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

Authors: Sandro da Silva Fernandes; Cleverson Maranhão Porto Marinho
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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 patched-conic 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.


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## Body

## 1. 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. It is assumed that the propulsion system delivers infinite thrusts during negligible times such that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses. The class of two impulse trajectories is considered: a first accelerating velocity impulse tangential to the space vehicle velocity relative to Earth is applied at a circular low Earth orbit and a second braking velocity impulse tangential to the space vehicle velocity relative to Moon is applied at a circular low Moon orbit [12]. The minimization of fuel consumption is equivalent to the minimization of the total characteristic velocity which is defined by the arithmetic sum of velocity changes [14].Three dynamical models are used to describe the motion of the space vehicle: the well-known patched-conic approximation [1] and two versions of the planar circular restricted three-body problem (PCR3BP). One version of PCR3BP assumes the Earth is fixed in space; this version will be referred as simplified version of PCR3BP, and it is same one used by Miele and Mancuso [12]. The second version of PCR3BP assumes the Earth moves around the center of mass of the Earth-Moon system, that is, it corresponds to the classical formulation [2, 3]. In the patched-conic approximation model, the parameters to be optimized are two: initial phase angle of space vehicle and the first velocity impulse. In this approach, the two-point boundary value problem involves only one final constraint. In this model, the flight time and the second velocity impulse are determined from the two-body dynamics after solving the two-point boundary value problem. 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 these formulations, the two-point boundary value problem involves three final constraints. In all cases, the optimization problem has one degree of freedom and can be solved by means of an algorithm based on gradient method [15] in conjunction with Newton-Raphson method [16]. The analysis of optimal trajectories is then carried out considering several final altitudes of a clockwise or counterclockwise circular low Moon orbit. All trajectories departure from a counterclockwise circular low Earth orbit corresponding to the altitude of the Space Station. Maneuvers with direct ascent and multiple revolutions around the Earth are considered in the analysis. The results for maneuvers with direct ascent are compared to the ones obtained by Miele and Mancuso [12] who used the sequential gradient-restoration algorithm for solving the optimization problem [15]. The results for maneuvers with multiple revolutions show that fuel can be saved if a lunar swing-by occurs.


## 2. Optimization Problem Based on Patched-Conic Approximation

In this section, the optimization problem based on the patched-conic approximation is formulated. A detailed presentation of the patched-conic approximation can be found in Bate et al. [1].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.(4)
The gravitational fields of Earth and Moon are central and obey the inverse square law.(5)
The trajectory has two distinct phases: geocentric and selenocentric trajectories. The geocentric phase corresponds to the portion of the trajectory which begins at the point of application of the first impulse and extends to the point of entering the 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:r0—radius of circular LEO; v0—velocity of the space vehicle at the point of application of the first impulse after the application of the impulse; φ0—flight path angle at the point of application of the first impulse; γ0—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 γ0 by the angle λ1 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. Figure1 shows the geometry of the geocentric phase.Figure 1
Geometry of the geocentric phase.For a given set of initial conditions(r0,v0,φ0), energy and angular momentum of the geocentric trajectory can be determined from the equations(2.1)E=12v02-μEr0,h=r0v0cos⁡φ0,
where μE is the Earth gravitational parameter.From the geometry of the geocentric phase (Figure1), one finds(2.2)r1=D2+RS2-2DRScos⁡λ1,sin⁡γ1=RSr1sin⁡λ1,
where D is the distance from the Earth to the Moon and RS 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)v1=2(E+μEr1),cos⁡φ1=hr1v1.The selenocentric phase begins at the point at which the geocentric trajectory crosses the Moon’s sphere of influence. Figure2 shows the geometry of the selenocentric phase for a clockwise arrival to LMO. Thus,(2.4)r2=RS,(2.5)v2=v1-vM,
where vM 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.Figure 2
Geometry of the selenocentric phase.From (2.5), one finds(2.6)v2=v12+vM2-2v1vMcos⁡(φ1-γ1),tan(λ1±φ2)=-v2sin⁡(φ1-γ1)vM-v2cos⁡(φ1-γ1).
The upper sign refers to clockwise arrival to LMO and the lower sign refers to counterclockwise to LMO.The semimajor axisaf and eccentricity ef of the selenocentric trajectory are given by(2.7)af=r22-Q2,ef=1+Q2(Q2-2)cos⁡2φ2,
where Q2=r2v22/μM and μM 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)rpM=af(1-ef),vpM=μM(1+ef)af(1-ef).Equations (2.1)–(2.8) lead to the following two-point boundary value problem: for a specified value of λ1 and a given set of initial parameters r0 and φ0=0° (the impulse is applied tangentially to the space vehicle velocity relative to Earth), determine v0 such that the final condition rpM=rf is satisfied, where r0 is the radius of LEO and rf 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 computingv0, the velocity changes at each impulse can be determined(2.9)Δv1=v0-μEr0,Δv2=μM(1+ef)af(1-ef)-μMrf.
The total characteristic velocity is then given by(2.10)ΔvTotal=Δv1+Δv2.Note that the total characteristic velocity is a function ofλ1 for a given set of parameters (r0,φ0=0°,rf). Accordingly, the following optimization problem can be formulated: determine λ1 to minimizeΔvTotal. 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(2.11)T=ΔtE+ΔtM,
where ΔtE is the flight time of the geocentric trajectory and ΔtM 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)ΔtE=a03μE(E1-e0sin⁡E1),ΔtM=(-af)3μM(efsinh⁡F2-F2),
with eccentric anomaly E1 and hyperbolic eccentric anomaly F2 obtained, respectively, from(2.13)cos⁡E1=1e0(1-r1a0),cosh⁡F2=1ef(1-r2af).
Since lunar arrival occurs prior to apoapsis of the geocentric trajectory, 0<E1≤180°, and F2 is positive. So, equations above define E1 and F2 without ambiguity. The semimajor axis and eccentricity of the geocentric trajectory are given by(2.14)a0=r02-Q0,e0=1+Q0(Q0-2)cos⁡2φ0,
where Q0=r0v02/μE. Recall that the impulses are applied at the periapses of the geocentric and selenocentric trajectories.


## 3. 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.(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 t0=0 and the y-axis is perpendicular to the x-axis. Figure 3 shows the inertial reference frame Exy.Figure 3
Inertial reference frame Exy.In the Exy reference frame, the motion of the space vehicle (P) is described by the following differential equations:(3.1)ẍP=-μErEP3xP-μMrMP3(xP-xM),ÿP=-μErEP3yP-μMrMP3(yP-yM),
where rEP and rMP are, respectively, the radial distances of space vehicle from Earth (E) and Moon (M), that is, rEP2=(xP-xE)2+(yP-yE)2 and rMP2=(xP-xM)2+(yP-yM)2. Because the origin of the inertial reference frame Exy is the Earth center, the position vector of the Earth is defined by rE=(0,0). The position vector of the Moon in the inertial reference frame Exy is defined by rM=(xM,yM). Since the eccentricity of the Moon orbit around Earth is neglected, the Moon inertial coordinates are given by(3.2)xM(t)=Dcos⁡(ωMt),yM(t)=Dsin⁡(ωMt),
where ωM=μE/D3 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(t0=0) can be written as follows:(3.3)xP(0)=xEP(0)=rEP(0)cos⁡θEP(0),yP(0)=yEP(0)=rEP(0)sin⁡θEP(0),ẋP(0)=ẋEP(0)=-[μErEP(0)+Δv1]sin⁡θEP(0),ẏP(0)=ẏEP(0)=[μErEP(0)+Δv1]cos⁡θEP(0),
where Δv1 is the velocity change at the first impulse, rEP(0)=rEP0, and θEP(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 rP forms with x-axis. It should be noted that rEP(0) and vEP(0) or, equivalently, rP(0) and vP(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(tf=T) can be written as follows:(3.4)xP(T)=xMP(T)+xM(T)=rMP(T)cos⁡θMP(T)+xM(T),(3.5)yP(T)=yMP(T)+yM(T)=rMP(T)sin⁡θMP(T)+yM(T),(3.6)ẋP(T)=ẋMP(T)+ẋM(T)=±[μMrMP(T)+Δv2]sin⁡θMP(T)+ẋM(T),(3.7)ẏP(T)=ẏMP(T)+ẏM(T)=∓[μMrMP(T)+Δv2]cos⁡θMP(T)+ẏM(T),
where Δv2 is the velocity change at the second impulse, rMP(T)=rMPf, and θMP(t) is the angle which the position vector rMP 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
(3.8)ẋM(T)=-DωMsin⁡(ωMT),ẏM(T)=DωMcos⁡(ωMT).The angleθMP(T) is free and can be eliminated. After the problem has been solved, the angle θMP(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:
(3.9)(xP(T)-xM(T))2+(yP(T)-yM(T))2=(rMP(T))2,(3.10)(ẋP(T)-ẋM(T))2+(ẏP(T)-ẏM(T))2=[μMrMP(T)+Δv2]2,(3.11)(xP(T)-xM(T))(ẏP(T)-ẏM(T))-(yP(T)-yM(T))(ẋP(T)-ẋM(T))=∓rMP(T)[μMrMP(T)+Δv2].
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 Δv1, Δv2, T, and θEP(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 Δv1, Δv2, T, and θEP(0) which satisfy the final constraints (3.9)–(3.11) and minimize the total characteristic velocity ΔvTotal=Δv1+Δv2. 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 θEP(0) has been chosen as the iteration variable in the gradient phase with Δv1, Δv2, and T calculated through Newton-Raphson method. The results are presented in Section 5.


## 4. 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 Section3, except for assumption (1) which must be replaced by the following one: Earth moves around the center of mass of the Earth-Moon system.


### 4.1. 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.Figure 4
Inertial reference frame Gxy.In the Gxy reference frame, the motion of the space vehicle (P) is described by the following differential equations:(4.1)ẍP=-μErEP3(xP-xE)-μMrMP3(xP-xM),ÿP=-μErEP3(yP-yE)-μMrMP3(yP-yM),
where rEP and rMP are, respectively, the radial distances of space vehicle from Earth (E) and Moon (M), that is, rEP2=(xP-xE)2+(yP-yE)2 and rMP2=(xP-xM)2+(yP-yM)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 rE=(xE,yE) and rM=(xM,yM). Since the eccentricity of the Moon orbit around Earth is neglected, the Earth and Moon inertial coordinates are given by(4.2)xE(t)=-μxM(t),yE(t)=-μyM(t),xM(t)=D1+μcos⁡(ωt),yM(t)=D1+μsin⁡(ωt),
where μ=μM/μE and ω=(μE+μM)/D3. Note that ω=ωM1+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 (t0=0) can be written as follows:(4.3)xP(0)=xEP(0)+xE(0)=rEP(0)cos⁡θEP(0)+xE(0),yP(0)=yEP(0)+yE(0)=rEP(0)sin⁡θEP(0)+yE(0),ẋP(0)=ẋEP(0)+ẋE(0)=-[μErEP(0)+Δv1]sin⁡θEP(0)+ẋE(0),ẏP(0)=ẏEP(0)+ẏE(0)=[μErEP(0)+Δv1]cos⁡θEP(0)+ẏE(0),
where Δv1, rEP(0), and θEP(t) have the same meaning previously defined in Section 3 and, from (4.2),(4.4)xE(0)=-μD1+μ,yE(0)=0,ẋE(0)=0,ẏE(0)=-μDω1+μ.
It should be noted that rEP and vEP 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)xM(T)=D1+μcos⁡(ωT),yM(T)=D1+μsin⁡(ωT),ẋM(T)=-Dω1+μsin⁡(ωT),ẏM(T)=Dω1+μcos⁡(ωT).
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 Section3, and it is solved by the same algorithm previously described. The results are presented in Section 5.


### 4.2. 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:(4.6)xk=ξkcos⁡(ωt)-ηksin⁡(ωt),yk=ξksin⁡(ωt)+ηkcos⁡(ωt),
or(4.7)ξk=xkcos⁡(ωt)+yksin⁡(ωt),ηk=-xksin⁡(ωt)+ykcos⁡(ωt),
where k=E,M,P. Thus, the new coordinates of the Earth and Moon are(4.8)ξE=xEcos⁡(ωt)+yEsin⁡(ωt),ηE=-xEsin⁡(ωt)+yEcos⁡(ωt),ξM=xMcos⁡(ωt)+yMsin⁡(ωt),ηM=-xMsin⁡(ωt)+yMcos⁡(ωt).Figure 5
Rotating reference frame Gxy.Substituting (4.2) into (4.8), one finds the fixed positions of the Earth and the Moon in the rotating reference frame:(4.9)ξE=-μD1+μ,ηE=0,ξM=D1+μ,ηM=0.
Now, consider the inertial coordinates of the space vehicle written in terms of the rotating coordinates(4.10)xP=ξPcos⁡(ωt)-ηPsin⁡(ωt),yP=ξPsin⁡(ωt)+ηPcos⁡(ωt).
Differentiating each of these equations twice and substituting into (4.1), one finds the new equations of motion:(4.11)ξ̈P-2ωη̇P-ω2ξP=-μErEP3(ξP-ξE)-μMrMP3(ξP-ξM),η̈P+2ωξ̇P-ω2ηP=-μErEP3(ηP-ηE)-μMrMP3(ηP-ηM),
where(4.12)rEP2=(ξP-ξE)2+(ηP-ηE)2,rMP2=(ξP-ξM)2+(ηP-ηM)2.The system of differential equations above has a constant of motion, the so-calledJacobi integral. In order to determine it, procede as follows. Multiply the first of (4.11) by ξ̇P, the second by η̇P and add. It results in(4.13)ξ̇Pξ̈P+η̇Pη̈P-ω2(ξ̇PξP+η̇PηP)=-μErEP3ξ̇P(ξP-ξE)-μMrMP3ξ̇P(ξP-ξM)-μErEP3η̇P(ηP-ηE)-μMrMP3η̇P(ηP-ηM).
This equation can be rewritten as(4.14)12ddt(ξ̇P2+η̇P2)-12ω2ddt(ξP2+ηP2)=ddt(μErEP+μMrMP).
Therefore,(4.15)v2=2Φ(ξP,ηP)-C,
where(4.16)v2=ξ̇P2+η̇P2,Φ(ξP,ηP)=12ω2(ξP2+ηP2)+μErEP+μMrMP,
and C is the so-called Jacobi constant. Therefore, from (4.15) it is seen that the Jacobi integral,(4.17)J(ξP,ηP,ξ̇P,η̇P)=2Φ(ξP,ηP)-v2,
is equal to C during the motion [17].The system of differential equations (4.11) has five equilibrium points. They are called Lagrange points and are labelled L1,…,L5. The points L1, L2, L3 are located on the ξ-axis while L4 and L5 form two equilateral triangles with the Earth and Moon in the plane of rotation. See Figure 6. At each Lk, the corresponding value of Jacobi constant Ck is given by (4.15) substituting rLk=(ξLk,ηLk) and vLk=(ξ̇Lk,η̇Lk)=(0,0). These Ck 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.Table 1
Lagrange points for the Earth-Moon system.
ξLkηLkCk(km)(km)(km2/s2)L13.2171×10503.3468L24.4424×10503.3298L3-3.8635×10503.1618L41.8753×1053.3290×1053.1365L51.8753×105-3.3290×1053.1365Lagrange points and Hill’s regions for the Earth-Moon system.
(a)
Lagrange points(b)
C1<C(c)
C2<C<C1(d)
C3<C<C2(e)
C4<C<C3(f)
C<C4Observe that the right-hand side of the (4.15) must be nonnegative, since v2≥0. Thus, an initial position (ξP0,ηP0) and an initial velocity (ξ̇P0,η̇P0) yield a Jacobi constant value C and the motion of the spacecraft is possible only in positions satisfying the relation(4.18)2Φ(ξP,ηP)≥C.
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 Ck 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 Section5.


## 4.1. 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.Figure 4
Inertial reference frame Gxy.In the Gxy reference frame, the motion of the space vehicle (P) is described by the following differential equations:(4.1)ẍP=-μErEP3(xP-xE)-μMrMP3(xP-xM),ÿP=-μErEP3(yP-yE)-μMrMP3(yP-yM),
where rEP and rMP are, respectively, the radial distances of space vehicle from Earth (E) and Moon (M), that is, rEP2=(xP-xE)2+(yP-yE)2 and rMP2=(xP-xM)2+(yP-yM)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 rE=(xE,yE) and rM=(xM,yM). Since the eccentricity of the Moon orbit around Earth is neglected, the Earth and Moon inertial coordinates are given by(4.2)xE(t)=-μxM(t),yE(t)=-μyM(t),xM(t)=D1+μcos⁡(ωt),yM(t)=D1+μsin⁡(ωt),
where μ=μM/μE and ω=(μE+μM)/D3. Note that ω=ωM1+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 (t0=0) can be written as follows:(4.3)xP(0)=xEP(0)+xE(0)=rEP(0)cos⁡θEP(0)+xE(0),yP(0)=yEP(0)+yE(0)=rEP(0)sin⁡θEP(0)+yE(0),ẋP(0)=ẋEP(0)+ẋE(0)=-[μErEP(0)+Δv1]sin⁡θEP(0)+ẋE(0),ẏP(0)=ẏEP(0)+ẏE(0)=[μErEP(0)+Δv1]cos⁡θEP(0)+ẏE(0),
where Δv1, rEP(0), and θEP(t) have the same meaning previously defined in Section 3 and, from (4.2),(4.4)xE(0)=-μD1+μ,yE(0)=0,ẋE(0)=0,ẏE(0)=-μDω1+μ.
It should be noted that rEP and vEP 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)xM(T)=D1+μcos⁡(ωT),yM(T)=D1+μsin⁡(ωT),ẋM(T)=-Dω1+μsin⁡(ωT),ẏM(T)=Dω1+μcos⁡(ωT).
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 Section3, and it is solved by the same algorithm previously described. The results are presented in Section 5.


## 4.2. 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:(4.6)xk=ξkcos⁡(ωt)-ηksin⁡(ωt),yk=ξksin⁡(ωt)+ηkcos⁡(ωt),
or(4.7)ξk=xkcos⁡(ωt)+yksin⁡(ωt),ηk=-xksin⁡(ωt)+ykcos⁡(ωt),
where k=E,M,P. Thus, the new coordinates of the Earth and Moon are(4.8)ξE=xEcos⁡(ωt)+yEsin⁡(ωt),ηE=-xEsin⁡(ωt)+yEcos⁡(ωt),ξM=xMcos⁡(ωt)+yMsin⁡(ωt),ηM=-xMsin⁡(ωt)+yMcos⁡(ωt).Figure 5
Rotating reference frame Gxy.Substituting (4.2) into (4.8), one finds the fixed positions of the Earth and the Moon in the rotating reference frame:(4.9)ξE=-μD1+μ,ηE=0,ξM=D1+μ,ηM=0.
Now, consider the inertial coordinates of the space vehicle written in terms of the rotating coordinates(4.10)xP=ξPcos⁡(ωt)-ηPsin⁡(ωt),yP=ξPsin⁡(ωt)+ηPcos⁡(ωt).
Differentiating each of these equations twice and substituting into (4.1), one finds the new equations of motion:(4.11)ξ̈P-2ωη̇P-ω2ξP=-μErEP3(ξP-ξE)-μMrMP3(ξP-ξM),η̈P+2ωξ̇P-ω2ηP=-μErEP3(ηP-ηE)-μMrMP3(ηP-ηM),
where(4.12)rEP2=(ξP-ξE)2+(ηP-ηE)2,rMP2=(ξP-ξM)2+(ηP-ηM)2.The system of differential equations above has a constant of motion, the so-calledJacobi integral. In order to determine it, procede as follows. Multiply the first of (4.11) by ξ̇P, the second by η̇P and add. It results in(4.13)ξ̇Pξ̈P+η̇Pη̈P-ω2(ξ̇PξP+η̇PηP)=-μErEP3ξ̇P(ξP-ξE)-μMrMP3ξ̇P(ξP-ξM)-μErEP3η̇P(ηP-ηE)-μMrMP3η̇P(ηP-ηM).
This equation can be rewritten as(4.14)12ddt(ξ̇P2+η̇P2)-12ω2ddt(ξP2+ηP2)=ddt(μErEP+μMrMP).
Therefore,(4.15)v2=2Φ(ξP,ηP)-C,
where(4.16)v2=ξ̇P2+η̇P2,Φ(ξP,ηP)=12ω2(ξP2+ηP2)+μErEP+μMrMP,
and C is the so-called Jacobi constant. Therefore, from (4.15) it is seen that the Jacobi integral,(4.17)J(ξP,ηP,ξ̇P,η̇P)=2Φ(ξP,ηP)-v2,
is equal to C during the motion [17].The system of differential equations (4.11) has five equilibrium points. They are called Lagrange points and are labelled L1,…,L5. The points L1, L2, L3 are located on the ξ-axis while L4 and L5 form two equilateral triangles with the Earth and Moon in the plane of rotation. See Figure 6. At each Lk, the corresponding value of Jacobi constant Ck is given by (4.15) substituting rLk=(ξLk,ηLk) and vLk=(ξ̇Lk,η̇Lk)=(0,0). These Ck 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.Table 1
Lagrange points for the Earth-Moon system.
ξLkηLkCk(km)(km)(km2/s2)L13.2171×10503.3468L24.4424×10503.3298L3-3.8635×10503.1618L41.8753×1053.3290×1053.1365L51.8753×105-3.3290×1053.1365Lagrange points and Hill’s regions for the Earth-Moon system.
(a)
Lagrange points(b)
C1<C(c)
C2<C<C1(d)
C3<C<C2(e)
C4<C<C3(f)
C<C4Observe that the right-hand side of the (4.15) must be nonnegative, since v2≥0. Thus, an initial position (ξP0,ηP0) and an initial velocity (ξ̇P0,η̇P0) yield a Jacobi constant value C and the motion of the spacecraft is possible only in positions satisfying the relation(4.18)2Φ(ξP,ηP)≥C.
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 Ck 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 Section5.


## 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 the second subsection, some maneuvers with multiple revolutions and flight time about 14, 24, 32, 40, and 58 days are considered. Three final altitudeshLMO of a clockwise or counterclockwise circular LMO and a specified altitude hLEO of a counterclockwise circular LEO, which corresponds to the altitude of the Space Station [12], are considered. The following data are used:(5.1)G=6.672×10-20km3kg-1s-2(universalconstantofgravitation),ME=5.9742×1024kg(massoftheEarth),MM=7.3483×1022kg(massoftheMoon),REM=384400km(meandistancefromtheEarthtotheMoon),RE=6378km(Earthradius),RM=1738km(Moonradius),hLEO=463km(altitudeofcircularLEO),hLMO=100,200,300km(altitudeofcircularLMO).


### 5.1. Direct Ascent Maneuvers

Table2 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 Δv1 and Δv2 at each impulse, the total characteristic velocity ΔvTotal=Δv1+Δv2, 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 θEP(0).Table 2
Lunar mission, counterclockwise LMO arrival, major parameters.
LMO altitudeModelΔvTotalΔv1Δv2TθEP(0)Feasibility(km)(km/s)(km/s)(km/s)(days)(deg)100Patched-conic3.84823.06550.78274.794−115.723YesSimplified PCR3BP3.87583.06490.81094.564−116.800YesClassical PCR3BP3.87773.06580.81194.573−116.410YesMiele and Mancuso [12]3.8763.0650.8114.37−118.98—200Patched-conic3.83313.06540.76774.794−115.750YesSimplified PCR3BP3.86143.06480.79664.562−116.832YesClassical PCR3BP3.86343.06580.79764.571−116.451YesMiele and Mancuso [12]3.8623.0650.7974.37−119.00—300Patched-conic3.81943.06540.75404.793−115.777YesSimplified PCR3BP3.84833.06480.78354.560−116.881YesClassical PCR3BP3.85023.06570.78454.569−116.491YesMiele and Mancuso [12]3.8493.0650.7844.37−119.03—Table 3
Lunar mission, clockwise LMO arrival, major parameters.
LMO altitudeModelΔvTotalΔv1Δv2TθEP(0)Feasibility(km)(km/s)(km/s)(km/s)(days)(deg)Patched-conic3.85283.06830.78454.936−113.681Yes100Simplified PCR3BP3.88113.06770.81344.750−114.215YesClassical PCR3BP3.88293.06860.81434.763−113.795YesMiele and Mancuso [12]3.8823.0680.8144.50−116.88—200Patched-conic3.83793.06830.76964.941−113.638YesSimplified PCR3BP3.86703.06770.79934.757−114.187YesClassical PCR3BP3.86883.06860.80024.769−113.742YesMiele and Mancuso [12]3.8683.0680.8004.50−116.88—300Patched-conic3.82433.06830.75604.944−113.608YesSimplified PCR3BP3.85413.06780.78634.760−114.116YesClassical PCR3BP3.85593.06870.78724.771−113.716YesMiele and Mancuso [12]3.8553.0680.7874.50−116.88—Results in Tables2 and 3 show good agreement. It should be noted the excellent results were obtained using the patched-conic approximation model. In all missions, the patched-conic 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 ΔvTotal, Δv1, Δv2, 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 timeT and in the angle θEP(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 θEP(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 = RE and 1 time unit =RE3/μE. 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 Tables2 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 time of each mission are small, they are approximately lesser tan 0.004 days (5.8 minutes);(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 velocityΔv1 is nearly independent of the LMO altitude;(5)
the second change velocityΔv2 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;(7)
for the PCR3BP and patched-conic approximation models, the angleθEP(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].ForhLMO=100km, 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.Direct ascent, counterclockwise LMO arrival,Δv1=3.0658 km/s, Δv2=0.8119 km/s, T=4.573 days, θEP(0)=-116.41 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameDirect ascent, clockwise LMO arrival,Δv1=3.0686 km/s, Δv2=0.8143 km/s, T=4.763 days, θEP(0)=-113.795 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frame


### 5.2. Multiple Revolution Ascent Maneuvers

Tables4, 5, and 6 show the results for lunar missions with clockwise and counterclockwise arrival at LMO, for hLMO=100,200,300km, 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, and 9.Table 4
Lunar mission, major parameters,hLEO=463km, hLMO=100km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.87583.06490.81094.564−116.800Yes3.87783.06530.812514.1759.119YesSimplifiedCounterclockwise3.87463.06470.809923.832135.661YesPCR3BP3.84113.05850.782632.108232.269Yes3.84443.05910.785340.871347.205Yes3.79363.04980.743858.701220.730Yes3.88113.06770.81344.750−114.215Yes3.88293.06810.814914.79117.307YesSimplifiedClockwise3.87853.06720.811324.916149.879YesPCR3BP3.83203.05860.773432.087232.573Yes3.83523.05920.776040.862347.608Yes3.78423.04980.744458.551220.998YesTable 5
Lunar mission, major parameters,hLEO=463km, hLMO=200km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.86143.06480.79664.562−116.832Yes3.86353.06530.798314.1689.002YesSimplifiedCounterclockwise3.86023.06460.795623.811135.376YesPCR3BP3.82633.05850.767832.110232.270Yes3.82963.05910.770540.873347.204Yes3.78263.05060.732058.398216.284Yes3.86703.06770.79934.753−114.187Yes3.86883.06810.800714.79817.390YesSimplifiedClockwise3.86433.06720.797124.930150.050YesPCR3BP3.81673.05860.758132.080232.475Yes3.82003.05920.760840.863347.613Yes3.76793.04980.718158.530220.726YesTable 6
Lunar mission, major parameters,hLEO=463km, hLMO=300km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.84833.06480.78354.560−116.881Yes3.85043.06520.785214.1578.851YesSimplifiedCounterclockwise3.84713.06460.782523.798135.197YesPCR3BP3.81273.05850.754232.111232.260Yes3.81613.05910.757040.874347.201Yes3.76333.04980.713558.685220.409Yes3.85413.06780.78634.760−114.110Yes3.85593.06810.787814.80917.504YesSimplifiedClockwise3.85133.06730.784024.948150.265YesPCR3BP3.80263.05860.744032.080232.475Yes3.80603.05920.746740.864347.620Yes3.75303.04980.703158.488220.014YesTable 7
Lunar mission, major parameters,hLEO=463km, hLMO=100km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.87773.06580.81194.573−116.4102.4784Yes3.87323.06500.808214.33012.4662.4965YesClassicalCounterclockwise3.87763.06580.811824.019140.0542.4788NoPCR3BP3.84283.05930.783531.910232.4642.6166Yes3.83793.05840.779540.742348.8822.6358Yes3.83003.05700.773058.415229.2392.6668Yes3.88293.06860.81434.763−113.7952.4187Yes3.87853.06780.810714.97021.0692.4363YesClassicalClockwise3.88213.06840.813625.103154.3152.4220NoPCR3BP3.83373.05950.774231.872232.6742.6138Yes3.82883.05860.770240.713349.3222.6331Yes3.78933.05130.738058.420224.4802.7871YesTable 8
Lunar mission, major parameters,hLEO=463km, hLMO=200km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.86343.06580.79764.571−116.4512.4793Yes3.85873.06490.793814.32212.3402.4974YesClassicalCounterclockwise3.86333.06570.797524.004139.8462.4797NoPCR3BP3.82803.05930.768631.912232.4612.6166Yes3.82303.05840.764540.743348.8762.6358Yes3.81493.05700.757958.417229.2182.6669Yes3.86883.06860.80024.769−113.7422.4178Yes3.86433.06780.796514.98121.1982.4355YesClassicalClockwise3.86793.06850.799425.121154.5422.4212NoPCR3BP3.81843.05950.758931.873232.6762.6137Yes3.81953.05970.759839.664353.1192.6094Yes3.77093.05090.719958.361223.8862.7957YesTable 9
Lunar mission, major parameters,hLEO=463km, hLMO=300km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.85023.06570.78454.569−116.4912.4802Yes3.84553.06490.780614.31312.2132.4983YesClassicalCounterclockwise3.85013.06570.784423.990139.6432.4806NoPCR3BP3.81443.05930.755131.913232.4612.6166Yes3.80933.05840.750940.745348.8702.6359Yes3.80103.05700.744158.422229.2352.6671Yes3.85593.06870.78724.771−113.7162.4170Yes3.85133.06780.783414.99221.3362.4346YesClassicalClockwise3.85503.06850.786525.139154.7572.4204NoPCR3BP3.80433.05950.744831.873232.6762.6137Yes3.80543.05970.745839.665353.1112.6094Yes3.75593.05090.705058.361223.9132.7957YesForhLMO=100km, the trajectories are shown in Figures 9 to 13 for counterclockwise LMO arrival and in Figures 14 to 18 for clockwise LMO arrival, considering the classical PCR3BP model. In both figures, trajectories are shown in the inertial reference frame Exy and in the rotating reference frame Gξη.Multiple revolution ascent, counterclockwise LMO arrival,Δv1=3.065 km/s, Δv2=0.8082 km/s, T=14.33 days, θEP(0)=12.466 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameFor maneuvers with multiple revolutions, major comments are as follows.(1)
All trajectories, regardless of the flight time, are feasible in the simplified PCR3BP model.(2)
In the classical PCR3BP model, all trajectories have Jacobi constant less thanC4. Therefore, there are not forbidden regions for the motion of the spacecraft.(3)
Trajectories with two revolutions around the Earth and flight time about 24.0 days collide with the Earth in the classical PCR3BP model. Figures10 and 15 depict the collision between the spacecraft and the Earth for lunar missions with hLMO=100km and counterclockwise or clockwise arrival to Moon, respectively. The remaining trajectories are feasible.(4)
According to the results in Tables4–9, 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 Tables4–9, 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.(6)
For lunar missions withhLMO=100km, Figures 11, 12, 13, 16, 17, and, 18 show that the spacecraft carries out one or two close approaches to the Moon before entering into the low circular orbit around the Moon. Such maneuvers, known as swing-by maneuvers, reduce the fuel consumption, although the flight time increases (see results in Table 7).(7)
According to the results in Tables4–9, the first change velocity Δv1 is nearly independent of the LMO altitude, and, the second change velocity Δv2 decreases as the LMO altitude increases, for all group of trajectories with similar flight time and same sense of arrival to Moon.(8)
The first change velocityΔv1 is nearly the same for maneuvers with flight time smaller than 25.5 days (the differences in Δv1 are approximately 0.6 m/s).(9)
For maneuvers with flight time larger than 30.0 days, the first change velocityΔv1 changes slightly (the differences in Δv1 are approximately 15.0 to 20.0 m/s).Multiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0658 km/s, Δv2=0.8118 km/s, T=24.019 days, θEP(0)=140.054 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0593 km/s, Δv2=0.7835 km/s, T=31.91 days, θEP(0)=232.464 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0584 km/s, Δv2=0.7795 km/s, T=40.742 days, θEP(0)=348.882 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.057 km/s, Δv2=0.7730 km/s, T=58.415 days, θEP(0)=229.239 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0678 km/s, Δv2=0.8107 km/s, T=14.97 days, θEP(0)=21.069 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0684 km/s, Δv2=0.8136 km/s, T=25.103 days, θEP(0)=154.315 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0595 km/s, Δv2=0.7742 km/s, T=31.872 days, θEP(0)=232.674 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0586 km/s, Δv2=0.7702 km/s, T=40.713 days, θEP(0)=349.322 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0513 km/s, Δv2=0.738 km/s, T=58.42 days, θEP(0)=224.48 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frame


## 5.1. Direct Ascent Maneuvers

Table2 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 Δv1 and Δv2 at each impulse, the total characteristic velocity ΔvTotal=Δv1+Δv2, 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 θEP(0).Table 2
Lunar mission, counterclockwise LMO arrival, major parameters.
LMO altitudeModelΔvTotalΔv1Δv2TθEP(0)Feasibility(km)(km/s)(km/s)(km/s)(days)(deg)100Patched-conic3.84823.06550.78274.794−115.723YesSimplified PCR3BP3.87583.06490.81094.564−116.800YesClassical PCR3BP3.87773.06580.81194.573−116.410YesMiele and Mancuso [12]3.8763.0650.8114.37−118.98—200Patched-conic3.83313.06540.76774.794−115.750YesSimplified PCR3BP3.86143.06480.79664.562−116.832YesClassical PCR3BP3.86343.06580.79764.571−116.451YesMiele and Mancuso [12]3.8623.0650.7974.37−119.00—300Patched-conic3.81943.06540.75404.793−115.777YesSimplified PCR3BP3.84833.06480.78354.560−116.881YesClassical PCR3BP3.85023.06570.78454.569−116.491YesMiele and Mancuso [12]3.8493.0650.7844.37−119.03—Table 3
Lunar mission, clockwise LMO arrival, major parameters.
LMO altitudeModelΔvTotalΔv1Δv2TθEP(0)Feasibility(km)(km/s)(km/s)(km/s)(days)(deg)Patched-conic3.85283.06830.78454.936−113.681Yes100Simplified PCR3BP3.88113.06770.81344.750−114.215YesClassical PCR3BP3.88293.06860.81434.763−113.795YesMiele and Mancuso [12]3.8823.0680.8144.50−116.88—200Patched-conic3.83793.06830.76964.941−113.638YesSimplified PCR3BP3.86703.06770.79934.757−114.187YesClassical PCR3BP3.86883.06860.80024.769−113.742YesMiele and Mancuso [12]3.8683.0680.8004.50−116.88—300Patched-conic3.82433.06830.75604.944−113.608YesSimplified PCR3BP3.85413.06780.78634.760−114.116YesClassical PCR3BP3.85593.06870.78724.771−113.716YesMiele and Mancuso [12]3.8553.0680.7874.50−116.88—Results in Tables2 and 3 show good agreement. It should be noted the excellent results were obtained using the patched-conic approximation model. In all missions, the patched-conic 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 ΔvTotal, Δv1, Δv2, 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 timeT and in the angle θEP(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 θEP(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 = RE and 1 time unit =RE3/μE. 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 Tables2 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 time of each mission are small, they are approximately lesser tan 0.004 days (5.8 minutes);(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 velocityΔv1 is nearly independent of the LMO altitude;(5)
the second change velocityΔv2 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;(7)
for the PCR3BP and patched-conic approximation models, the angleθEP(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].ForhLMO=100km, 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.Direct ascent, counterclockwise LMO arrival,Δv1=3.0658 km/s, Δv2=0.8119 km/s, T=4.573 days, θEP(0)=-116.41 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameDirect ascent, clockwise LMO arrival,Δv1=3.0686 km/s, Δv2=0.8143 km/s, T=4.763 days, θEP(0)=-113.795 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frame


## 5.2. Multiple Revolution Ascent Maneuvers

Tables4, 5, and 6 show the results for lunar missions with clockwise and counterclockwise arrival at LMO, for hLMO=100,200,300km, 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, and 9.Table 4
Lunar mission, major parameters,hLEO=463km, hLMO=100km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.87583.06490.81094.564−116.800Yes3.87783.06530.812514.1759.119YesSimplifiedCounterclockwise3.87463.06470.809923.832135.661YesPCR3BP3.84113.05850.782632.108232.269Yes3.84443.05910.785340.871347.205Yes3.79363.04980.743858.701220.730Yes3.88113.06770.81344.750−114.215Yes3.88293.06810.814914.79117.307YesSimplifiedClockwise3.87853.06720.811324.916149.879YesPCR3BP3.83203.05860.773432.087232.573Yes3.83523.05920.776040.862347.608Yes3.78423.04980.744458.551220.998YesTable 5
Lunar mission, major parameters,hLEO=463km, hLMO=200km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.86143.06480.79664.562−116.832Yes3.86353.06530.798314.1689.002YesSimplifiedCounterclockwise3.86023.06460.795623.811135.376YesPCR3BP3.82633.05850.767832.110232.270Yes3.82963.05910.770540.873347.204Yes3.78263.05060.732058.398216.284Yes3.86703.06770.79934.753−114.187Yes3.86883.06810.800714.79817.390YesSimplifiedClockwise3.86433.06720.797124.930150.050YesPCR3BP3.81673.05860.758132.080232.475Yes3.82003.05920.760840.863347.613Yes3.76793.04980.718158.530220.726YesTable 6
Lunar mission, major parameters,hLEO=463km, hLMO=300km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)Feasibility(km/s)(km/s)(km/s)(days)(deg)3.84833.06480.78354.560−116.881Yes3.85043.06520.785214.1578.851YesSimplifiedCounterclockwise3.84713.06460.782523.798135.197YesPCR3BP3.81273.05850.754232.111232.260Yes3.81613.05910.757040.874347.201Yes3.76333.04980.713558.685220.409Yes3.85413.06780.78634.760−114.110Yes3.85593.06810.787814.80917.504YesSimplifiedClockwise3.85133.06730.784024.948150.265YesPCR3BP3.80263.05860.744032.080232.475Yes3.80603.05920.746740.864347.620Yes3.75303.04980.703158.488220.014YesTable 7
Lunar mission, major parameters,hLEO=463km, hLMO=100km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.87773.06580.81194.573−116.4102.4784Yes3.87323.06500.808214.33012.4662.4965YesClassicalCounterclockwise3.87763.06580.811824.019140.0542.4788NoPCR3BP3.84283.05930.783531.910232.4642.6166Yes3.83793.05840.779540.742348.8822.6358Yes3.83003.05700.773058.415229.2392.6668Yes3.88293.06860.81434.763−113.7952.4187Yes3.87853.06780.810714.97021.0692.4363YesClassicalClockwise3.88213.06840.813625.103154.3152.4220NoPCR3BP3.83373.05950.774231.872232.6742.6138Yes3.82883.05860.770240.713349.3222.6331Yes3.78933.05130.738058.420224.4802.7871YesTable 8
Lunar mission, major parameters,hLEO=463km, hLMO=200km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.86343.06580.79764.571−116.4512.4793Yes3.85873.06490.793814.32212.3402.4974YesClassicalCounterclockwise3.86333.06570.797524.004139.8462.4797NoPCR3BP3.82803.05930.768631.912232.4612.6166Yes3.82303.05840.764540.743348.8762.6358Yes3.81493.05700.757958.417229.2182.6669Yes3.86883.06860.80024.769−113.7422.4178Yes3.86433.06780.796514.98121.1982.4355YesClassicalClockwise3.86793.06850.799425.121154.5422.4212NoPCR3BP3.81843.05950.758931.873232.6762.6137Yes3.81953.05970.759839.664353.1192.6094Yes3.77093.05090.719958.361223.8862.7957YesTable 9
Lunar mission, major parameters,hLEO=463km, hLMO=300km.
ModelManeuverΔvTotalΔv1Δv2TθEP(0)CFeasibility(km/s)(km/s)(km/s)(days)(deg)(km2/s2)3.85023.06570.78454.569−116.4912.4802Yes3.84553.06490.780614.31312.2132.4983YesClassicalCounterclockwise3.85013.06570.784423.990139.6432.4806NoPCR3BP3.81443.05930.755131.913232.4612.6166Yes3.80933.05840.750940.745348.8702.6359Yes3.80103.05700.744158.422229.2352.6671Yes3.85593.06870.78724.771−113.7162.4170Yes3.85133.06780.783414.99221.3362.4346YesClassicalClockwise3.85503.06850.786525.139154.7572.4204NoPCR3BP3.80433.05950.744831.873232.6762.6137Yes3.80543.05970.745839.665353.1112.6094Yes3.75593.05090.705058.361223.9132.7957YesForhLMO=100km, the trajectories are shown in Figures 9 to 13 for counterclockwise LMO arrival and in Figures 14 to 18 for clockwise LMO arrival, considering the classical PCR3BP model. In both figures, trajectories are shown in the inertial reference frame Exy and in the rotating reference frame Gξη.Multiple revolution ascent, counterclockwise LMO arrival,Δv1=3.065 km/s, Δv2=0.8082 km/s, T=14.33 days, θEP(0)=12.466 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameFor maneuvers with multiple revolutions, major comments are as follows.(1)
All trajectories, regardless of the flight time, are feasible in the simplified PCR3BP model.(2)
In the classical PCR3BP model, all trajectories have Jacobi constant less thanC4. Therefore, there are not forbidden regions for the motion of the spacecraft.(3)
Trajectories with two revolutions around the Earth and flight time about 24.0 days collide with the Earth in the classical PCR3BP model. Figures10 and 15 depict the collision between the spacecraft and the Earth for lunar missions with hLMO=100km and counterclockwise or clockwise arrival to Moon, respectively. The remaining trajectories are feasible.(4)
According to the results in Tables4–9, 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 Tables4–9, 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.(6)
For lunar missions withhLMO=100km, Figures 11, 12, 13, 16, 17, and, 18 show that the spacecraft carries out one or two close approaches to the Moon before entering into the low circular orbit around the Moon. Such maneuvers, known as swing-by maneuvers, reduce the fuel consumption, although the flight time increases (see results in Table 7).(7)
According to the results in Tables4–9, the first change velocity Δv1 is nearly independent of the LMO altitude, and, the second change velocity Δv2 decreases as the LMO altitude increases, for all group of trajectories with similar flight time and same sense of arrival to Moon.(8)
The first change velocityΔv1 is nearly the same for maneuvers with flight time smaller than 25.5 days (the differences in Δv1 are approximately 0.6 m/s).(9)
For maneuvers with flight time larger than 30.0 days, the first change velocityΔv1 changes slightly (the differences in Δv1 are approximately 15.0 to 20.0 m/s).Multiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0658 km/s, Δv2=0.8118 km/s, T=24.019 days, θEP(0)=140.054 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0593 km/s, Δv2=0.7835 km/s, T=31.91 days, θEP(0)=232.464 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.0584 km/s, Δv2=0.7795 km/s, T=40.742 days, θEP(0)=348.882 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, counterclockwise LMO arrival,Δv1=3.057 km/s, Δv2=0.7730 km/s, T=58.415 days, θEP(0)=229.239 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0678 km/s, Δv2=0.8107 km/s, T=14.97 days, θEP(0)=21.069 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0684 km/s, Δv2=0.8136 km/s, T=25.103 days, θEP(0)=154.315 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0595 km/s, Δv2=0.7742 km/s, T=31.872 days, θEP(0)=232.674 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0586 km/s, Δv2=0.7702 km/s, T=40.713 days, θEP(0)=349.322 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frameMultiple revolution ascent, clockwise LMO arrival,Δv1=3.0513 km/s, Δv2=0.738 km/s, T=58.42 days, θEP(0)=224.48 deg.
(a)
Earth-Moon trajectory, inertial coordinate frame(b)
LEO departure, inertial coordinate frame(c)
LMO arrival, inertial coordinate frame(d)
Earth-Moon trajectory, rotating coordinate frame(e)
LEO departure, rotating coordinate frame(f)
LMO arrival, rotating coordinate frame


## 6. 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.



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*Source: 971983-2011-11-24.xml*