Optimal Two-Impulse Trajectories with Moderate Flight Time for Earth-Moon Missions

A study of optimal two-impulse trajectories with moderate flight time for Earth-Moon missions is presented. The optimization criterion is the total characteristic velocity. Three dynamical models are used to describe the motion of the space vehicle: the well-known patched-conic approximation and two versions of the planar circular restricted three-body problem PCR3BP . In the patchedconic approximation model, the parameters to be optimized are two: initial phase angle of space vehicle and the first velocity impulse. In the PCR3BP models, the parameters to be optimized are four: initial phase angle of space vehicle, flight time, and the first and the second velocity impulses. In all cases, the optimization problem has one degree of freedom and can be solved by means of an algorithm based on gradient method in conjunction with Newton-Raphson method.


Introduction
In the last two decades, new types of trajectories have been proposed to transfer a spacecraft from an Earth orbit to a Moon orbit which reduce the cost of the traditional Hohmann transfer based on the two-body dynamics 1 .The new trajectories are designed using more realistic models of the motion of the spacecraft such as the PCR3BP 2, 3 or the planar bicircular four body problem 4 .These models describing the motion of the spacecraft exhibit very complex dynamics that are used to design new Earth-to-Moon trajectories 5-10 .The most of the proposed approaches to calculate the new trajectories are based on the concept of the weak stability boundary introduced by Belbruno 11 , and, usually, involve large flight time.Only few works consider the minimization of the total cost or the time for two-impulse trajectories 8-10, 12, 13 .In this paper, the problem of transferring a space vehicle from a circular low Earth orbit LEO to a circular low Moon orbit LMO with minimum fuel consumption is considered.Moon's sphere of influence.The selenocentric phase corresponds to the portion of trajectory in the Moon's sphere of influence and ends at the point of application of the second impulse.In each one of these phases, the space vehicle is under the gravitational attraction of only one body, Earth or Moon.6 The class of two impulse trajectories is considered.The impulses are applied tangentially to the space vehicle velocity relative to Earth first impulse and Moon second impulse .
An Earth-Moon trajectory is completely specified by four quantities: r 0 -radius of circular LEO; v 0 -velocity of the space vehicle at the point of application of the first impulse after the application of the impulse; ϕ 0 -flight path angle at the point of application of the first impulse; γ 0 -phase angle at departure.These quantities must be determined such that the space vehicle is injected into a circular LMO with specified altitude after the application of the second impulse.It is particularly convenient to replace γ 0 by the angle λ 1 which specifies the point at which the geocentric trajectory crosses the Moon's sphere of influence.
Equations describing each phase of an Earth-Moon trajectory are briefly presented in what follows.It is assumed that the geocentric trajectory is direct and that lunar arrival occurs prior to apoapsis of the geocentric orbit.Figure 1 shows the geometry of the geocentric phase.
For a given set of initial conditions r 0 , v 0 , ϕ 0 , energy and angular momentum of the geocentric trajectory can be determined from the equations where μ E is the Earth gravitational parameter.

Moon at time t 1
Moon's sphere of influence From the geometry of the geocentric phase Figure 1 , one finds where D is the distance from the Earth to the Moon and R S is the radius of the Moon's sphere of influence.Subscript 1 denotes quantities of the geocentric trajectory calculated at the edge of the Moon's sphere of influence.
From energy and angular momentum of the geocentric trajectory, one finds

2.3
The selenocentric phase begins at the point at which the geocentric trajectory crosses the Moon's sphere of influence.Figure 2 shows the geometry of the selenocentric phase for a clockwise arrival to LMO.Thus, where v M is the velocity vector of the Moon relative to the center of the Earth.Subscript 2 denotes quantities of the selenocentric trajectory calculated at the edge of the Moon's sphere of influence.
From 2.5 , one finds

2.6
The upper sign refers to clockwise arrival to LMO and the lower sign refers to counterclockwise to LMO.The semimajor axis a f and eccentricity e f of the selenocentric trajectory are given by where Q 2 r 2 v 2 2 /μ M and μ M is the Moon gravitational parameter.The second impulse is applied at the periselenium of the selenocentric trajectory such that the terminal conditions, before the impulse, are defined by

2.8
Equations 2.1 -2.8 lead to the following two-point boundary value problem: for a specified value of λ 1 and a given set of initial parameters r 0 and ϕ 0 0 • the impulse is applied tangentially to the space vehicle velocity relative to Earth , determine v 0 such that the final condition r p M r f is satisfied, where r 0 is the radius of LEO and r f is the radius of LMO both orbits, LEO and LMO, are circular .This boundary value problem can be solved by means of Newton-Raphson method 16 .
After computing v 0 , the velocity changes at each impulse can be determined

2.9
The total characteristic velocity is then given by Δv Total Δv 1 Δv 2 .

2.10
Note that the total characteristic velocity is a function of λ 1 for a given set of parameters r 0 , ϕ 0 0 • , r f .Accordingly, the following optimization problem can be formulated: 6 Mathematical Problems in Engineering determine λ 1 to minimize Δv Total .This minimization problem was solved using a classic gradient method 15 .The results are presented in Section 5.The total flight time of an Earth-Moon trajectory is given by where Δt E is the flight time of the geocentric trajectory and Δt M is the flight time of the selenocentric trajectory.These flight times are calculated from the well-known time of flight equations of two-body dynamics as follows: 2.12 with eccentric anomaly E 1 and hyperbolic eccentric anomaly F 2 obtained, respectively, from

2.13
Since lunar arrival occurs prior to apoapsis of the geocentric trajectory, 0 < E 1 ≤ 180 • , and F 2 is positive.So, equations above define E 1 and F 2 without ambiguity.The semimajor axis and eccentricity of the geocentric trajectory are given by where Recall that the impulses are applied at the periapses of the geocentric and selenocentric trajectories.

Optimization Problem Based on the Simplified Version of PCR3BP
In this section, the optimization problem based on the simplified PCR3BP is formulated.A detailed presentation of this problem can be found in Miele and Mancuso 12 .
The following assumptions are employed.
1 The Earth is fixed in space.
2 The eccentricity of the Moon orbit around the Earth is neglected.
3 The flight of the space vehicle takes place in the Moon orbital plane.
Mathematical Problems in Engineering 7 4 The space vehicle is subject to only the gravitational fields of Earth and Moon.
5 The gravitational fields of Earth and Moon are central and obey the inverse square law.
6 The class of two impulse trajectories is considered.The impulses are applied tangentially to the space vehicle velocity relative to Earth first impulse and Moon second impulse .
Consider an inertial reference frame Exy contained in the Moon orbital plane: its origin is the Earth center; the x-axis points towards the Moon position at the initial time t 0 0 and the y-axis is perpendicular to the x-axis.Figure 3 shows the inertial reference frame Exy.
In the Exy reference frame, the motion of the space vehicle P is described by the following differential equations: where ω M μ E /D 3 is the angular velocity of the Moon.The initial conditions of the system of differential equations correspond to the position and velocity vectors of the space vehicle after the application of the first impulse.The initial conditions t 0 0 can be written as follows: where Δv 1 is the velocity change at the first impulse, r EP 0 r EP 0 , and θ EP t is the angle defining the position of the space vehicle in the inertial reference frame Exy at time t, more precisely the angle which the position vector r P forms with x-axis.It should be noted that r EP 0 and v EP 0 or, equivalently, r P 0 and v P 0 are orthogonal, because the impulse is applied tangentially to the circular LEO.
The final conditions of the system of differential equations correspond to the position and velocity vectors of the space vehicle before the application of the second impulse.The final conditions t f T can be written as follows: where Δv 2 is the velocity change at the second impulse, r MP T r MP f , and θ MP t is the angle which the position vector r MP forms with x-axis.The upper sign refers to clockwise arrival to LMO and the lower sign refers to counterclockwise to LMO.Since the eccentricity of the Moon orbit around Earth is neglected, it follows from 3.2 that the components of the Moon inertial velocity at time T are given by ẋM T −Dω M sin ω M T , ẏM T Dω M cos ω M T .

3.8
The angle θ MP T is free and can be eliminated.After the problem has been solved, the angle θ MP T can be calculated from 3.4 and 3.5 .So, combining 3.4 -3.7 , the final conditions can be put in the following form: 3.10

3.11
As before, the upper sign refers to clockwise arrival to LMO and the lower sign refers to counterclockwise to LMO.It should be noted that constraint defined by 3.11 is derived from the angular momentum considering a direct counterclockwise arrival or a retrograde clockwise arrival orbit around the Moon.The problem defined by 3.1 -3.11 involves four unknowns Δv 1 , Δv 2 , T , and θ EP 0 that must be determined in order to satisfy the three final conditions 3.9 -3.11 .Since this problem has one degree of freedom, an optimization problem can be formulated as follows: determine Δv 1 , Δv 2 , T , and θ EP 0 which satisfy the final constraints 3.9 -3.11 and minimize the total characteristic velocity Δv Total Δv 1 Δv 2 .This problem was solved by Miele and Mancuso 12 using the sequential gradient-restoration algorithm for mathematical programming problems developed by Miele et al. 15 .
In this paper, the optimization problem described above is solved by means of an algorithm based on gradient method 15 in conjunction with Newton-Raphson method 16 , similarly to the one described in the previous section for the problem based on the patched-conic approximation.The angle θ EP 0 has been chosen as the iteration variable in the gradient phase with Δv 1 , Δv 2 , and T calculated through Newton-Raphson method.The results are presented in Section 5.

Optimization Problem Based on the Classical Version of PCR3BP
In this section, the optimization problem based on the classical version of PCR3BP is formulated.The assumptions employed in this formulation are the same ones previously presented in Section 3, except for assumption 1 which must be replaced by the following one: Earth moves around the center of mass of the Earth-Moon system.

Problem Formulation in Inertial Frame
Consider an inertial reference frame Gxy contained in the Moon orbital plane: its origin is the center of mass of the Earth-Moon system; the x-axis points towards the Moon position at the initial time and the y-axis is perpendicular to the x-axis.Figure 4 shows the inertial reference frame Gxy.In the Gxy reference frame, the motion of the space vehicle P is described by the following differential equations: where r EP and r MP are, respectively, the radial distances of space vehicle from Earth E and Moon M , that is, r 2 Because the origin of the inertial reference frame Gxy is the center of mass of Earth-Moon system, the position vectors of the Earth and the Moon are, respectively, defined by r E x E , y E and r M x M , y M .Since the eccentricity of the Moon orbit around Earth is neglected, the Earth and Moon inertial coordinates are given by where μ μ M /μ E and ω μ E μ M /D 3 .Note that ω ω M 1 μ.
The initial conditions of the system of differential equations correspond to the position and velocity vectors of the space vehicle after the application of the first impulse.The initial conditions t 0 0 can be written as follows: where Δv 1 , r EP 0 , and θ EP t have the same meaning previously defined in Section 3 and, from 4.2 , It should be noted that r EP and v EP are orthogonal because the impulse is applied tangentially to LEO, assumed circular.The final conditions of the system of differential equations correspond to the position and velocity vectors of the space vehicle before the application of the second impulse and they are given by 3.4 -3.7 , with the final position and velocity vectors of Moon obtained from 4.2 , that is, given by

4.5
Accordingly, the final conditions can be put in the same form defined by 3.9 -3.11 .Therefore, the optimization problem is the same one defined in Section 3, and it is solved by the same algorithm previously described.The results are presented in Section 5.

Transformation to Rotating Frame
Consider a rotating reference frame Gξη contained in the Moon orbital plane: its origin is the center of mass of the Earth-Moon; the ξ-axis points towards the Moon position at any time t and the η-axis is perpendicular to the ξ-axis.In this rotating reference frame the Earth and the Moon are at rest. Figure 5 shows the inertial and rotating reference frames, Gxy and Gξη.To write the differential equations of motion of the space vehicle P in the rotating reference frame, consider the coordinate transformation equations: where k E, M, P .Thus, the new coordinates of the Earth and Moon are

4.9
Now, consider the inertial coordinates of the space vehicle written in terms of the rotating coordinates x P ξ P cos ωt − η P sin ωt , y P ξ P sin ωt η P cos ωt .

4.10
Differentiating each of these equations twice and substituting into 4.1 , one finds the new equations of motion: where

4.11
The system of differential equations above has a constant of motion, the so-called Jacobi integral.In order to determine it, procede as follows.Multiply the first of 4.11 by ξP , the second by ηP and add.It results in

4.13
This equation can be rewritten as The system of differential equations 4.11 has five equilibrium points.They are called Lagrange points and are labelled L 1 , . . ., L 5 .The points L 1 , L 2 , L 3 are located on the ξ-axis while L 4 and L 5 form two equilateral triangles with the Earth and Moon in the plane of rotation.See Figure 6.At each L k , the corresponding value of Jacobi constant C k is given by 4.15 substituting r L k ξ L k , η L k and v L k ξL k , ηL k 0, 0 .These C k are related to the regions in the rotating reference frame Gξη where the spacecraft can move.See in Table 1 the position and the correspondent value of the Jacobi constant per unit mass of each Lagrange point.
Observe that the right-hand side of the 4.15 must be nonnegative, since v 2 ≥ 0. Thus, an initial position ξ P 0 , η P 0 and an initial velocity ξP0 , ηP0 yield a Jacobi constant value C and the motion of the spacecraft is possible only in positions satisfying the relation 2Φ ξ P , η P ≥ C.

4.18
The set of points in the ξ, η -plane defined by the inequality 4.18 is called Hill's regions.See in Figure 6 how the Hill's regions white areas are related to the C k values.The shaded areas are the forbidden regions.Finally, we note that the transformation to the rotating frame gives a better insight about swing-by maneuvers with the Moon, as described in the results presented in Section 5.

Results
In this section, results are presented for some lunar missions using the three formulations described in the preceding sections.Analysis of the results is discussed in two subsections: in the first one, direct ascent maneuvers with flight time about 4.7 days are considered; in

Direct Ascent Maneuvers
Table 2 shows the results for lunar missions with counterclockwise arrival at LMO, and Table 3 shows the results for lunar missions with clockwise arrival at LMO. Recall that the departure from LEO is counterclockwise for all missions.The major parameters that are presented in these tables are the velocity changes Δv 1 and Δv 2 at each impulse, the total characteristic velocity Δv Total Δv 1 Δv 2 , the flight time of lunar mission T , and the angular position of the space vehicle with respect to Earth at the initial time defined by the angle θ EP 0 .  2 and 3 show good agreement.It should be noted the excellent results were obtained using the patched-conic approximation model.In all missions, the patchedconic approximation model yields very accurate estimate for the first impulse in comparison to the results obtained using the PCR3BP models.For the second impulse, there exists a small difference between the results given by the patched-conic approximation model and the PCR3BP models.For all lunar missions, the values of the major parameters Δv Total , Δv 1 , Δv 2 , and T obtained using the simplified PCR3BP model are a little lesser than the values obtained using the classical PCR3BP.In all cases, the trajectories are feasible, that is, the spacecraft does not collide with the Moon.As described in the next subsection, collisions can occur for maneuvers with multiple revolutions.
Tables also show a small difference in the flight time T and in the angle θ EP 0 calculated by the three approaches.We suppose that the difference between the values obtained in this paper and the values presented by 12 for the flight time T and the angle θ EP 0 calculated using the simplified PCR3BP model should be related to the accuracy in the integration of differential equations and in the solution of the terminal constraints.The algorithm based on gradient algorithm in conjunction with Newton-Raphson method, described in this paper, uses a Runge-Kutta-Fehlberg method of orders 4 and 5, with step-size control and relative error tolerance of 10 −10 and absolute error tolerance of 10 −11 , as described in Stoer and Bulirsch 16 and Forsythe et al. 18 .The terminal constraints are satisfied with an error lesser than 10 −8 .In all simulations the following canonical units are used: 1 distance unit R E and 1 time unit R 3 E /μ E .On the other hand, the paper by Miele and Mancuso 12 does not describe the accuracy used in the calculations.
According to the results presented in Tables 2 and 3, we note, regardless the dynamical model used in the analysis, that 1 lunar missions with clockwise LMO arrival spend more fuel than lunar missions with counterclockwise LMO arrival; 2 the flight time is nearly the same for all lunar missions with clockwise LMO arrival, independently on the final altitude of LMO.The differences between the flight 7 for the PCR3BP and patched-conic approximation models, the angle θ EP 0 varies with the LMO altitude for all lunar missions.
We note that some of these general results are quite similar to the ones described by 12 .
For h LMO 100 km, the trajectory is shown in Figure 7 for counterclockwise LMO arrival and in Figure 8 for clockwise LMO arrival.In both figures, trajectories are shown in the inertial reference frame Exy and in the rotating reference frame Gξη.Only results obtained through the classical PCR3BP model are depicted.

Multiple Revolution Ascent Maneuvers
Tables 4, 5, and 6 show the results for lunar missions with clockwise and counterclockwise arrival at LMO, for h LMO 100, 200, 300 km, respectively, considering the simplified PCR3BP model.Tables 7, 8, and 9 show similar results considering the classical PCR3BP model.The major parameters that are presented in these tables are the same ones presented in Tables 3  and 2. The value of the Jacobi constant per unit mass for each mission is presented in Tables 7, 8  For maneuvers with multiple revolutions, major comments are as follows.depict the collision between the spacecraft and the Earth for lunar missions with h LMO 100 km and counterclockwise or clockwise arrival to Moon, respectively.The remaining trajectories are feasible.4 According to the results in Tables 4-9, simplified and classical PCR3BP models show that the group of trajectories with counterclockwise arrival to Moon is slightly superior to the group of trajectories with clockwise arrival to Moon in terms of total characteristic velocity and flight time for maneuvers with flight time smaller than 25.5 days.
5 According to the results in Tables 4-9, simplified and classical PCR3BP models show that the group of trajectories with clockwise arrival to Moon is slightly superior to the group of trajectories with counterclockwise arrival to Moon in terms of total characteristic velocity and flight time excepting trajectories with flight time about 58.5 days for maneuvers with flight time larger than 30.0 days.

Conclusion
In this paper, a systematic study of optimal trajectories for Earth-Moon flight of a space vehicle is presented.The optimization criterion is the total characteristic velocity.The optimization problem has been formulated using the patched-conic approximation and two versions of the planar circular restricted three-body problem PCR3BP and has been solved using a gradient algorithm in conjunction with Newton-Raphson method.Results for direct ascent maneuvers and for maneuvers with multiple revolutions around the Earth are presented.For direct ascent maneuvers, all models show that lunar missions with clockwise   LMO arrival spend more fuel than lunar missions with counterclockwise LMO arrival.For maneuvers with multiple revolutions, fuel can be saved if the spacecraft accomplishes a swing-by maneuver with the Moon before the arrival at LMO.

4 . 8
Substituting 4.2 into 4.8 , one finds the fixed positions of the Earth and the Moon in the rotating reference frame:

Figure 6 :
Figure 6: Lagrange points and Hill's regions for the Earth-Moon system.

Table 1 :
Lagrange points for the Earth-Moon system.
and C is the so-called Jacobi constant.Therefore, from 4.15 it is seen that the Jacobi integral, J ξ P , η P , ξP , ηP 2Φ ξ P , η P − v 2 , 4.17 is equal to C during the motion 17 .
the second subsection, some maneuvers with multiple revolutions and flight time about 14, 24, 32, 40, and 58 days are considered.Three final altitudes h LMO of a clockwise or counterclockwise circular LMO and a specified altitude h LEO of a counterclockwise circular LEO, which corresponds to the altitude of the Space Station 12 , are considered.The following data are used: G 6.672 × 10 −20 km 3 kg −1 s −2 universal constant of gravitation , M E 5.9742 × 10 24 kg mass of the Earth , M M 7.3483 × 10 22 kg mass of the Moon , R EM 384 400 km mean distance from the Earth to the Moon , R E 6 378 km Earth radius , R M 1 738 km Moon radius , h
3 the flight time is nearly the same for all lunar missions with counterclockwise LMO arrival, independently on the final altitude of LMO.The differences between the flight time of each mission are small; they are approximately lesser than 0.010 days 14.4 minutes ; 4 the first change velocity Δv 1 is nearly independent of the LMO altitude; 5 the second change velocity Δv 2 decreases with the LMO altitude; 6 the flight time for lunar missions with clockwise LMO arrival is larger than the flight time for lunar missions with counterclockwise LMO arrival;

Table 6 :
Lunar mission, major parameters, h LEO 463 km, h LMO 300 km.In both figures, trajectories are shown in the inertial reference frame Exy and in the rotating reference frame Gξη.