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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 971983, 34 pages
http://dx.doi.org/10.1155/2012/971983
Research Article

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

Departamento de Matemática, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil

Received 10 June 2011; Accepted 4 September 2011

Academic Editor: Antonio F. Bertachini A. Prado

Copyright © 2012 Sandro da Silva Fernandes and Cleverson Maranhão Porto Marinho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [510]. 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 [810, 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. Figure 1 shows the geometry of the geocentric phase.

971983.fig.001
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 equations1=2𝑣20𝜇𝐸𝑟0,=𝑟0𝑣0cos𝜑0,(2.1) where 𝜇𝐸 is the Earth gravitational parameter.

From the geometry of the geocentric phase (Figure 1), one finds𝑟1=𝐷2+𝑅2𝑆2𝐷𝑅𝑆cos𝜆1,sin𝛾1=𝑅𝑆𝑟1sin𝜆1,(2.2) 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 finds𝑣1=2𝜇+𝐸𝑟1,cos𝜑1=𝑟1𝑣1.(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,𝑟2=𝑅𝑆,𝐯(2.4)2=𝐯1𝐯𝑀,(2.5) 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.

971983.fig.002
Figure 2: Geometry of the selenocentric phase.

From (2.5), one finds𝑣2=𝑣21+𝑣2𝑀2𝑣1𝑣𝑀𝜑cos1𝛾1,𝜆tan1±𝜑2𝑣=2𝜑sin1𝛾1𝑣𝑀𝑣2𝜑cos1𝛾1.(2.6) The upper sign refers to clockwise arrival to LMO and the lower sign refers to counterclockwise to LMO.

The semimajor axis 𝑎𝑓 and eccentricity 𝑒𝑓 of the selenocentric trajectory are given by𝑎𝑓=𝑟22𝑄2,𝑒𝑓=1+𝑄2𝑄22cos2𝜑2,(2.7) 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 by𝑟𝑝𝑀=𝑎𝑓1𝑒𝑓,𝑣𝑝𝑀=𝜇𝑀1+𝑒𝑓𝑎𝑓1𝑒𝑓.(2.8)

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 computing 𝑣0, the velocity changes at each impulse can be determinedΔ𝑣1=𝑣0𝜇𝐸𝑟0,Δ𝑣2=𝜇𝑀1+𝑒𝑓𝑎𝑓1𝑒𝑓𝜇𝑀𝑟𝑓.(2.9) The total characteristic velocity is then given byΔ𝑣Total=Δ𝑣1+Δ𝑣2.(2.10)

Note that the total characteristic velocity is a function of 𝜆1 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 by𝑇=Δ𝑡𝐸+Δ𝑡𝑀,(2.11) 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:Δ𝑡𝐸=𝑎30𝜇𝐸𝐸1𝑒0sin𝐸1,Δ𝑡𝑀=𝑎𝑓3𝜇𝑀𝑒𝑓sinh𝐹2𝐹2,(2.12) with eccentric anomaly 𝐸1 and hyperbolic eccentric anomaly 𝐹2 obtained, respectively, fromcos𝐸1=1𝑒0𝑟11𝑎0,cosh𝐹2=1𝑒𝑓𝑟12𝑎𝑓.(2.13) Since lunar arrival occurs prior to apoapsis of the geocentric trajectory, 0<𝐸1180, and 𝐹2 is positive. So, equations above define 𝐸1 and 𝐹2 without ambiguity. The semimajor axis and eccentricity of the geocentric trajectory are given by𝑎0=𝑟02𝑄0,𝑒0=1+𝑄0𝑄02cos2𝜑0,(2.14) where 𝑄0=𝑟0𝑣20/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.

971983.fig.003
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:̈𝑥𝑃𝜇=𝐸𝑟3𝐸𝑃𝑥𝑃𝜇𝑀𝑟3𝑀𝑃𝑥𝑃𝑥𝑀,̈𝑦𝑃𝜇=𝐸𝑟3𝐸𝑃𝑦𝑃𝜇𝑀𝑟3𝑀𝑃𝑦𝑃𝑦𝑀,(3.1) where 𝑟𝐸𝑃 and 𝑟𝑀𝑃 are, respectively, the radial distances of space vehicle from Earth (𝐸) and Moon (𝑀), that is, 𝑟2𝐸𝑃=(𝑥𝑃𝑥𝐸)2+(𝑦𝑃𝑦𝐸)2 and 𝑟2𝑀𝑃=(𝑥𝑃𝑥𝑀)2+(𝑦𝑃𝑦𝑀)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 by𝑥𝑀𝜔(𝑡)=𝐷cos𝑀𝑡,𝑦𝑀(𝜔𝑡)=𝐷sin𝑀𝑡,(3.2) 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:𝑥𝑃(0)=𝑥𝐸𝑃(0)=𝑟𝐸𝑃(0)cos𝜃𝐸𝑃𝑦(0),𝑃(0)=𝑦𝐸𝑃(0)=𝑟𝐸𝑃(0)sin𝜃𝐸𝑃(0),̇𝑥𝑃(0)=̇𝑥𝐸𝑃(0)=𝜇𝐸𝑟𝐸𝑃(0)+Δ𝑣1sin𝜃𝐸𝑃(0),̇𝑦𝑃(0)=̇𝑦𝐸𝑃(0)=𝜇𝐸𝑟𝐸𝑃(0)+Δ𝑣1cos𝜃𝐸𝑃(0),(3.3) 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:𝑥𝑃(𝑇)=𝑥𝑀𝑃(𝑇)+𝑥𝑀(𝑇)=𝑟𝑀𝑃(𝑇)cos𝜃𝑀𝑃(𝑇)+𝑥𝑀𝑦(𝑇),(3.4)𝑃(𝑇)=𝑦𝑀𝑃(𝑇)+𝑦𝑀(𝑇)=𝑟𝑀𝑃(𝑇)sin𝜃𝑀𝑃(𝑇)+𝑦𝑀(𝑇),(3.5)̇𝑥𝑃(𝑇)=̇𝑥𝑀𝑃(𝑇)+̇𝑥𝑀(𝑇)=±𝜇𝑀𝑟𝑀𝑃(𝑇)+Δ𝑣2sin𝜃𝑀𝑃(𝑇)+̇𝑥𝑀(𝑇),(3.6)̇𝑦𝑃(𝑇)=̇𝑦𝑀𝑃(𝑇)+̇𝑦𝑀(𝑇)=𝜇𝑀𝑟𝑀𝑃(𝑇)+Δ𝑣2cos𝜃𝑀𝑃(𝑇)+̇𝑦𝑀(𝑇),(3.7) 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 ̇𝑥𝑀(𝑇)=𝐷𝜔𝑀𝜔sin𝑀𝑇,̇𝑦𝑀(𝑇)=𝐷𝜔𝑀𝜔cos𝑀𝑇.(3.8)

The angle 𝜃𝑀𝑃(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: 𝑥𝑃(𝑇)𝑥𝑀(𝑇)2+𝑦𝑃(𝑇)𝑦𝑀(𝑇)2=𝑟𝑀𝑃(𝑇)2,(3.9)̇𝑥𝑃(𝑇)̇𝑥𝑀(𝑇)2+̇𝑦𝑃(𝑇)̇𝑦𝑀(𝑇)2=𝜇𝑀𝑟𝑀𝑃(𝑇)+Δ𝑣22,𝑥(3.10)𝑃(𝑇)𝑥𝑀(𝑇)̇𝑦𝑃(𝑇)̇𝑦𝑀𝑦(𝑇)𝑃(𝑇)𝑦𝑀(𝑇)̇𝑥𝑃(𝑇)̇𝑥𝑀(𝑇)=𝑟𝑀𝑃(𝑇)𝜇𝑀𝑟𝑀𝑃(𝑇)+Δ𝑣2.(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 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 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.

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.

971983.fig.004
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:̈𝑥𝑃𝜇=𝐸𝑟3𝐸𝑃𝑥𝑃𝑥𝐸𝜇𝑀𝑟3𝑀𝑃𝑥𝑃𝑥𝑀,̈𝑦𝑃𝜇=𝐸𝑟3𝐸𝑃𝑦𝑃𝑦𝐸𝜇𝑀𝑟3𝑀𝑃𝑦𝑃𝑦𝑀,(4.1) where 𝑟𝐸𝑃 and 𝑟𝑀𝑃 are, respectively, the radial distances of space vehicle from Earth (𝐸) and Moon (𝑀), that is, 𝑟2𝐸𝑃=(𝑥𝑃𝑥𝐸)2+(𝑦𝑃𝑦𝐸)2 and 𝑟2𝑀𝑃=(𝑥𝑃𝑥𝑀)2+(𝑦𝑃𝑦𝑀)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 by𝑥𝐸(𝑡)=𝜇𝑥𝑀𝑦(𝑡),𝐸(𝑡)=𝜇𝑦𝑀𝑥(𝑡),𝑀𝐷(𝑡)=𝑦1+𝜇cos(𝜔𝑡),𝑀𝐷(𝑡)=1+𝜇sin(𝜔𝑡),(4.2) 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:𝑥𝑃(0)=𝑥𝐸𝑃(0)+𝑥𝐸(0)=𝑟𝐸𝑃(0)cos𝜃𝐸𝑃(0)+𝑥𝐸𝑦(0),𝑃(0)=𝑦𝐸𝑃(0)+𝑦𝐸(0)=𝑟𝐸𝑃(0)sin𝜃𝐸𝑃(0)+𝑦𝐸(0),̇𝑥𝑃(0)=̇𝑥𝐸𝑃(0)+̇𝑥𝐸(0)=𝜇𝐸𝑟𝐸𝑃(0)+Δ𝑣1sin𝜃𝐸𝑃(0)+̇𝑥𝐸(0),̇𝑦𝑃(0)=̇𝑦𝐸𝑃(0)+̇𝑦𝐸(0)=𝜇𝐸𝑟𝐸𝑃(0)+Δ𝑣1cos𝜃𝐸𝑃(0)+̇𝑦𝐸(0),(4.3) where Δ𝑣1, 𝑟𝐸𝑃(0), and 𝜃𝐸𝑃(𝑡) have the same meaning previously defined in Section 3 and, from (4.2),𝑥𝐸(0)=𝜇𝐷1+𝜇,𝑦𝐸(0)=0,̇𝑥𝐸(0)=0,̇𝑦𝐸(0)=𝜇𝐷𝜔.1+𝜇(4.4) 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 by𝑥𝑀𝐷(𝑇)=1+𝜇cos(𝜔𝑇),𝑦𝑀𝐷(𝑇)=1+𝜇sin(𝜔𝑇),̇𝑥𝑀(𝑇)=𝐷𝜔1+𝜇sin(𝜔𝑇),̇𝑦𝑀(𝑇)=𝐷𝜔1+𝜇cos(𝜔𝑇).(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.

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:𝑥𝑘=𝜉𝑘cos(𝜔𝑡)𝜂𝑘𝑦sin(𝜔𝑡),𝑘=𝜉𝑘sin(𝜔𝑡)+𝜂𝑘cos(𝜔𝑡),(4.6) or𝜉𝑘=𝑥𝑘cos(𝜔𝑡)+𝑦𝑘𝜂sin(𝜔𝑡),𝑘=𝑥𝑘sin(𝜔𝑡)+𝑦𝑘cos(𝜔𝑡),(4.7) where 𝑘=𝐸,𝑀,𝑃. Thus, the new coordinates of the Earth and Moon are𝜉𝐸=𝑥𝐸cos(𝜔𝑡)+𝑦𝐸𝜂sin(𝜔𝑡),𝐸=𝑥𝐸sin(𝜔𝑡)+𝑦𝐸𝜉cos(𝜔𝑡),𝑀=𝑥𝑀cos(𝜔𝑡)+𝑦𝑀𝜂sin(𝜔𝑡),𝑀=𝑥𝑀sin(𝜔𝑡)+𝑦𝑀cos(𝜔𝑡).(4.8)

971983.fig.005
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:𝜉𝐸=𝜇𝐷,𝜂1+𝜇𝐸𝜉=0,𝑀=𝐷,𝜂1+𝜇𝑀=0.(4.9) Now, consider the inertial coordinates of the space vehicle written in terms of the rotating coordinates𝑥𝑃=𝜉𝑃cos(𝜔𝑡)𝜂𝑃𝑦sin(𝜔𝑡),𝑃=𝜉𝑃sin(𝜔𝑡)+𝜂𝑃cos(𝜔𝑡).(4.10) Differentiating each of these equations twice and substituting into (4.1), one finds the new equations of motion:̈𝜉𝑃2𝜔̇𝜂𝑃𝜔2𝜉𝑃𝜇=𝐸𝑟3𝐸𝑃𝜉𝑃𝜉𝐸𝜇𝑀𝑟3𝑀𝑃𝜉𝑃𝜉𝑀,̈𝜂𝑃̇𝜉+2𝜔𝑃𝜔2𝜂𝑃𝜇=𝐸𝑟3𝐸𝑃𝜂𝑃𝜂𝐸𝜇𝑀𝑟3𝑀𝑃𝜂𝑃𝜂𝑀,(4.11) where𝑟2𝐸𝑃=𝜉𝑃𝜉𝐸2+𝜂𝑃𝜂𝐸2,𝑟2𝑀𝑃=𝜉𝑃𝜉𝑀2+𝜂𝑃𝜂𝑀2.(4.12)

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 iṅ𝜉𝑃̈𝜉𝑃+̇𝜂𝑃̈𝜂𝑃𝜔2̇𝜉𝑃𝜉𝑃+̇𝜂𝑃𝜂𝑃𝜇=𝐸𝑟3𝐸𝑃̇𝜉𝑃𝜉𝑃𝜉𝐸𝜇𝑀𝑟3𝑀𝑃̇𝜉𝑃𝜉𝑃𝜉𝑀𝜇𝐸𝑟3𝐸𝑃̇𝜂𝑃𝜂𝑃𝜂𝐸𝜇𝑀𝑟3𝑀𝑃̇𝜂𝑃𝜂𝑃𝜂𝑀.(4.13) This equation can be rewritten as12𝑑̇𝜉𝑑𝑡2𝑃+̇𝜂2𝑃12𝜔2𝑑𝜉𝑑𝑡2𝑃+𝜂2𝑃=𝑑𝜇𝑑𝑡𝐸𝑟𝐸𝑃+𝜇𝑀𝑟𝑀𝑃.(4.14) Therefore,𝑣2𝜉=2Φ𝑃,𝜂𝑃𝐶,(4.15) where𝑣2=̇𝜉2𝑃+̇𝜂2𝑃,Φ𝜉𝑃,𝜂𝑃=12𝜔2𝜉2𝑃+𝜂2𝑃+𝜇𝐸𝑟𝐸𝑃+𝜇𝑀𝑟𝑀𝑃,(4.16) and 𝐶 is the so-called Jacobi constant. Therefore, from (4.15) it is seen that the Jacobi integral,𝐽𝜉𝑃,𝜂𝑃,̇𝜉𝑃,̇𝜂𝑃𝜉=2Φ𝑃,𝜂𝑃𝑣2,(4.17) 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 𝐯𝐿𝑘̇𝜉=(𝐿𝑘,̇𝜂𝐿𝑘)=(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.

tab1
Table 1: Lagrange points for the Earth-Moon system.
fig6
Figure 6: Lagrange points and Hill’s regions for the Earth-Moon system.

Observe that the right-hand side of the (4.15) must be nonnegative, since 𝑣20. Thus, an initial position (𝜉𝑃0,𝜂𝑃0) 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 relation𝜉2Φ𝑃,𝜂𝑃𝐶.(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 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 Section 5.

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 altitudes LMO of a clockwise or counterclockwise circular LMO and a specified altitude LEO of a counterclockwise circular LEO, which corresponds to the altitude of the Space Station [12], are considered. The following data are used:𝐺=6.672×1020km3kg1s2𝑀(universalconstantofgravitation),𝐸=5.9742×1024𝑀kg(massoftheEarth),𝑀=7.3483×1022𝑅kg(massoftheMoon),𝐸𝑀𝑅=384400km(meandistancefromtheEarthtotheMoon),𝐸𝑅=6378km(Earthradius),𝑀=1738km(Moonradius),LEO=463km(altitudeofcircularLEO),LMO=100,200,300km(altitudeofcircularLMO).(5.1)

5.1. 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 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).

tab2
Table 2: Lunar mission, counterclockwise LMO arrival, major parameters.
tab3
Table 3: Lunar mission, clockwise LMO arrival, major parameters.

Results in Tables 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 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 time 𝑇 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 1010 and absolute error tolerance of 1011, as described in Stoer and Bulirsch [16] and Forsythe et al. [18]. The terminal constraints are satisfied with an error lesser than 108. In all simulations the following canonical units are used: 1 distance unit = 𝑅𝐸 and 1 time unit =𝑅3𝐸/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 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 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 velocity Δ𝑣1 is nearly independent of the LMO altitude;(5)the second change velocity Δ𝑣2 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 angle 𝜃𝐸𝑃(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 LMO=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.

fig7
Figure 7: Direct ascent, counterclockwise LMO arrival, Δ𝑣1=3.0658 km/s, Δ𝑣2=0.8119 km/s, 𝑇=4.573 days, 𝜃𝐸𝑃(0)=116.41 deg.
fig8
Figure 8: Direct ascent, clockwise LMO arrival, Δ𝑣1=3.0686 km/s, Δ𝑣2=0.8143 km/s, 𝑇=4.763 days, 𝜃𝐸𝑃(0)=113.795 deg.
5.2. Multiple Revolution Ascent Maneuvers

Tables 4, 5, and 6 show the results for lunar missions with clockwise and counterclockwise arrival at LMO, for LMO=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.

tab4
Table 4: Lunar mission, major parameters, LEO=463km, LMO=100km.
tab5
Table 5: Lunar mission, major parameters, LEO=463km, LMO=200km.
tab6
Table 6: Lunar mission, major parameters, LEO=463km, LMO=300km.
tab7
Table 7: Lunar mission, major parameters, LEO=463km, LMO=100km.
tab8
Table 8: Lunar mission, major parameters, LEO=463km, LMO=200km.
tab9
Table 9: Lunar mission, major parameters, LEO=463km, LMO=300km.

For L𝑀𝑂=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.

fig9
Figure 9: 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.

For 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 than 𝐶4. 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. Figures 10 and 15 depict the collision between the spacecraft and the Earth for lunar missions with LMO=100km and counterclockwise or clockwise arrival to Moon, respectively. The remaining trajectories are feasible. (4)According to the results in Tables 49, 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 49, 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 with LMO=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 Tables 49, 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 velocity Δ𝑣1 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 velocity Δ𝑣1 changes slightly (the differences in Δ𝑣1 are approximately 15.0 to 20.0 m/s).

fig10
Figure 10: 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.
fig11
Figure 11: Multiple revolution ascent, counterclockwise LMO arrival, Δ𝑣1=3.0593 km/s, Δ𝑣2=0.7835 km/s, 𝑇=31.91 days, 𝜃𝐸𝑃(0)=232.464 deg.
fig12
Figure 12: Multiple revolution ascent, counterclockwise LMO arrival, Δ𝑣1=3.0584 km/s, Δ𝑣2=0.7795 km/s, 𝑇=40.742 days, 𝜃𝐸𝑃(0)=348.882 deg.
fig13
Figure 13: Multiple revolution ascent, counterclockwise LMO arrival, Δ𝑣1=3.057 km/s, Δ𝑣2=0.7730 km/s, 𝑇=58.415 days, 𝜃𝐸𝑃(0)=229.239 deg.
fig14
Figure 14: Multiple revolution ascent, clockwise LMO arrival, Δ𝑣1=3.0678 km/s, Δ𝑣2=0.8107 km/s, 𝑇=14.97 days, 𝜃𝐸𝑃(0)=21.069 deg.
fig15
Figure 15: Multiple revolution ascent, clockwise LMO arrival, Δ𝑣1=3.0684 km/s, Δ𝑣2=0.8136 km/s, 𝑇=25.103 days, 𝜃𝐸𝑃(0)=154.315 deg.
fig16
Figure 16: Multiple revolution ascent, clockwise LMO arrival, Δ𝑣1=3.0595 km/s, Δ𝑣2=0.7742 km/s, 𝑇=31.872 days, 𝜃𝐸𝑃(0)=232.674 deg.
fig17
Figure 17: Multiple revolution ascent, clockwise LMO arrival, Δ𝑣1=3.0586 km/s, Δ𝑣2=0.7702 km/s, 𝑇=40.713 days, 𝜃𝐸𝑃(0)=349.322 deg.
fig18
Figure 18: Multiple revolution ascent, clockwise LMO arrival, Δ𝑣1=3.0513 km/s, Δ𝑣2=0.738 km/s, 𝑇=58.42 days, 𝜃𝐸𝑃(0)=224.48 deg.

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.

Acknowledgments

This research was supported by CNPq under contract 302949/2009-7 and by FAPESP under contract 2008/56241-8.

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