended Lawden’s work to determine when an additional impulse would improve on the fuel cost of the transfer. Jezewski [14] later gives a summary of primer vector theory, and introduces an estimation of the magnitude of an intermediate impulse that would optimize the transfer. Primer vector theory has been used to evaluate the optimality of several different types of transfers. Optimal transfers in an inverse square gravitational field have been analyzed for various types of missions. Rendezvous in the vicinity of a circular orbit has been explored by Prussing [15], direct ascent rendezvous by Gross and Prussing [16], and rendezvous between circular orbits by Prussing and Chiu [17]. Jezewski [14] has studied the
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
problem of multiple impulse transfers and transfers with inequality constraints. The primer vector in the three body problem was studied by Hiday-Johnston and Howell [18] to evaluate the optimality of transfers between libration point orbits, by D’Amario and Edelbaum [19] in the transfer from a collinear libration point to a lunar orbit, and by Ocampo [20] in the problem of insertion into distant retrograde orbits. The use of primer vector theory in the four body problem, the dynamics of which is required for Lunar WSB transfers, has been less comprehensive. In one example, Pu and Edelbaum [21] created a numerical method to compute 2-impulse and 3-impulse optimal transfers in the four body problem. Once the trajectories were numerically optimized, the primer vector history of the transfers was evaluated to verify their optimality. A similar method is employed here. The goals of the current research are twofold. First, we apply primer vector theory to derive new necessary conditions for a type of transfer that had not yet been considered in the context of primer vector theory, namely ballistic capture transfers. Previous implementations of primer vector theory have been insufficient to deal with ballistic capture trajectories. Necessary conditions have been traditionally based on the magnitude and orientation of the primer vector at the times of impulse. In a ballistic capture scenario with only one impulse occurring during the entire transfer, necessary conditions relating to the impulse provide an insufficient number of conditions to determine an optimal transfer. New conditions must be derived specifically for a ballistic capture to produce the required number of necessary conditions to define local optimality. Ballistic capture trajectories can be used in a number of contexts and offer advantages over traditional Hohmann-based approaches. As an example, the BepiColumbo mission cites the engine fault tolerance of a ballistic capture arc as a motivation for using a ballistic capture transfer to Mercury [22]. As these types of transfers become better understood, an analysis of their optimality becomes necessary. The second goal is to demonstrate, through the examination of BLCTs, an application of the new necessary conditions. Particular transfers in the Sun–Earth–Moon system will be found using the targeting algorithm and then analyzed to determine whether the transfers can be improved with an intermediate impulse. Multiple impulse fixed-time ballistic capture transfers will then be constructed to determine the tradeoff between optimal fuel consumption and transfer time.
optimal control of this spacecraft is based on Hull’s [23] formulation of the optimal control problem. Where the derivation overlaps with the orbit-to-orbit problem, it has equivalent results to those of Lawden, Jezewski, and Lion and Handelsman [12–14]. 2.1. Cost function An optimal control will minimize the total impulse provided to the spacecraft, represented by the following performance index: Z t2 J¼ Dv1 dðtt 1 Þ dt, ð1Þ t 1
where nv1 is the magnitude of the initial impulses, t2 is defined to be the final time of the transfer, and d(t t1) is an impulse function in which the impulse occurs at the beginning of the transfer, t ¼t1. The impulse function is used as a way to represent a finite change in the velocity of the spacecraft due to an infinite acceleration. By containing the limit functions within the impulse function, the necessary conditions for optimal control can be derived without taking the limit of the performance index itself. The delta function used here is defined by the relationship:
dðttc Þ ¼ lim dn ðttc Þ, n-1
Consider the orbit-to-capture transfer problem for an impulsively maneuvering spacecraft in a general force field. A spacecraft orbiting a body is to be transferred to any captured orbit of another body via a single impulse. In this situation, capture is defined by an instantaneous negative Keplerian energy level with respect to the second body, assuring an instantaneous elliptical orbit around the body. The derivation of the necessary conditions for an
ð2Þ
where dn(t tc) is a delta sequence that may be defined by the following piecewise continuous function [25]: 8 1 0 for t o t c 2n > > < 1 1 ð3Þ dn ðttc Þ ¼ n for tc 2n ot o tc þ 2n > > 1 : 0 for t c þ ot 2n When used in the cost function, the delta function represents an impulsive maneuver. The velocity change and fuel consumption associated with the maneuver can be determined through the integration of the function. The integral of the delta function has the property [24] Z 1 f ðtÞdðtt c Þ dt ¼ f ðt c Þ ð4Þ 1
The resulting velocity change due to a maneuver at time c can be determined explicitly using Eq. (4): Z tcþ DVc ¼ Dvc dðttc Þlc dt t c
Z
1
Dvc dðttc Þlc dt
¼ 2. Primer vector theory for an impulsive low energy transfer
3
1
¼ Dvc lc
ð5Þ
In this expression, lc is the steering vector that orients the impulse. The mass consumed by a maneuver can be calculated from the mass state equation. If thrust acceleration, ath, is represented by the expression: ath ¼
_ mc m
ð6Þ
_ is the mass flow rate of the propellant and c is its where m characteristic velocity, then the following equation holds
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for an impulsive maneuver occurring at time t1: _ mc Dv1 dðtt1 Þ ¼ m
ð7Þ
or _ ¼ m
mDv1 dðtt 1 Þ c
ð8Þ
2.2. Optimal control problem The controls in the orbit-to-capture problem, shown in Eq. (9), are the magnitude and direction of the transfer insertion maneuver at time t1 and the location of the spacecraft on a specified orbit around a gravitational body, represented by the time-like parameter t1. Following the methodology of Ocampo [20], t1 is introduced into the problem to define a plane in phase space that contains the initial orbit. This variable is defined such that r(t1) and v(t1) lie on the initial orbit for any t1. In this way, t1 is similar to the true anomaly of an orbit, or any other variable that defines a spacecraft’s location on a particular orbit. Both t1 and t2 are free parameters, but are not included among the controls according to the methodology of Hull: u ¼ ð Dv1
l1
T
t1 ÞT
ð9Þ
A control constraint exists on the thrust direction vector l1 so that its magnitude is unity: T
C ¼ ðl1 l1 1Þ ¼ 0
ð10Þ
The differential constraint for the problem is the state equation: ! v r_ ¼ , ð11Þ x_ ¼ g þ dðtt 1 ÞWv1 l1 v_ where r and v are the position vector and the velocity vector, respectively. The gravitational acceleration vector g represents a general force field, and although the low energy lunar transfer is typically calculated with a four body system, no assumptions about the composition of the gravity field will be made in the derivation. The acceleration vector for the spacecraft consists of the gravitational acceleration and the acceleration due to the impulsive maneuver. The initial conditions constrain the initial state of the spacecraft to lie on a pre-established orbit. The constraint is written as ! r1 rðt1 Þ h¼ ¼0 ð12Þ v1 vðt1 Þ The final conditions are generalized to require the spacecraft to arrive at the second body with a specific energy state with respect to the second body less than or equal to a stated value. For a spacecraft to be captured in a two body system, the stated energy level would be less than zero. In a multi-body system, however, a stronger constraint may be desired to increase the likelihood of the spacecraft orbiting the second body before it escapes. In the derivation, the desired energy level will be left unspecified.
An inequality constraint is used to allow for the possibility of an optimal transfer arriving at the secondary body at a more strongly captured state than specified. The resulting final condition is
c ¼ En
vs 2 ms r 0, 2 rs
ð13Þ
where rs and vs are the radius and the velocity with respect to the secondary body, respectively, ms is the mass parameter of the secondary body, and E* is the desired energy state. Following the methodology of Hull [23], a slack variable is introduced to convert the inequality constraint in Eq. (13) into an equality constraint. A consequence of the introduction of a slack variable into the constraint is an additional parameter in the problem. The final condition becomes
c¼
vs 2 m s En þ a2 ¼ 0 2 rs
ð14Þ
where a is the slack variable. The boundary constraint function is T
G0 ¼ uc þ n h
ð15Þ
where u and x are a scalar and a vector, respectively, of constant Lagrange multipliers adjoined to the endpoint constraints. An extended Hamiltonian is formed using a Lagrange multiplier m to adjoin the control constraint in Eq. (10) to the Hamiltonian: ^ ¼ kr T v þ kv T ðg þ Dv1 dðtt 1 Þl1 Þ þ Dv1 dðtt 1 Þ þ mC H
ð16Þ
The extended Hamiltonian is constant if the gravitational acceleration vector is not a function of time. It is not necessarily an integral of the problem for the general force field. The performance index and constraints described above form a time-free optimal control problem. The necessary conditions from the first differential are given by the Euler–Lagrange equations: x_ ¼ f
ð17Þ
k_ ¼ H^ x
T
^ uT 0¼H
ð18Þ ð19Þ
and the natural boundary conditions: ^ 2 ¼ G0 H t2 ^ 1 ¼ G0 H t1
k2 ¼ G0x2 T k1 ¼ G0x1 T
ð20Þ
The conditions of optimality pertaining to the impulse in the orbit-to-capture problem are equivalent to the orbit-to-orbit necessary conditions derived in the work of Lawden, Jezewski, and others. As previously stated, the primer vector must be aligned with the impulse at the time of the impulse, the time derivative of its magnitude must be zero, and its magnitude must be unity: p1 ¼
DV1 Dv1
ð21Þ
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
p_ 1 ¼ 0,
ð22Þ
where p ¼ kv p_ ¼ kr
ð23Þ
At time t2, there are two complications that call for necessary conditions that differ from the conventional derivation. First, the inequality endpoint constraint does not fit neatly into the optimal control theory formulation of Hull. After extending Hull’s formulation to account for an endpoint inequality constraint, the inclusion of the slack variable produces two separate cases, one when the solution is on the boundary of the constraint, and one when it is off of the boundary. Secondly, the traditional impulsive orbit-to-orbit problem contains at least two impulses. A first impulse initiates the transfer, and a second inserts the spacecraft from a transfer trajectory into the target orbit. This paradigm is insufficient when dealing with ballistic capture problems. New conditions need to be derived from primer vector theory to account for the possibility of reaching the desired state without an impulse to conclude the transfer. Hull’s derivation allows for the endpoint function to depend on the initial and final time, the state variables, and constant Lagrange multipliers. The slack variable associated with the inequality constraint cannot be considered to belong to any class of these variables. It can be dealt with outside of Hull’s formulation by treating it as a separate class of variable and including it in a modified endpoint function, G0 . The modified endpoint function shown in Eq. (15) depends on the initial and final time, the state variables, constant Lagrange multipliers, and the slack variable, a. Hull derives the first order conditions of optimality by taking the first differential of the extended cost function, which includes the endpoint function. By adding the slack variable to the parameters of the problem, a term appears in the differential that depends on da: 0
dG ¼ 2ua da
ð24Þ
In order for a trajectory to be optimal, the expression in Eq. (24) must be equal to zero. This optimality condition can be satisfied in two cases. In the first case, a is equal to zero and the solution lies on the boundary of the inequality constraint. In the second case, the Lagrange multiplier is equal to zero. This condition applies to the situation where the optimality conditions are satisfied on a trajectory that terminates with an energy level less than the targeted energy level. All other conditions on G0 are equivalent to the conditions on G in Hull’s formulation. To determine necessary conditions appropriate for an orbit to capture trajectory at the terminal time, consider the endpoint conditions on from Eq. (20), along with the definition of the primer vector given in Eq. (23): ms p_ 2 ¼ G0T r2 ¼ u r 3 rs
at the final time is aligned with the velocity vector of the spacecraft with respect to the second gravitational body, and the rate of change of the primer vector is aligned with the radius vector of the spacecraft with respect to the gravitational body. Furthermore, because the constant Lagrange multiplier u appears in both expressions, its value is not arbitrary. The parameter u establishes a relationship between the magnitude of the primer vector and its time derivative. It is either zero in the off boundary case, or it can be solved for by post-multiplying the equation for the Lagrange multipliers associated with the velocity by the relative velocity vector: u¼
kTv2 vs
ð26Þ
v3s
kv2 and vs must be parallel to satisfy Eq. (25), so their inner product can be reduced to the product of their two magnitudes, resulting in a simplified expression for u: u¼ 7
lv2
ð27Þ
vs 2
The sign of Eq. (27) is left to be determined by analyzing the results of a particular problem. Together, Eqs.(24) and (25) define the necessary conditions at the final time for a ballistic capture trajectory. They apply to the magnitude and direction of both the primer vector and the time derivative of the primer vector, and represent trajectories that terminate with an energy level either equal to that targeted or less than the targeted value. Additionally, it can be shown that the magnitude of the primer vector must be less than unity to preclude an additional impulse. 2.3. Additional impulses A separate analysis can be made to determine whether an intermediate impulse has the potential to reduce the cost of the maneuver. Consider a trajectory that has an additional maneuver at time tk. The performance index of the trajectory would be changed to Z tþ 2 J¼ Dv1 dðtt1 Þ þ Dvk dðttk Þ dt, ð28Þ t 1
and the vector of controls would include the magnitude, time, and direction of the intermediate maneuver: u ¼ ð Dv1
l1
t1 Dvk lk tk ÞT
ð25Þ
Eq. (25) establishes new necessary conditions for the orbit-to-capture problem. It dictates that the primer vector
ð29Þ
The unit thrust vectors are subject to the constraint: ! T l1 l1 1 C¼ ¼ 0, ð30Þ T lk lk 1 and the extended Hamiltonian is ^ ¼ kT v þ kT ðg þ Dv1 dðtt 1 Þl1 þ Dvk dðtt k Þlk Þ H r v þ Dv1 dðtt 1 Þ þ Dvk dðtt k Þ þ lT C
s
p2 ¼ G0T v2 ¼ uvs vs
5
ð31Þ
Applying the optimality condition in Eq. (19) to Eq. (31) gives the following result when considering the magnitude of the primer vector at an intermediate
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P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
impulse: ^ Dv ¼ dðtt k ÞkT lk þ dðtt k Þ ¼ 0 H v k
ð32Þ
At time tk Eq. (32) reduces to kvk ¼ 1
ð33Þ
The result obtained from applying the same condition at the initial time is kv1 ¼ 1 ð34Þ It can be seen from comparing Eq. (33) to Eq. (34) that a necessary condition for an intermediate impulse is p ¼ p ð35Þ k
1
Eq. (35) requires that an optimal trajectory with an intermediate impulse have primer vectors with equal magnitudes at each impulse. As stated previously, it is common to assign a unit magnitude to the primer vector at the initial impulse. Consequently, if the primer vector never again returns to unit magnitude between times t1 and t2, there is no potential for an improvement in the performance index via an intermediate impulse. 3. Numerical analysis The necessary conditions on the state and co-state vectors described above can be used in a targeting algorithm to verify the optimality of transfers or optimize near-optimal trajectories. The boundary conditions and the necessary conditions of the optimal control problem define a two point boundary value problem that can be numerically solved to yield a control that satisfies the conditions of optimality. The unknown and constraint vectors associated with the problem are shown in Eq. (36). The differential equations to be satisfied are the state and co-state equations shown in Eqs.(17) and (18): a11x1 ¼ ð t 1
^ c11x1 ¼ H1
p_ 1
t2
t1
^2 H
EEn þ a2
Dv1 2am
u
used for this analysis is to remove the initial and final times from the vector of free parameters in the optimal control problem, and thus fix the transfer time. With the addition of a second impulse, optimal transfers can be constructed that are initiated by a transfer insertion burn and have a second burn that places the spacecraft on a ballistic arc that transitions to a captured state at the final time. By constructing transfers using this methodology, the transfer time can be set anywhere between an upper bound of the transfer time of a single impulse low energy transfer and lower bound of the transfer time of a direct two impulse transfer. The necessary conditions for an optimal two impulse transfer to a ballistic arc terminating at a desired energy state mirror those presented in Section 2 for an optimal ballistic lunar transfer. The two modifications of the problem are the removal of the initial and final times from the vector of free parameters and the addition of an intermediate impulse. The two point boundary value problem that is used to target optimal trajectories shown in Eq. (36) can be modified in the following way to target optimal time-fixed trajectories. The initial and final times are removed from the parameter vector, and the time, magnitude, and direction of the intermediate impulse are added. The constraint vector is modified by eliminating the constraints on the initial and final Hamiltonians and adding the optimality conditions on the intermediate impulse (Eq. (35)). The resulting two point boundary value problem, shown in Eq. (37), is thirteen dimensional and can be solved by iterating from an initial guess using a numerical nonlinear equation solver: a13x1 ¼ ð tk
Dvk
f
p_ 1
Dv1
u
a ÞT ,
T m Dv n T 2 c13x1 ¼ pk jDvkk j pk p_ k EE þ a 2am kr2 u r3ss rs kv2 uvs vs p_ 1
ð37Þ
T
aÞ ,
kr2 u mr3s rs s
kv2 uvs vs
p_ 1
T
3.2. The restricted four body problem
ð36Þ This system, with an appropriate initial guess, can be solved by a numerical nonlinear equation solver such as NS-12 [25]. The initial state vector is formed from the prescribed initial orbit and the parameter t1. The initial co-state vector is determined from the parameter p_ 1 and by aligning the primer vector in the direction of the impulse and giving it a magnitude of unity. A solution to this system is not guaranteed to meet the necessary conditions of optimality, as there are no path constraints to ensure the magnitude of the primer vector remains below unity. An inspection of the primer vector history to verify this condition is required to ensure that the trajectory meets the necessary conditions once a solution to the boundary value problem is found. 3.1. Multiple impulses In this section, analysis is presented with the intent to provide a means of analyzing the trade space between transfer time and fuel cost in lunar transfers. The method
The two point boundary value problem shown above can be solved for any dynamical system. However, single impulse ballistic lunar transfer trajectories rely on the gravitational attraction of the Sun, Earth, and Moon [2]. In this paper, the governing dynamics of the spacecraft are generated by assuming these three primary bodies act as point masses attracting the spacecraft. In the numerical analysis, their orbits about each other are defined by the JPL DE405 ephemeris [26]. A spacecraft moving through their gravitational field is assumed to be infinitesimally small, so that it will not perturb the orbits of the primary masses. A non-rotating coordinate system is established that is centered on the Earth–Moon barycenter. Assuming Newtonian gravity, the gravitational acceleration of a spacecraft in this four-body system is defined in Eq. (38) [27], where the vectors are defined in Fig. 1. r€ ¼
me jr-re j3
ðrre Þ
mm jr-rm j3
ðrrm Þ
ms jr-rs j3
ðrrs Þ
ms r 3s
rs ð38Þ
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
convergence. The algorithm used to target optimal trajectories requires partial derivatives that relate the constraints, c, of the problem to the free parameters, a. For the constraints that are applied at the initial time, partial derivatives can be derived directly from the initial state. For the constraints at the final time, the state transition matrix can be used in conjunction with the chain rule to determine partial derivatives that relate parameters at the initial time to the final constraints. It should be noted, however, that the state transition matrix can only relate changes in the state that occur at fixed times. Therefore, when relating constraint values to the variation of the initial time, an additional step must be taken. The changes in the state at t ¼ t1 must be related to the change in the initial time by calculating a numerical derivative that relates to change in the state at a fixed time t1 to a change in the initial time. The time derivative of the state vector does not apply to this calculation, as discussed by Zimmer [28]. This derivative can be calculated with the forward difference method by simply integrating the state from a perturbed time back to t1. The partial derivatives used in the numerical routine are shown below:
Fig. 1. Restricted four body problem.
Table 1 Gravitational parameter values. Gravitational parameter
Value (km3/s2)
ms me mm
1.32715 1011 3.986004 105 4.9029 103
@Xðt 2 Þ @Xðt 1 Þ ¼ Fðt 2 ,t 1 Þ @t 1 @t 1
The gravitational parameters ms, me, and mm are defined in Table 1. Eq. (38) contains three direct gravitational accelerations followed by an indirect term. The indirect term accounts for the acceleration of the reference frame around the Sun–EarthEarth–Moon barycenter. 0
1 kTv @g @t1
B B @H^ B 2 Fðt 2 ,t 1 Þ @X1 @t 1 B @X2 B B @c3 Fðt ,t Þ @X1 2 1 @t1 B @X2 @c B 0 ¼B B @a B @c 1 B @X52 Fðt 2 ,t 1 Þ @X @t1 B B @c6 1 B @X Fðt 2 ,t 1 Þ @X @t1 B 2 @ 0
^ 1 @X1 @H @X1 @f
0
7
ð39Þ
where q[x(t1)]/qt1 is calculated using the forward difference method by varying the initial time by a small amount and numerically integrating back to t1.
0
0
0
0
^ 2 @X2 @H @X2 @t 2
^2 @H @X2
1 Fðt2 ,t1 Þ @X @f
^2 @H @X2
1 Fðt2 ,t1 Þ @X @p_ 1
^2 @H @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
0
@c3 @X2 @X2 @t 2
@c3 @X2
1 Fðt2 ,t1 Þ @X @f
@c3 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c3 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
0
@c5 @X2 @X2 @t 2
@c5 @X2
1 Fðt2 ,t1 Þ @X @f
@c5 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c5 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
mr3s rs
@c6 @X2 @X2 @t 2
@c6 @X2
1 Fðt2 ,t1 Þ @X @f
@c6 @X2
1 Fðt2 ,t1 Þ @X @p_ 1
@c6 @X2
1 Fðt 2 ,t1 Þ @@X Dv 1
vs vs
0
0
0
0
0
1 p_ 1
3.3. State transition matrix derivatives It should be stressed that convergence is an issue when working with these chaotic trajectories in the four body problem. Jacobian matrices are poorly conditioned and numerical targeting algorithms that rely on linear approximations to the complex dynamical systems have small convergence envelopes. A reasonable first guess is required to converge upon an optimal solution. A particularly problematic case for convergence is the case of a lunar flyby on the outbound portion of the transfer, which yield Jacobian matrices that are more poorly conditioned. Although transfers with these flybys can be more efficient, they will not be considered here. As documented by Zimmer and Ocampo [28] the use of analytical derivatives in targeting algorithms that require gradients increases both the speed and likelihood of
0
T p_ 1
0 s
0
0
1
C C 0 C C C 2a C C C 2m C C C 0 C C C 0 C C A 0
ð40Þ
where ^ 2 T @g @H ¼ kv2 @r22 @X2
kTr2
gT2
vT2
ð41Þ
@ðEEn þ a2 Þ mm rT vT 0 0 ¼ rm 3 m m @X2 @c5 T 1 1 0 ¼ umm rm 3 I þ3 rm 5 rm rm @X2 @c6 ¼ @X2
0
@X2 ¼ ð vT @t 2 @X1 ¼ 0 @p_ 1
u vm I þ v1m vm vm T
k_ r
gT 0
I
T
0
T
k_ v Þ
I
ð42Þ 0
0
I
ð43Þ
ð44Þ
ð45Þ
ð46Þ
8
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
@X1 ¼ 0 @Dv1
I
0
0
ð47Þ
0
1 aðcos O sin f þ sin O cos f cos iÞ B C B aðsin O sin f þ cos O cos f cos iÞ C B C C a cos f sin i @X1 B B C ¼B C ð O cos f sin O sin f cos i Þ v cos B C @f B C B vðsin O cos f þ cos O sin f cos iÞ C @ A v sin f sin i
ð48Þ
Note that in these equations the initial orbit around the Earth is circular, where a is the semi-major axis of the orbit, f is the true anomaly of the orbit, i is the inclination, v is the magnitude of the velocity, and O is the right ascension of the ascending node. F(t2,t1) is the state transition matrix associated with the four body equations of motion given in Eq. (38). These partial derivatives are employed in the targeting code to iterate from an initial guess to an optimal solution. The guesses are propagated using a numerical predictor-corrector integration method [29]. 4. Results An optimal transfer is sought from the low Earth orbit and detailed in Table 2 to a captured state around the Moon using a single impulsive maneuver. A previously targeted nonoptimal transfer is used as an initial guess, and improved upon by targeting the necessary conditions developed above. Fig. 2 shows a converged solution in non-rotating coordinates centered at the Earth–Moon barycenter. Table 2 Parking orbit properties at t0. Orbit property
Value
epoch (JED) a (km) e i (deg.) o (deg.) O (deg.)
2,455,658.47677 7200 0 47.188 0 1.09948
In an effort to find a suitable initial guess for the targeting of an optimal transfer, parameters from a non-optimal ballistic lunar capture trajectory were calculated using a method detailed in previous work [30]. The parameters of the transfer are listed in Table 3, where the subscript (v) refers to the component of the vector in the direction of the velocity, the subscript (h) refers to the component in the direction of the angular momentum, and the subscript (r) refers to the direction that completes the right-handed coordinate system. The final state of the transfer lies on an elliptical two body orbit about the Moon. The parameters of this transfer are used as an initial guess in the two point boundary value problem described above. The initial values of the co-state associated with the velocity are assigned to align the primer vector with the impulse and give it a unit magnitude. The remaining initial values, those in the co-state associated with the radius, are assigned very small numbers. The solution to the two point boundary value problem in Eq. (36) yields a transfer that meets all of the necessary conditions of optimality. The targeted energy state is E* ¼ 0.02 MJ/kg in accordance with the experience gained in targeting states that result in a physical orbit of the Moon [30]. The time history of the magnitude of the primer vector for the converged solution to the two point boundary value problem is shown in Fig. 3. This primer vector history satisfies the necessary conditions detailed above: it begins with a magnitude of unity and a slope of zero, shown in detail in Fig. 4, and never again returns to a magnitude of unity. Additionally, this solution lies on the boundary of the inequality constraint, with slack variable, a, having a value of zero for this transfer and satisfying the condition in Eq. (24). Through the computation of the necessary conditions, the maneuver was improved from 3.050475 km/s to 3.050474 km/s. It should be noted that this modest reduction is with respect to a numerically optimized result. The role of this analysis is to verify the optimality of the previous findings. Finally, it is not evident in Fig. 3, but the primer vector at the final time is aligned with the velocity vector and its time derivative is aligned with the radius vector with respect to the moon. Not only are the vectors aligned but their magnitudes are related through the Lagrange multiplier u. Table 4 shows the angles of the related vectors relative to their counterparts. 4.1. Two impulse ballistic capture transfers The methodology of Section 3.1 is applied to determine the tradeoff between time of flight and fuel consumption for BLCTs. In this approach, transfer time is a fixed Table 3 Transfer properties. Transfer property
Value
DVv (km/s) DVr (km/s) DVh (km/s) Dt (days)
3.05048 0.0 0.0 117.32581 22.6881
f (deg.) Fig. 2. Single burn low energy transfer to lunar orbit.
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
9
Fig. 3. Primer vector magnitude of a single burn transfer.
Fig. 4. Detail of the primer vector magnitude near the initial time.
Table 4 Primer vector properties at tf. Primer vector angle with the velocity vector (deg.) Primer rate vector angle with the radius vector (deg.)
0.01 179.95
variable, and the two point boundary value problem is slightly modified to the form of Eq. (37). E* remains 0.02 MJ/kg. The necessary conditions on the primer vector for this problem are that the magnitude begins at unity and returns to a maximum of unity at the time of the impulse. A very good initial guess near the solution in both the state space and the co-state space must be provided to the nonlinear targeting algorithm. A numerically optimized transfer can be used as such a guess and can provide a solution within the convergence envelope. Based on the previous results of optimal orbit-to-orbit transfers, it would be expected that an optimal low energy transfer would terminate at or near a periapse with respect to
Fig. 5. Time-fixed reference trajectory in non-rotating coordinates.
the Moon. Therefore, it will be assumed that an optimal transfer to periselene is the desired target for generating an initial guess. A transfer of this type can be found by adding a second impulse at the apogee of a low energy orbit-to-orbit transfer and then targeting a minimum Dv transfer to the Earth–Moon L2 point as the transfer time is reduced. Fig. 5 shows a transfer that was formed to meet these conditions with a transfer time of 54 days. The numerical optimization of this transfer is continued with the constraint vector altered to target a periselene. Fig. 6 shows the results of the numerical optimization. This trajectory is the initial guess for the two point boundary value problem described in Eq. (37). The initial guess for the co-state parameters is formed by satisfying two of the necessary conditions. The initial primer vector is chosen to align with the velocity vector and has a unit magnitude. The primer rate vector is calculated based on the necessary condition that the primer vector be aligned with the second impulse.
10
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
1.1 1
primer magnitude
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2
Fig. 6. Fixed time initial guess in non-rotating coordinates.
3
3.5
4 4.5 time
5
5.5
6
6.5 x 106
Fig. 8. Primer vector history of the converged optimal solution.
3.8
5
3.7 delta V (km/s)
4.5 4 primer magnitude
2.5
3.5 3 2.5
3.6 3.5 3.4 3.3 3.2
2
3.1 0
1.5
10
20
1
30 40 50 transfer time (days)
60
70
80
70
80
Fig. 9. Optimal time-fixed transfer parameters.
0.5 0 2.5
3
3.5
4 4.5 time
5
5.5
6
6.5 x 106
Fig. 7. Primer vector history of the initial guess.
Fig. 7 shows the primer vector history of the initial guess, with the impulse applied at the time signified by the overlaid circle. The trajectory does not meet the conditions of optimality due to the segment near the end of the trajectory where the magnitude exceeds unity. The initial guess is iterated to an optimal solution using a numerical non-linear equation solving routine until the two point boundary value problem is solved. The primer vector history of the converged solution is shown in Fig. 8. In the process of solving the optimization problem, the magnitude of the cost was reduced from 3.30535 km/s to 3.288372 km/s. The method described above is repeated for different transfer times spanning the space between the Hohmann transfer time and the single impulse low energy transfer time. Fig. 9 shows the results of the trade-space study. It displays a linear increase in fuel cost as transfer time is shortened. Figs. 10 and 11 show the change in magnitude of the first and second maneuver, respectively, for the
3.06 3.05 delta V1 (km/s)
2
3.04 3.03 3.02 3.01 3 0
10
20
30 40 50 transfer time (days)
60
Fig. 10. Time-fixed transfer initial maneuver magnitude.
transfers included in Fig. 9. The cost of shortening the transfer time is shown to be entirely contained in the second impulse. The initial impulse actually decreases in magnitude as the transfer is shortened. This result can be interpreted by assuming the second impulse does the work of the Sun as the transfer time is shortened and the spacecraft does not travel far enough from the Earth to be influenced by the Sun’s dynamics.
P. Ricord Griesemer et al. / Acta Astronautica 76 (2012) 1–12
consumption is explored for ballistic transfers. Optimal multiple impulse transfers are computed that show the cost of the initial impulse is reduced, but offset by the addition of a second impulse when shortening the travel time.
0.8 0.7 delta V2 (km/s)
11
0.6 0.5 0.4 0.3
Acknowledgments
0.2 0.1 0 0
10
20
30 40 50 transfer time (days)
60
70
80
Fig. 11. Time-fixed transfer intermediate maneuver magnitude.
5. Conclusions A method of targeting optimal low energy transfers, relying on multi-body dynamics, from an orbit about one body to another is presented. The targeting algorithm applies necessary conditions derived from primer vector theory to the orbit-to-capture transfer problem. With the help of precise derivatives obtained from the chain rule and the state transition matrix, optimal solutions can be converged using the non-linear equation solving techniques. The necessary conditions for an impulsive single impulse orbit-to-capture transfer offer a tool for evaluating low energy missions such as lunar tour missions of the outer planets or ballistic lunar capture trajectories. The use of primer vector theory to obtain necessary conditions for the problem offers an extension of previous work with primer vector theory. The resulting conditions can be seen as slight modifications of, but still consistent with, previous results. For example, for the off boundary case, the primer vector of an optimal transfer must align with the velocity vector of the spacecraft with respect to the secondary mass, and the primer rate vector must align with the radius vector. Previously, it has been shown that the primer vector in an optimal orbit-to-orbit transfer must align with the impulse vector at the final time, and the time derivative of its magnitude must be zero. The new result is equivalent to previous findings as long as the second orbit insertion burn occurs at perigee and the burn is in the direction of the spacecraft’s relative velocity, two conditions previously associated with optimal transfers. At the perigee point, the velocity vector is perpendicular to the radius vector. At this point on an optimal trajectory, both the radius vector and the primer vector have magnitudes that are unchanging. Using the new necessary conditions, a ballistic lunar capture transfer was optimized and then analyzed to determine whether a second maneuver could improve the result. The transfer was found to be a local optimal with no available savings from an additional burn. It was therefore shown that a single burn transfer from the Earth to the Moon can meet the necessary conditions of optimality from primer vector theory. The methods are applied to multiple impulse transfers in which the tradeoff between transfer time and fuel
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