# Design and Analysis of Direct Abort Orbits in the Earth-Moon Transfer Phase of Crewed Lunar Exploration Missions

Authors: Lin Lu; Haiyang Li; Wanmeng Zhou; Xinfeng Wu; Huiru Cui
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

A direct abort orbit design method is presented for direct abort missions in the Earth-Moon transfer phase of crewed lunar exploration missions. First, according to the demand of an emergency rescue in the Earth-Moon transfer phase, two direct abort orbit schemes are introduced. Then, a serial orbit design method is proposed for a high-fidelity direct abort orbit. An analytical model is established for the calculation of initial values, and the optimization design is performed in the high-fidelity orbit model to determine a single-impulse abort orbit. A hybrid optimization design process is proposed to generate a two-impulse abort orbit. The results of simulation examples verify the validity and feasibility of the proposed direct abort orbit design method. Finally, extensive simulations are carried out to analyze the characteristics of abort impulse and abort return time and reveal the general rules of direct abort orbits. The research conclusions can provide a reference for the design of emergency rescue schemes in future crewed lunar exploration missions.


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## Body

## 1. Introduction

The Moon is the first choice for humans to explore the mysteries of space, and it is also the most suitable transition station to enter deep space for expanding the living space of humans. The lunar exploration areas of the early American Apollo missions were mainly concentrated near the equator. However, due to the special location environment of the Moon’s polar regions and the discovery of water ice [1, 2], humans are paying increasing attention to the middle and high latitudes of the Moon and regard them as hot spot areas for the next exploration, even considering the establishment of a lunar base in the future [3, 4]. At the beginning of the 21st century, the “Constellation Program” announced by the United States clearly stated the mission goals of “full global access” and “return anytime”. In recent years, the “Artemis Project” to return to the Moon has been proposed [5], which has set off a worldwide upsurge of crewed lunar exploration [6–8]. With the successful implementation of the Chang′E-5 mission [9, 10], an increasing number of Chinese people also expect China to carry people to the Moon as soon as possible. Crewed lunar exploration is a complex and systematic engineering practice that demands high-technology and has high-risk. Ensuring the safety of astronauts is the primary factor considered in engineering design. In the process of mission implementation, if a crewed spacecraft fails and the safety of the astronauts is threatened, the astronauts can abort the mission at any time through a predesigned emergency rescue strategy and return to the Earth safely. Orbit design is an important part of the top-level design of space missions. Therefore, an emergency return orbit scheme is also a vital part of the emergency rescue strategy. The design and analysis of the emergency return orbit are of great significance to the design and implementation of the entire mission.In the Earth-Moon transfer phase of crewed lunar exploration missions, a free return orbit with high safety can be generally adopted, and a crewed spacecraft can safely return to the Earth along the free return orbit without perilune injection [11]. At present, many scholars have studied the free return orbit. Zhang et al. [12] proposed a fast prediction algorithm based on linear approximation for the design of free return orbit, which can effectively balance calculation accuracy and efficiency. He et al. [13] investigated an adaptive LEO-phase free-return orbit design method. Bao et al. [14] designed a two-segment free return orbit using pseudostate theory. Yim and Baoyin [15] designed free return orbits launching from lower-latitude launch sites and landing at high-latitude landing sites in a high-fidelity orbit model. Li et al. [16] established an analytical model of multisegment free return orbits based on the patched-conic method. However, the accessible lunar surface region of a free return orbit is very limited, confined to near the Moon’s equator and unable to support the exploration of lunar high-latitude regions [17, 18]. A hybrid orbit is composed of a free return orbit segment and a nonfree return orbit segment. Its accessible lunar surface region is not restricted, so it has been widely used [19, 20]. There are also some studies on the design and analysis of hybrid orbits. Bass [21] described the design and optimization of a hybrid orbit considering the constraints of mission abort but did not give a specific calculation model. Bai et al. [22] used a differential correction algorithm to describe the design process of a hybrid orbit. Based on the double two-body hypothesis, Huang et al. [23] analyzed the characteristics of a hybrid orbit, mainly including the characteristics of energy consumption and flight time. In addition, Peng [24] investigated the reachable domain characteristic of a hybrid orbit. However, a hybrid orbit has autonomous return ability for the free return orbit segment, and the other segment does not have the ability of autonomous return. Therefore, to ensure the safety of astronauts, during the engineering demonstration stage, the issue of abort return must be considered, and an abort orbit scheme must be designed in advance.Scholars have performed much analysis and design work related to the abort issues of crewed lunar exploration missions. Hyle et al. [25] summarized the abort planning of crewed spacecraft during each flight phase of the Apollo missions. Baker [26] explored the abort issues of the Earth-Moon transfer phase and circumlunar phase. Kelley and Adornato [27] presented a method to select and determine a way station of an abort orbit in the circumlunar phase. Merrick and Callas [28] studied the abort orbit with the shortest time in the Earth-Moon transfer phase. These investigations play an important role in promoting early crewed lunar exploration missions, but they only describe some conclusions and do not give specific models and algorithms. Xi et al. [29] reviewed the issues related to the abort orbit in crewed lunar exploration missions. Based on the two-body problem model, Bond [30] estimated the velocity increment required for an abort during the Earth-Moon transfer phase. Huang et al. [31, 32] analyzed the characteristics of near Earth and near Moon abort orbits for crewed lunar exploration missions. Ocampo and Saudemont [33] established an initial calculation model of a multi-impulse trans-Earth abort orbit. Although these studies put forward some specific calculation models of an abort orbit, they all perform the design in a low-fidelity orbit model, and the calculation accuracy needs to be improved.On the basis of the above research results, this paper mainly studies a direct abort orbit during the Earth-Moon transfer phase of crewed lunar exploration missions. The main contributions are as follows: On the one hand, a serial design method is proposed for high-fidelity direct abort orbits, which provides an effective tool for accurately determining two kinds of direct abort orbits. An analytical model of initial values calculation is established for a single-impulse abort orbit, while a hybrid optimization design process is proposed for a two-impulse abort orbit. On the other hand, the general rules of direct abort orbits are revealed, which provides a reference for engineering practice. By adopting the proposed design method of direct abort orbits, the characteristics of direct abort orbits are discussed, including the abort impulse characteristic and abort return time characteristic.The structure of this paper is as follows. Section2 describes the issue of a direct abort orbit in the Earth-Moon transfer phase of crewed lunar exploration missions. Section 3 proposes a design method for direct abort orbits and obtains high-fidelity solutions of two kinds of direct abort orbits. Section 4 verifies the validity and feasibility of the orbit design method using numerical simulations and analyzes the characteristics of the direct abort orbits. Some conclusions of this study are summarized in Section 5.


## 2. Statement of the Problem

To achieve the crewed lunar exploration mission goal of “full global access” given by the new era and effectively support scientific research in lunar middle- and high-latitude regions, this paper adopts a hybrid orbit which is adopted during the Earth-Moon transfer phase of crewed lunar exploration missions. It is necessary to consider the capability of safely returning to the Earth if a failure occurs at any time during flight to ensure the safety of the astronauts.In this way, once a crewed spacecraft fails in the free return segment of a hybrid orbit, if the situation is not urgent, it can continue to fly along the free return orbit and finally return to the Earth safely. If the situation is quite urgent, the mission must be aborted directly and it can return to the Earth along the direct abort orbit. Once the crewed spacecraft fails in the nonfree return segment of the hybrid orbit before reaching the Moon, if the situation is not urgent, it can enter the free return orbit again through a maneuver and return to the Earth safely after circling the Moon. If the situation is urgent, the mission must be aborted directly and it can return to the Earth along the direct abort orbit. This paper studies a direct abort orbit in the Earth-Moon transfer phase and mainly focuses on the design and analysis of a direct abort orbit in the nonfree return segment of a hybrid orbit.There are two main ways of direct abort in the Earth-Moon transfer phase, which are single-impulse abort and two-impulse abort, as shown in Figure1. The process of a single-impulse abort is described as follows. After a crewed spacecraft fails, an abort impulse is applied at the abort point and the crewed spacecraft enters the single-impulse abort orbit. It reaches the reentry point after serval hours and lands on the Earth’s surface finally. The process of a two-impulse abort is depicted as follows. After a crewed spacecraft fails, a first abort impulse is applied at the abort point to enable the crewed spacecraft to enter the first segment of the two-impulse abort orbit. After serval hours, the crewed spacecraft reaches the maneuver point and applies a second abort impulse. Then, it reaches the reentry point along the second segment of the two-impulse abort orbit and lands on the Earth’s surface finally.Figure 1
Diagram of direct abort orbits in the Earth-Moon transfer phase.


## 3. Direct Abort Orbit Design Method

Aiming at a direct abort orbit in the Earth-Moon transfer phase of crewed lunar exploration missions, this section proposes an orbit design method from preliminary calculation to accurate design, which is used to design two kinds of direct abort orbits. For a single-impulse abort orbit, an analytical model is established for calculating the initial value, and the accurate design is performed in the high-fidelity orbit model. For a two-impulse abort orbit, a hybrid optimization design process is introduced to generate an accurate two-impulse abort orbit.


### 3.1. Preliminary Calculation

In the preliminary calculation stage, based on the two-body orbit assumption, preliminary calculation models are established for two kinds of direct abort orbits.


#### 3.1.1. Preliminary Calculation Model of a Single-Impulse Abort Orbit

If the position vectorR0 and the velocity vector V0 of the abort point are known, the flight-path angle before applying the abort impulse is calculated as
(1)ΘI=arccoshR0⋅V0,where h is the magnitude of the orbital angular momentum.The flight-path angle after applying the abort impulse is defined as(2)ΘF=KΘ·ΘI,where KΘ is the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse.When the distance of the reentry point with respect to the EarthRReen and the flight-path angle of the reentry point ΘReen are given, according to the law of the conservation of angular momentum and the law of the conservation of energy, the following equation can be obtained:
(3)VReen2=VF2+2μe1RReen−1RF,RFVFcosΘF=RReenVReencosΘReen,where VReen is the velocity magnitude of the reentry point, VF is the velocity magnitude after applying the abort impulse, μe is the gravitational parameter of the Earth, and RF is the distance of the abort point with respect to the Earth after applying the abort impulse.By rearranging Equation (3), the following equation is obtained:
(4)VF2=2μe1/RReen−1/RFRF2cos2ΘF/RReen2cos2ΘReen‐1.Assuming that the abort impulse does not change the orbital plane, in the local vertical local horizon (LVLH) coordinate system, the magnitude of the components of the abort impulse in each direction is given by(5)ΔVLVLHx=VFsinΘF−V0sinΘI,ΔVLVLHy=VFcosΘF−V0cosΘI,ΔVLVLHz=0.Through coordinate system transformation, the abort impulse vector in the geocentric J2000 coordinate system is expressed as(6)ΔVEJ2=MLVLHEJ2ΔVLVLH,where ΔVLVLH is the abort impulse vector in the LVLH coordinate system and MLVLHEJ2 is the transformation matrix from the LVLH coordinate system to the geocentric J2000 coordinate system at the abort point, which is described as
(7)MLVLHEJ2=M3−Ω0M1−i0M3−ω0−θ0,where Ω0, i0, ω0, and θ0 are the right ascension of the ascending node, orbital inclination, argument of perigee, and true anomaly of the hybrid orbit at the abort point, respectively. M3 and M1 are the rotation matrices that rotate a certain angle around the Z-axis and the X-axis, respectively.Thus, the abort impulse vector can be computed, and then the orbital elements of the single-impulse abort orbit can be obtained. The true anomaly of the reentry point is expressed as(8)θReen=2π−arccos1eSAOrpSAORReen1+eSAO−1,rpSAO≤RReen,2π,rpSAO>RReen,where eSAO is the eccentricity of the single-impulse abort orbit and rpSAO is the perigee radius of the single-impulse abort orbit.The true anomaly of the abort point and the reentry point are determined, so the abort return time after the single-impulse abort is calculated by(9)Tre=aSAO3μeEReen−eSAOsinEReen−EAP−eSAOsinEAP,where EAP and EReen are the eccentric anomalies of the abort point and the reentry point, respectively, which can be obtained by the corresponding true anomaly.In summary, when the position vectorR0 and the velocity vector V0 of the abort point are known, once KΘ, the distance of the reentry point with respect to the Earth RReen and the flight-path angle of the reentry point ΘReen are given, the abort impulse and abort return time required for the single-impulse abort are determined, and then the single-impulse abort orbit can be preliminarily generated.


#### 3.1.2. Preliminary Calculation Model of a Two-Impulse Abort Orbit

When the fuel of a crewed spacecraft is limited, in some cases, it is impossible to perform a single-impulse abort. In the case that a single-impulse abort cannot be carried out, a two-impulse abort can be adopted to optimize the total abort impulse and realize an abort return.From the preliminary calculation model of a single-impulse abort orbit, it is known that if the position vectorR0 and the velocity vector V0 of the abort point, the distance with respect to the Earth RReen, and the flight-path angle ΘReen of the reentry point are given and the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse is denoted by KΘ1, the first abort impulse can be obtained.When the time interval between the two abort impulses is given, the eccentric anomalyEAP2 at the maneuver point can be calculated iteratively using the following equation:
(10)Tre1=aTAO13μeEAP2−eTAO1EAP2−EAP1−eTAO1EAP1,where eTAO1 is the eccentricity of the first segment of the two-impulse abort orbit after applying the first abort impulse and EAP1 is the eccentric anomaly of the abort point after applying the first abort impulse.The true anomaly of the maneuver point is computed from the eccentric anomaly of this point, and then, the orbit elements at this point can be obtained. The position vector and velocity vector before applying the second abort impulse can be acquired through transformation. Assuming that the ratio of the flight-path angle after applying the second abort impulse to the flight-path angle before applying the second abort impulse isKΘ2, the second abort impulse and flight time of the second segment of the two-impulse abort orbit are calculated by referring to the preliminary calculation model of the single-impulse abort orbit.Thus, the two-impulse abort orbit design problem is transformed into a constrained nonlinear programming problem. The design variables includeKΘ1, Tre1, and KΘ2. Once these design variables have been determined, a two-impulse abort orbit can be generated, which can be expressed as
(11)YTAO=fKΘ1,Tre1,KΘ2.The constraint conditions consider abort return time and the reentry corridor constraints, that is,(12)Tre≤Tmax,RReen=Rreq,ΘReen=Θreq,where Tmax is the maximum abort return time allowed, Rreq is the required distance of the reentry point with respect to the Earth, and Θreq is the required flight-path angle of the reentry point.In actual abort return missions, it is generally expected that the energy consumption is minimal in the two-impulse abort return process. In the preliminary calculation, the optimization objective function is set to be the minimum total abort impulse, that is,(13)JTAO0=ΔVmin,where ΔV is the magnitude of the total abort impulse.To solve this nonlinear programming problem, this study adopts the improved differential evolution algorithm DE_CMSBHS (Differential Evolution Algorithm with Combined Mutation Strategies and Boundary Handing Scheme). Based on the classical differential evolution algorithm framework, DE_CMSBHS introduces combined mutation strategies and combined boundary-handing schemes and adopts an adaptive parameter control method, which has better global search ability and optimization stability [34].


### 3.2. Accurate Design

During the accurate design stage, based on a high-fidelity orbit model, with the result of the preliminary calculation as the initial value, an accurate direct abort orbit is designed.In the geocentric J2000 coordinate system, considering the relevant perturbation factors, the high-fidelity orbit model is described as(14)d2Rdt2=−μeRR3+AN+ANSE+ANSM+AR+AD+APwhere R is the position vector of the spacecraft with respect to the geocentric J2000 coordinate system; AN is the gravitational perturbation of the N-body, where only the solar and lunar perturbations are considered and the relative positions between the stars are obtained from the JPL DE405 ephemeris; ANSE is the Earth nonspherical perturbation, here 6 degrees by 6 orders is adopted using the WGS84 model; ANSM is the Moon nonspherical perturbation, in which here 6 degrees by 6 orders is adopted using the LP165P model; AR is the solar radiation pressure perturbation that is related to the spacecraft area to mass ratio and the reflection factor; AD is the atmospheric drag perturbation; and AP is the thrust acceleration. The perturbations of Jupiter, Venus, and other large planets, the Earth tides, and the relativistic effect are ignored. The RKF78 integrator with variable pitch is selected as the integrator to perform the numerical integration.


#### 3.2.1. Accurate Design Algorithm of a Single-Impulse Abort Orbit

In the accurate design of a single-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The design variables are the abort impulse vectorΔVEJ2 and the abort return time Tre. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. To improve the convergence speed of the high-fidelity solution, the equality constraints are set as the objective function [35]:
(15)JSAO=σ1RReen′−Rreq+σ2ΘReen′−Θreqwhere σ1 and σ2 are weight factors and RReen′ and ΘReen′ are variation values of the distance with respect to the Earth and the flight-path angle of the reentry point in the iteration process, respectively.The SQP_Snopt algorithm is adopted to optimize the single-impulse abort orbit. The SQP_Snopt algorithm is an advanced SQP algorithm [36], in which the finite quasi-Newton method is set to approximate the Hessian matrix of the Lagrange multiplier, thereby reducing the calculation of the Hessian matrix greatly and effectively improving the calculation speed. Hence, it is usually used to solve smooth nonlinear programming problems.The single-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure2.Figure 2
Flow chart of the single-impulse abort orbit design method.


#### 3.2.2. Accurate Design Algorithm of a Two-Impulse Abort Orbit

In the accurate design of a two-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The following variables are selected as the design variables: the first abort impulse vectorΔV1EJ2, the time interval between two abort impulses Tre1, the second abort impulse vector ΔV2EJ2, and the time interval between the maneuver point and the reentry point Tre2. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. The objective function is set to be the same as Equation (15).The two-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure3.Figure 3
Flow chart of the two-impulse abort orbit design method.


## 3.1. Preliminary Calculation

In the preliminary calculation stage, based on the two-body orbit assumption, preliminary calculation models are established for two kinds of direct abort orbits.


### 3.1.1. Preliminary Calculation Model of a Single-Impulse Abort Orbit

If the position vectorR0 and the velocity vector V0 of the abort point are known, the flight-path angle before applying the abort impulse is calculated as
(1)ΘI=arccoshR0⋅V0,where h is the magnitude of the orbital angular momentum.The flight-path angle after applying the abort impulse is defined as(2)ΘF=KΘ·ΘI,where KΘ is the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse.When the distance of the reentry point with respect to the EarthRReen and the flight-path angle of the reentry point ΘReen are given, according to the law of the conservation of angular momentum and the law of the conservation of energy, the following equation can be obtained:
(3)VReen2=VF2+2μe1RReen−1RF,RFVFcosΘF=RReenVReencosΘReen,where VReen is the velocity magnitude of the reentry point, VF is the velocity magnitude after applying the abort impulse, μe is the gravitational parameter of the Earth, and RF is the distance of the abort point with respect to the Earth after applying the abort impulse.By rearranging Equation (3), the following equation is obtained:
(4)VF2=2μe1/RReen−1/RFRF2cos2ΘF/RReen2cos2ΘReen‐1.Assuming that the abort impulse does not change the orbital plane, in the local vertical local horizon (LVLH) coordinate system, the magnitude of the components of the abort impulse in each direction is given by(5)ΔVLVLHx=VFsinΘF−V0sinΘI,ΔVLVLHy=VFcosΘF−V0cosΘI,ΔVLVLHz=0.Through coordinate system transformation, the abort impulse vector in the geocentric J2000 coordinate system is expressed as(6)ΔVEJ2=MLVLHEJ2ΔVLVLH,where ΔVLVLH is the abort impulse vector in the LVLH coordinate system and MLVLHEJ2 is the transformation matrix from the LVLH coordinate system to the geocentric J2000 coordinate system at the abort point, which is described as
(7)MLVLHEJ2=M3−Ω0M1−i0M3−ω0−θ0,where Ω0, i0, ω0, and θ0 are the right ascension of the ascending node, orbital inclination, argument of perigee, and true anomaly of the hybrid orbit at the abort point, respectively. M3 and M1 are the rotation matrices that rotate a certain angle around the Z-axis and the X-axis, respectively.Thus, the abort impulse vector can be computed, and then the orbital elements of the single-impulse abort orbit can be obtained. The true anomaly of the reentry point is expressed as(8)θReen=2π−arccos1eSAOrpSAORReen1+eSAO−1,rpSAO≤RReen,2π,rpSAO>RReen,where eSAO is the eccentricity of the single-impulse abort orbit and rpSAO is the perigee radius of the single-impulse abort orbit.The true anomaly of the abort point and the reentry point are determined, so the abort return time after the single-impulse abort is calculated by(9)Tre=aSAO3μeEReen−eSAOsinEReen−EAP−eSAOsinEAP,where EAP and EReen are the eccentric anomalies of the abort point and the reentry point, respectively, which can be obtained by the corresponding true anomaly.In summary, when the position vectorR0 and the velocity vector V0 of the abort point are known, once KΘ, the distance of the reentry point with respect to the Earth RReen and the flight-path angle of the reentry point ΘReen are given, the abort impulse and abort return time required for the single-impulse abort are determined, and then the single-impulse abort orbit can be preliminarily generated.


### 3.1.2. Preliminary Calculation Model of a Two-Impulse Abort Orbit

When the fuel of a crewed spacecraft is limited, in some cases, it is impossible to perform a single-impulse abort. In the case that a single-impulse abort cannot be carried out, a two-impulse abort can be adopted to optimize the total abort impulse and realize an abort return.From the preliminary calculation model of a single-impulse abort orbit, it is known that if the position vectorR0 and the velocity vector V0 of the abort point, the distance with respect to the Earth RReen, and the flight-path angle ΘReen of the reentry point are given and the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse is denoted by KΘ1, the first abort impulse can be obtained.When the time interval between the two abort impulses is given, the eccentric anomalyEAP2 at the maneuver point can be calculated iteratively using the following equation:
(10)Tre1=aTAO13μeEAP2−eTAO1EAP2−EAP1−eTAO1EAP1,where eTAO1 is the eccentricity of the first segment of the two-impulse abort orbit after applying the first abort impulse and EAP1 is the eccentric anomaly of the abort point after applying the first abort impulse.The true anomaly of the maneuver point is computed from the eccentric anomaly of this point, and then, the orbit elements at this point can be obtained. The position vector and velocity vector before applying the second abort impulse can be acquired through transformation. Assuming that the ratio of the flight-path angle after applying the second abort impulse to the flight-path angle before applying the second abort impulse isKΘ2, the second abort impulse and flight time of the second segment of the two-impulse abort orbit are calculated by referring to the preliminary calculation model of the single-impulse abort orbit.Thus, the two-impulse abort orbit design problem is transformed into a constrained nonlinear programming problem. The design variables includeKΘ1, Tre1, and KΘ2. Once these design variables have been determined, a two-impulse abort orbit can be generated, which can be expressed as
(11)YTAO=fKΘ1,Tre1,KΘ2.The constraint conditions consider abort return time and the reentry corridor constraints, that is,(12)Tre≤Tmax,RReen=Rreq,ΘReen=Θreq,where Tmax is the maximum abort return time allowed, Rreq is the required distance of the reentry point with respect to the Earth, and Θreq is the required flight-path angle of the reentry point.In actual abort return missions, it is generally expected that the energy consumption is minimal in the two-impulse abort return process. In the preliminary calculation, the optimization objective function is set to be the minimum total abort impulse, that is,(13)JTAO0=ΔVmin,where ΔV is the magnitude of the total abort impulse.To solve this nonlinear programming problem, this study adopts the improved differential evolution algorithm DE_CMSBHS (Differential Evolution Algorithm with Combined Mutation Strategies and Boundary Handing Scheme). Based on the classical differential evolution algorithm framework, DE_CMSBHS introduces combined mutation strategies and combined boundary-handing schemes and adopts an adaptive parameter control method, which has better global search ability and optimization stability [34].


## 3.1.1. Preliminary Calculation Model of a Single-Impulse Abort Orbit

If the position vectorR0 and the velocity vector V0 of the abort point are known, the flight-path angle before applying the abort impulse is calculated as
(1)ΘI=arccoshR0⋅V0,where h is the magnitude of the orbital angular momentum.The flight-path angle after applying the abort impulse is defined as(2)ΘF=KΘ·ΘI,where KΘ is the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse.When the distance of the reentry point with respect to the EarthRReen and the flight-path angle of the reentry point ΘReen are given, according to the law of the conservation of angular momentum and the law of the conservation of energy, the following equation can be obtained:
(3)VReen2=VF2+2μe1RReen−1RF,RFVFcosΘF=RReenVReencosΘReen,where VReen is the velocity magnitude of the reentry point, VF is the velocity magnitude after applying the abort impulse, μe is the gravitational parameter of the Earth, and RF is the distance of the abort point with respect to the Earth after applying the abort impulse.By rearranging Equation (3), the following equation is obtained:
(4)VF2=2μe1/RReen−1/RFRF2cos2ΘF/RReen2cos2ΘReen‐1.Assuming that the abort impulse does not change the orbital plane, in the local vertical local horizon (LVLH) coordinate system, the magnitude of the components of the abort impulse in each direction is given by(5)ΔVLVLHx=VFsinΘF−V0sinΘI,ΔVLVLHy=VFcosΘF−V0cosΘI,ΔVLVLHz=0.Through coordinate system transformation, the abort impulse vector in the geocentric J2000 coordinate system is expressed as(6)ΔVEJ2=MLVLHEJ2ΔVLVLH,where ΔVLVLH is the abort impulse vector in the LVLH coordinate system and MLVLHEJ2 is the transformation matrix from the LVLH coordinate system to the geocentric J2000 coordinate system at the abort point, which is described as
(7)MLVLHEJ2=M3−Ω0M1−i0M3−ω0−θ0,where Ω0, i0, ω0, and θ0 are the right ascension of the ascending node, orbital inclination, argument of perigee, and true anomaly of the hybrid orbit at the abort point, respectively. M3 and M1 are the rotation matrices that rotate a certain angle around the Z-axis and the X-axis, respectively.Thus, the abort impulse vector can be computed, and then the orbital elements of the single-impulse abort orbit can be obtained. The true anomaly of the reentry point is expressed as(8)θReen=2π−arccos1eSAOrpSAORReen1+eSAO−1,rpSAO≤RReen,2π,rpSAO>RReen,where eSAO is the eccentricity of the single-impulse abort orbit and rpSAO is the perigee radius of the single-impulse abort orbit.The true anomaly of the abort point and the reentry point are determined, so the abort return time after the single-impulse abort is calculated by(9)Tre=aSAO3μeEReen−eSAOsinEReen−EAP−eSAOsinEAP,where EAP and EReen are the eccentric anomalies of the abort point and the reentry point, respectively, which can be obtained by the corresponding true anomaly.In summary, when the position vectorR0 and the velocity vector V0 of the abort point are known, once KΘ, the distance of the reentry point with respect to the Earth RReen and the flight-path angle of the reentry point ΘReen are given, the abort impulse and abort return time required for the single-impulse abort are determined, and then the single-impulse abort orbit can be preliminarily generated.


## 3.1.2. Preliminary Calculation Model of a Two-Impulse Abort Orbit

When the fuel of a crewed spacecraft is limited, in some cases, it is impossible to perform a single-impulse abort. In the case that a single-impulse abort cannot be carried out, a two-impulse abort can be adopted to optimize the total abort impulse and realize an abort return.From the preliminary calculation model of a single-impulse abort orbit, it is known that if the position vectorR0 and the velocity vector V0 of the abort point, the distance with respect to the Earth RReen, and the flight-path angle ΘReen of the reentry point are given and the ratio of the flight-path angle after applying the abort impulse to the flight-path angle before applying the abort impulse is denoted by KΘ1, the first abort impulse can be obtained.When the time interval between the two abort impulses is given, the eccentric anomalyEAP2 at the maneuver point can be calculated iteratively using the following equation:
(10)Tre1=aTAO13μeEAP2−eTAO1EAP2−EAP1−eTAO1EAP1,where eTAO1 is the eccentricity of the first segment of the two-impulse abort orbit after applying the first abort impulse and EAP1 is the eccentric anomaly of the abort point after applying the first abort impulse.The true anomaly of the maneuver point is computed from the eccentric anomaly of this point, and then, the orbit elements at this point can be obtained. The position vector and velocity vector before applying the second abort impulse can be acquired through transformation. Assuming that the ratio of the flight-path angle after applying the second abort impulse to the flight-path angle before applying the second abort impulse isKΘ2, the second abort impulse and flight time of the second segment of the two-impulse abort orbit are calculated by referring to the preliminary calculation model of the single-impulse abort orbit.Thus, the two-impulse abort orbit design problem is transformed into a constrained nonlinear programming problem. The design variables includeKΘ1, Tre1, and KΘ2. Once these design variables have been determined, a two-impulse abort orbit can be generated, which can be expressed as
(11)YTAO=fKΘ1,Tre1,KΘ2.The constraint conditions consider abort return time and the reentry corridor constraints, that is,(12)Tre≤Tmax,RReen=Rreq,ΘReen=Θreq,where Tmax is the maximum abort return time allowed, Rreq is the required distance of the reentry point with respect to the Earth, and Θreq is the required flight-path angle of the reentry point.In actual abort return missions, it is generally expected that the energy consumption is minimal in the two-impulse abort return process. In the preliminary calculation, the optimization objective function is set to be the minimum total abort impulse, that is,(13)JTAO0=ΔVmin,where ΔV is the magnitude of the total abort impulse.To solve this nonlinear programming problem, this study adopts the improved differential evolution algorithm DE_CMSBHS (Differential Evolution Algorithm with Combined Mutation Strategies and Boundary Handing Scheme). Based on the classical differential evolution algorithm framework, DE_CMSBHS introduces combined mutation strategies and combined boundary-handing schemes and adopts an adaptive parameter control method, which has better global search ability and optimization stability [34].


## 3.2. Accurate Design

During the accurate design stage, based on a high-fidelity orbit model, with the result of the preliminary calculation as the initial value, an accurate direct abort orbit is designed.In the geocentric J2000 coordinate system, considering the relevant perturbation factors, the high-fidelity orbit model is described as(14)d2Rdt2=−μeRR3+AN+ANSE+ANSM+AR+AD+APwhere R is the position vector of the spacecraft with respect to the geocentric J2000 coordinate system; AN is the gravitational perturbation of the N-body, where only the solar and lunar perturbations are considered and the relative positions between the stars are obtained from the JPL DE405 ephemeris; ANSE is the Earth nonspherical perturbation, here 6 degrees by 6 orders is adopted using the WGS84 model; ANSM is the Moon nonspherical perturbation, in which here 6 degrees by 6 orders is adopted using the LP165P model; AR is the solar radiation pressure perturbation that is related to the spacecraft area to mass ratio and the reflection factor; AD is the atmospheric drag perturbation; and AP is the thrust acceleration. The perturbations of Jupiter, Venus, and other large planets, the Earth tides, and the relativistic effect are ignored. The RKF78 integrator with variable pitch is selected as the integrator to perform the numerical integration.


### 3.2.1. Accurate Design Algorithm of a Single-Impulse Abort Orbit

In the accurate design of a single-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The design variables are the abort impulse vectorΔVEJ2 and the abort return time Tre. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. To improve the convergence speed of the high-fidelity solution, the equality constraints are set as the objective function [35]:
(15)JSAO=σ1RReen′−Rreq+σ2ΘReen′−Θreqwhere σ1 and σ2 are weight factors and RReen′ and ΘReen′ are variation values of the distance with respect to the Earth and the flight-path angle of the reentry point in the iteration process, respectively.The SQP_Snopt algorithm is adopted to optimize the single-impulse abort orbit. The SQP_Snopt algorithm is an advanced SQP algorithm [36], in which the finite quasi-Newton method is set to approximate the Hessian matrix of the Lagrange multiplier, thereby reducing the calculation of the Hessian matrix greatly and effectively improving the calculation speed. Hence, it is usually used to solve smooth nonlinear programming problems.The single-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure2.Figure 2
Flow chart of the single-impulse abort orbit design method.


### 3.2.2. Accurate Design Algorithm of a Two-Impulse Abort Orbit

In the accurate design of a two-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The following variables are selected as the design variables: the first abort impulse vectorΔV1EJ2, the time interval between two abort impulses Tre1, the second abort impulse vector ΔV2EJ2, and the time interval between the maneuver point and the reentry point Tre2. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. The objective function is set to be the same as Equation (15).The two-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure3.Figure 3
Flow chart of the two-impulse abort orbit design method.


## 3.2.1. Accurate Design Algorithm of a Single-Impulse Abort Orbit

In the accurate design of a single-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The design variables are the abort impulse vectorΔVEJ2 and the abort return time Tre. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. To improve the convergence speed of the high-fidelity solution, the equality constraints are set as the objective function [35]:
(15)JSAO=σ1RReen′−Rreq+σ2ΘReen′−Θreqwhere σ1 and σ2 are weight factors and RReen′ and ΘReen′ are variation values of the distance with respect to the Earth and the flight-path angle of the reentry point in the iteration process, respectively.The SQP_Snopt algorithm is adopted to optimize the single-impulse abort orbit. The SQP_Snopt algorithm is an advanced SQP algorithm [36], in which the finite quasi-Newton method is set to approximate the Hessian matrix of the Lagrange multiplier, thereby reducing the calculation of the Hessian matrix greatly and effectively improving the calculation speed. Hence, it is usually used to solve smooth nonlinear programming problems.The single-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure2.Figure 2
Flow chart of the single-impulse abort orbit design method.


## 3.2.2. Accurate Design Algorithm of a Two-Impulse Abort Orbit

In the accurate design of a two-impulse abort orbit, with the result of the preliminary calculation as the initial value, forward numerical integration is performed in the high-fidelity orbit model. The following variables are selected as the design variables: the first abort impulse vectorΔV1EJ2, the time interval between two abort impulses Tre1, the second abort impulse vector ΔV2EJ2, and the time interval between the maneuver point and the reentry point Tre2. After forward extrapolation in the high-fidelity orbit model, the state parameters of the reentry point can be obtained. The constraint condition is the constraint of the reentry corridor, that is, the constraints of the distance with respect to the Earth and the flight-path angle of the reentry point. The objective function is set to be the same as Equation (15).The two-impulse abort orbit design method from the preliminary calculation to the accurate design is shown in Figure3.Figure 3
Flow chart of the two-impulse abort orbit design method.


## 4. Simulation Verification

This section verifies the feasibility of the direct abort orbit design method proposed in this paper using numerical simulations and analyzes the characteristics of the direct abort orbit.


### 4.1. Simulation Example

First, a simulation example is used to examine the validity of the direct abort orbit design method. The simulation parameters are set as follows. A hybrid orbit is selected as the orbit of the Earth-Moon transfer phase for crewed lunar exploration missions, and the maneuver that transfers the spacecraft from the free return segment to the nonfree return segment is called the hybrid maneuver, which is performed 10 hours after departing the Earth. Table1 provides the orbit parameters of the hybrid orbit. Referring to the analysis and statistics results of the Apollo missions [37], the reentry point altitude of the direct abort orbit is set to 122 km, and the flight-path angle of the reentry point is set to ‐6∘.Table 1
Orbital parameters of the hybrid orbit.
ParameterValueDeparture time2029-05-18 11:17:07.3131Semimajor axis (km)249547.1395Eccentricity0.9738Orbital inclination (deg)27.9922Right ascension of the ascending node (deg)143.7595Argument of perigee (deg)199.3091True anomaly (deg)28.4130Hybrid maneuver time2029-05-18 21:17:07.3131Hybrid maneuver impulse vector (m/s)(16.7923, 30.3731, -5.5991)


#### 4.1.1. Single-Impulse Abort Orbit Example

In the single-impulse abort case, it is assumed that an abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. WhenKΘ is 0.4, by adopting the single-impulse abort orbit design method proposed in this paper, the single-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as shown in Table 2. When KΘ is 0.8, the results of the preliminary calculation and accurate design are obtained as shown in Table 3. In the tables, T is the time relative to the hybrid maneuver. ΔVEJ2 is the abort impulse vector in the geocentric J2000 coordinate system. ΔV is the abort impulse magnitude.Table 2
The results of the single-impulse abort orbit (whenKΘ is 0.4).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1263.92, -978.84, 19.71)1598.7535.21Accurate design4(1264.82, -980.59, 20.12)1600.5435.21Error(0.90, -1.75, 0.41)1.790.00Preliminary calculation8(1159.82, -839.36, -8.37)1431.7143.72Accurate design8(1161.75, -842.22, -7.72)1434.9443.72Error(1.93, -2.86, 0.65)3.230.00Preliminary calculation12(1073.05, -737.20, -25.48)1302.1351.83Accurate design12(1076.54, -741.80, -25.18)1307.6051.83Error(3.49, -4.60, 0.30)5.470.00Table 3
The results of the single-impulse abort orbit (whenKΘ is 0.8).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(803.70, -629.91, 15.89)1021.2655.12Accurate design4(804.02, -634.27, 17.43)1024.2355.12Error(0.32, -4.36, 1.54)2.970.00Preliminary calculation8(747.14, -547.12, -2.50)926.0566.08Accurate design8(747.83, -554.93, -1.87)931.2466.08Error(0.69, -7.81, 0.63)5.190.00Preliminary calculation12(696.21, -484.11, -13.91)848.0976.27Accurate design12(716.77, -471.00, 31.80)858.2676.27Error(20.56, 13.11, 45.71)10.170.00It can be seen from the tables that in the six cases, the results of the single-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. WhenKΘ is 0.4, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(a). When KΘ is 0.8, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, and the red orbits represent the single-impulse abort orbits of the three cases. Thus, the single-impulse abort orbit design method proposed in this paper can effectively generate a single-impulse abort orbit and the results of preliminary calculation can provide good initial values for accurate design.Figure 4
Accurate single-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.4)(b)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.8)


#### 4.1.2. Two-Impulse Abort Orbit Example

In the two-impulse abort case, it is assumed that the abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. The maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit. By adopting the two-impulse abort orbit design method proposed in this paper, the two-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as described in Table4. When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the results of the preliminary calculation and accurate design are obtained as described in Table 5. In the tables, ΔV1EJ is the first abort impulse vector in the geocentric J2000 coordinate system. ΔV1 is the magnitude of the first abort impulse. ΔV2EJ is the second abort impulse vector in the geocentric J2000 coordinate system. ΔV2 is the magnitude of the second abort impulse.Table 4
The results of the two-impulse abort orbits (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(987.67, -769.25, 17.35)1252.019.57(23.41, -14.80, -1.13)27.721279.7344.56Accurate design4(988.30, -770.57, 18.18)1253.339.57(23.55, -16.02, -0.62)28.491281.8244.56Error(0.63, -1.32, 0.83)1.320.00(0.14, -1.22, 0.51)0.772.090.00Preliminary calculation8(909.20, -661.83, -4.83)1124.5813.56(2.45, -1.43, -0.17)2.841127.4255.15Accurate design8(910.68, -664.02, -4.24)1127.0713.56(2.86, -3.37, 1.82)4.781131.8555.15Error(1.48, -2.19, 0.59)2.490.00(0.41, -1.94, 1.99)1.944.430.00Preliminary calculation12(795.42, -550.75, -16.96)967.631.93(14.88, -9.77, -0.56)17.81985.4467.00Accurate design12(799.84, -565.67, -14.91)979.771.93(12.10, -3.33, -1.56)12.64992.4167.00Error(4.32, -14.92, 2.05)12.140.00(-2.78, 6.44, -1.00)-5.176.970.00Table 5
The results of the two-impulse abort orbits (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1175.60, -911.82, 18.95)1487.891.91(31.14, -22.76, -0.12)38.571526.4637.03Accurate design4(1175.85, -912.75, 19.39)1488.671.91(31.32, -23.67, 0.23)39.251527.9237.03Error(0.25, -0.93, 0.44)0.780.00(0.18, -0.91, 0.35)0.681.460.00Preliminary calculation8(1081.84, -784.20, -7.24)1336.197.57(0.51, -0.32, -0.02)0.601336.7946.76Accurate design8(1082.84, -785.77, -5.82)1337.917.57(0.93, -1.75, 0.49)2.041339.9546.76Error(1.00, -1.57, 1.42)1.720.00(0.42, -1.43, 0.51)1.443.160.00Preliminary calculation12(987.37, -679.81, -22.79)1198.987.87(5.11, -3.13, -0.29)6.001204.9855.79Accurate design12(988.97, -682.49, -22.06)1201.817.87(6.00, -5.58, 1.14)8.281210.0955.79Error(1.60, -2.68, 0.73)2.830.00(0.89, 2.45, 1.43)2.285.110.00It can be observed from the tables that in the six cases, the results of the two-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. When the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure5(a). When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure 5(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, the pink orbits represent the first segments of two-impulse abort orbits in the three cases, and the red orbits represent the second segments of two-impulse abort orbits in the three cases. Thus, the two-impulse abort orbit design method proposed in this paper can effectively determine the two-impulse abort orbit, and the results of preliminary calculation can provide a good initial value for accurate design.Figure 5
Accurate two-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit)(b)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit)


### 4.2. Orbit Characteristic Analysis

During the engineering demonstration stage, the design of a specified direct abort orbit is not often given much attention, and much work is concentrated on grasping the relevant characteristics of this kind of orbit. Following the verification in the above subsection of the effectiveness and feasibility of the direct abort orbit design method proposed in this paper, this section adopts the method to perform extensive simulations and analyze the characteristics of the direct abort orbits.


#### 4.2.1. Single-Impulse Abort Orbit Characteristic

First, the single-impulse abort orbit characteristics are analyzed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth and the flight-path angle of the reentry point are consistent with those in Section4.1. Figure 6 illustrates the curve of the required abort impulse changing with the abort time under different flight-path angle ratios. The figure indicates that the variation tendency of the required abort impulse is the same and that the required abort impulse decreases with the delay of the abort time under different flight-path angles. In addition, compared to the flight-path angle before applying the abort impulse, the larger the variation of the flight-path angle after applying the abort impulse is, the larger the required abort impulse is.Figure 6
Abort impulse versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.Figure7 presents the variation in the abort return time corresponding to the abort time in the cases of different flight-path angle ratios. The required abort return time has the same variation tendency in the cases of different flight-path angles. The change in the required abort return time corresponding to the abort time is approximately linear and increases with the delay of the abort time. In addition, the larger the change in flight-path angle after applying the abort impulse is, the smaller the required abort return time is. When the flight-path angle is 0, the required abort return time is the minimum. In this case, the single-impulse abort is regarded as the minimum return time abort.Figure 7
Abort return time versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.


#### 4.2.2. Two-Impulse Abort Orbit Characteristic

Second, the two-impulse abort orbit characteristics are discussed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth, and the flight-path angle of the reentry point are consistent with those in Section4.1. The variation in the total abort impulse is depicted in Figure 8 corresponding to the abort time in the cases of different maximum abort return time constraints. In the figure, KTre is the ratio of the maximum abort return time constraint to the minimum return time of the single-impulse abort. The required total abort impulse has the same variation tendency and decreases with the delay of the abort time. The larger the maximum abort return time constraint is, the smaller the required total abort impulse for two-impulse abort is. Compared with the single-impulse abort case of the minimum abort return time, the required total abort impulse for the two-impulse abort is obviously reduced, which can save 350~1100 m/s.Figure 8
Total abort impulse versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.Figure9 depicts the variation in the required abort return time corresponding to the abort time in the cases of different maximum abort return time constraints. The change tendency of the required abort return time is similar under different maximum abort return time constraints. With the delay of the abort time, the required abort return time is increasing and changes approximately linearly. The larger the maximum abort return time constraint is, the longer the required abort return time is.Figure 9
Abort return time versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.


## 4.1. Simulation Example

First, a simulation example is used to examine the validity of the direct abort orbit design method. The simulation parameters are set as follows. A hybrid orbit is selected as the orbit of the Earth-Moon transfer phase for crewed lunar exploration missions, and the maneuver that transfers the spacecraft from the free return segment to the nonfree return segment is called the hybrid maneuver, which is performed 10 hours after departing the Earth. Table1 provides the orbit parameters of the hybrid orbit. Referring to the analysis and statistics results of the Apollo missions [37], the reentry point altitude of the direct abort orbit is set to 122 km, and the flight-path angle of the reentry point is set to ‐6∘.Table 1
Orbital parameters of the hybrid orbit.
ParameterValueDeparture time2029-05-18 11:17:07.3131Semimajor axis (km)249547.1395Eccentricity0.9738Orbital inclination (deg)27.9922Right ascension of the ascending node (deg)143.7595Argument of perigee (deg)199.3091True anomaly (deg)28.4130Hybrid maneuver time2029-05-18 21:17:07.3131Hybrid maneuver impulse vector (m/s)(16.7923, 30.3731, -5.5991)


### 4.1.1. Single-Impulse Abort Orbit Example

In the single-impulse abort case, it is assumed that an abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. WhenKΘ is 0.4, by adopting the single-impulse abort orbit design method proposed in this paper, the single-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as shown in Table 2. When KΘ is 0.8, the results of the preliminary calculation and accurate design are obtained as shown in Table 3. In the tables, T is the time relative to the hybrid maneuver. ΔVEJ2 is the abort impulse vector in the geocentric J2000 coordinate system. ΔV is the abort impulse magnitude.Table 2
The results of the single-impulse abort orbit (whenKΘ is 0.4).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1263.92, -978.84, 19.71)1598.7535.21Accurate design4(1264.82, -980.59, 20.12)1600.5435.21Error(0.90, -1.75, 0.41)1.790.00Preliminary calculation8(1159.82, -839.36, -8.37)1431.7143.72Accurate design8(1161.75, -842.22, -7.72)1434.9443.72Error(1.93, -2.86, 0.65)3.230.00Preliminary calculation12(1073.05, -737.20, -25.48)1302.1351.83Accurate design12(1076.54, -741.80, -25.18)1307.6051.83Error(3.49, -4.60, 0.30)5.470.00Table 3
The results of the single-impulse abort orbit (whenKΘ is 0.8).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(803.70, -629.91, 15.89)1021.2655.12Accurate design4(804.02, -634.27, 17.43)1024.2355.12Error(0.32, -4.36, 1.54)2.970.00Preliminary calculation8(747.14, -547.12, -2.50)926.0566.08Accurate design8(747.83, -554.93, -1.87)931.2466.08Error(0.69, -7.81, 0.63)5.190.00Preliminary calculation12(696.21, -484.11, -13.91)848.0976.27Accurate design12(716.77, -471.00, 31.80)858.2676.27Error(20.56, 13.11, 45.71)10.170.00It can be seen from the tables that in the six cases, the results of the single-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. WhenKΘ is 0.4, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(a). When KΘ is 0.8, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, and the red orbits represent the single-impulse abort orbits of the three cases. Thus, the single-impulse abort orbit design method proposed in this paper can effectively generate a single-impulse abort orbit and the results of preliminary calculation can provide good initial values for accurate design.Figure 4
Accurate single-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.4)(b)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.8)


### 4.1.2. Two-Impulse Abort Orbit Example

In the two-impulse abort case, it is assumed that the abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. The maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit. By adopting the two-impulse abort orbit design method proposed in this paper, the two-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as described in Table4. When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the results of the preliminary calculation and accurate design are obtained as described in Table 5. In the tables, ΔV1EJ is the first abort impulse vector in the geocentric J2000 coordinate system. ΔV1 is the magnitude of the first abort impulse. ΔV2EJ is the second abort impulse vector in the geocentric J2000 coordinate system. ΔV2 is the magnitude of the second abort impulse.Table 4
The results of the two-impulse abort orbits (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(987.67, -769.25, 17.35)1252.019.57(23.41, -14.80, -1.13)27.721279.7344.56Accurate design4(988.30, -770.57, 18.18)1253.339.57(23.55, -16.02, -0.62)28.491281.8244.56Error(0.63, -1.32, 0.83)1.320.00(0.14, -1.22, 0.51)0.772.090.00Preliminary calculation8(909.20, -661.83, -4.83)1124.5813.56(2.45, -1.43, -0.17)2.841127.4255.15Accurate design8(910.68, -664.02, -4.24)1127.0713.56(2.86, -3.37, 1.82)4.781131.8555.15Error(1.48, -2.19, 0.59)2.490.00(0.41, -1.94, 1.99)1.944.430.00Preliminary calculation12(795.42, -550.75, -16.96)967.631.93(14.88, -9.77, -0.56)17.81985.4467.00Accurate design12(799.84, -565.67, -14.91)979.771.93(12.10, -3.33, -1.56)12.64992.4167.00Error(4.32, -14.92, 2.05)12.140.00(-2.78, 6.44, -1.00)-5.176.970.00Table 5
The results of the two-impulse abort orbits (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1175.60, -911.82, 18.95)1487.891.91(31.14, -22.76, -0.12)38.571526.4637.03Accurate design4(1175.85, -912.75, 19.39)1488.671.91(31.32, -23.67, 0.23)39.251527.9237.03Error(0.25, -0.93, 0.44)0.780.00(0.18, -0.91, 0.35)0.681.460.00Preliminary calculation8(1081.84, -784.20, -7.24)1336.197.57(0.51, -0.32, -0.02)0.601336.7946.76Accurate design8(1082.84, -785.77, -5.82)1337.917.57(0.93, -1.75, 0.49)2.041339.9546.76Error(1.00, -1.57, 1.42)1.720.00(0.42, -1.43, 0.51)1.443.160.00Preliminary calculation12(987.37, -679.81, -22.79)1198.987.87(5.11, -3.13, -0.29)6.001204.9855.79Accurate design12(988.97, -682.49, -22.06)1201.817.87(6.00, -5.58, 1.14)8.281210.0955.79Error(1.60, -2.68, 0.73)2.830.00(0.89, 2.45, 1.43)2.285.110.00It can be observed from the tables that in the six cases, the results of the two-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. When the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure5(a). When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure 5(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, the pink orbits represent the first segments of two-impulse abort orbits in the three cases, and the red orbits represent the second segments of two-impulse abort orbits in the three cases. Thus, the two-impulse abort orbit design method proposed in this paper can effectively determine the two-impulse abort orbit, and the results of preliminary calculation can provide a good initial value for accurate design.Figure 5
Accurate two-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit)(b)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit)


## 4.1.1. Single-Impulse Abort Orbit Example

In the single-impulse abort case, it is assumed that an abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. WhenKΘ is 0.4, by adopting the single-impulse abort orbit design method proposed in this paper, the single-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as shown in Table 2. When KΘ is 0.8, the results of the preliminary calculation and accurate design are obtained as shown in Table 3. In the tables, T is the time relative to the hybrid maneuver. ΔVEJ2 is the abort impulse vector in the geocentric J2000 coordinate system. ΔV is the abort impulse magnitude.Table 2
The results of the single-impulse abort orbit (whenKΘ is 0.4).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1263.92, -978.84, 19.71)1598.7535.21Accurate design4(1264.82, -980.59, 20.12)1600.5435.21Error(0.90, -1.75, 0.41)1.790.00Preliminary calculation8(1159.82, -839.36, -8.37)1431.7143.72Accurate design8(1161.75, -842.22, -7.72)1434.9443.72Error(1.93, -2.86, 0.65)3.230.00Preliminary calculation12(1073.05, -737.20, -25.48)1302.1351.83Accurate design12(1076.54, -741.80, -25.18)1307.6051.83Error(3.49, -4.60, 0.30)5.470.00Table 3
The results of the single-impulse abort orbit (whenKΘ is 0.8).
ParameterT (h)ΔVEJ2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(803.70, -629.91, 15.89)1021.2655.12Accurate design4(804.02, -634.27, 17.43)1024.2355.12Error(0.32, -4.36, 1.54)2.970.00Preliminary calculation8(747.14, -547.12, -2.50)926.0566.08Accurate design8(747.83, -554.93, -1.87)931.2466.08Error(0.69, -7.81, 0.63)5.190.00Preliminary calculation12(696.21, -484.11, -13.91)848.0976.27Accurate design12(716.77, -471.00, 31.80)858.2676.27Error(20.56, 13.11, 45.71)10.170.00It can be seen from the tables that in the six cases, the results of the single-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. WhenKΘ is 0.4, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(a). When KΘ is 0.8, the accurate single-impulse abort orbits in the three cases are drawn in Figure 4(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, and the red orbits represent the single-impulse abort orbits of the three cases. Thus, the single-impulse abort orbit design method proposed in this paper can effectively generate a single-impulse abort orbit and the results of preliminary calculation can provide good initial values for accurate design.Figure 4
Accurate single-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.4)(b)
Accurate single-impulse abort orbits in the three cases (whenKΘ is 0.8)


## 4.1.2. Two-Impulse Abort Orbit Example

In the two-impulse abort case, it is assumed that the abort return occurs 4 hours, 8 hours, and 12 hours after the hybrid maneuver. The maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit. By adopting the two-impulse abort orbit design method proposed in this paper, the two-impulse abort orbits in the three cases are calculated and the results of the preliminary calculation and accurate design are obtained as described in Table4. When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the results of the preliminary calculation and accurate design are obtained as described in Table 5. In the tables, ΔV1EJ is the first abort impulse vector in the geocentric J2000 coordinate system. ΔV1 is the magnitude of the first abort impulse. ΔV2EJ is the second abort impulse vector in the geocentric J2000 coordinate system. ΔV2 is the magnitude of the second abort impulse.Table 4
The results of the two-impulse abort orbits (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(987.67, -769.25, 17.35)1252.019.57(23.41, -14.80, -1.13)27.721279.7344.56Accurate design4(988.30, -770.57, 18.18)1253.339.57(23.55, -16.02, -0.62)28.491281.8244.56Error(0.63, -1.32, 0.83)1.320.00(0.14, -1.22, 0.51)0.772.090.00Preliminary calculation8(909.20, -661.83, -4.83)1124.5813.56(2.45, -1.43, -0.17)2.841127.4255.15Accurate design8(910.68, -664.02, -4.24)1127.0713.56(2.86, -3.37, 1.82)4.781131.8555.15Error(1.48, -2.19, 0.59)2.490.00(0.41, -1.94, 1.99)1.944.430.00Preliminary calculation12(795.42, -550.75, -16.96)967.631.93(14.88, -9.77, -0.56)17.81985.4467.00Accurate design12(799.84, -565.67, -14.91)979.771.93(12.10, -3.33, -1.56)12.64992.4167.00Error(4.32, -14.92, 2.05)12.140.00(-2.78, 6.44, -1.00)-5.176.970.00Table 5
The results of the two-impulse abort orbits (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit).
ParameterT (h)ΔV1EJ (m/s)ΔV1 (m/s)Tre1 (h)ΔV2EJ (m/s)ΔV2 (m/s)ΔV (m/s)Tre (h)Preliminary calculation4(1175.60, -911.82, 18.95)1487.891.91(31.14, -22.76, -0.12)38.571526.4637.03Accurate design4(1175.85, -912.75, 19.39)1488.671.91(31.32, -23.67, 0.23)39.251527.9237.03Error(0.25, -0.93, 0.44)0.780.00(0.18, -0.91, 0.35)0.681.460.00Preliminary calculation8(1081.84, -784.20, -7.24)1336.197.57(0.51, -0.32, -0.02)0.601336.7946.76Accurate design8(1082.84, -785.77, -5.82)1337.917.57(0.93, -1.75, 0.49)2.041339.9546.76Error(1.00, -1.57, 1.42)1.720.00(0.42, -1.43, 0.51)1.443.160.00Preliminary calculation12(987.37, -679.81, -22.79)1198.987.87(5.11, -3.13, -0.29)6.001204.9855.79Accurate design12(988.97, -682.49, -22.06)1201.817.87(6.00, -5.58, 1.14)8.281210.0955.79Error(1.60, -2.68, 0.73)2.830.00(0.89, 2.45, 1.43)2.285.110.00It can be observed from the tables that in the six cases, the results of the two-impulse abort orbit obtained from the preliminary calculation are close to those obtained from the accurate design. When the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure5(a). When the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit, the accurate two-impulse abort orbits in these cases are shown in Figure 5(b). In the figure, the green orbit represents the free return orbit, the blue orbit represents the hybrid orbit, the pink orbits represent the first segments of two-impulse abort orbits in the three cases, and the red orbits represent the second segments of two-impulse abort orbits in the three cases. Thus, the two-impulse abort orbit design method proposed in this paper can effectively determine the two-impulse abort orbit, and the results of preliminary calculation can provide a good initial value for accurate design.Figure 5
Accurate two-impulse abort orbits in the three cases of two simulation scenarios.
(a)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.5 times the minimum return time of a single-impulse abort orbit)(b)
Accurate two-impulse abort orbits in the three cases (when the maximum abort return time is 1.25 times the minimum return time of a single-impulse abort orbit)


## 4.2. Orbit Characteristic Analysis

During the engineering demonstration stage, the design of a specified direct abort orbit is not often given much attention, and much work is concentrated on grasping the relevant characteristics of this kind of orbit. Following the verification in the above subsection of the effectiveness and feasibility of the direct abort orbit design method proposed in this paper, this section adopts the method to perform extensive simulations and analyze the characteristics of the direct abort orbits.


### 4.2.1. Single-Impulse Abort Orbit Characteristic

First, the single-impulse abort orbit characteristics are analyzed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth and the flight-path angle of the reentry point are consistent with those in Section4.1. Figure 6 illustrates the curve of the required abort impulse changing with the abort time under different flight-path angle ratios. The figure indicates that the variation tendency of the required abort impulse is the same and that the required abort impulse decreases with the delay of the abort time under different flight-path angles. In addition, compared to the flight-path angle before applying the abort impulse, the larger the variation of the flight-path angle after applying the abort impulse is, the larger the required abort impulse is.Figure 6
Abort impulse versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.Figure7 presents the variation in the abort return time corresponding to the abort time in the cases of different flight-path angle ratios. The required abort return time has the same variation tendency in the cases of different flight-path angles. The change in the required abort return time corresponding to the abort time is approximately linear and increases with the delay of the abort time. In addition, the larger the change in flight-path angle after applying the abort impulse is, the smaller the required abort return time is. When the flight-path angle is 0, the required abort return time is the minimum. In this case, the single-impulse abort is regarded as the minimum return time abort.Figure 7
Abort return time versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.


### 4.2.2. Two-Impulse Abort Orbit Characteristic

Second, the two-impulse abort orbit characteristics are discussed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth, and the flight-path angle of the reentry point are consistent with those in Section4.1. The variation in the total abort impulse is depicted in Figure 8 corresponding to the abort time in the cases of different maximum abort return time constraints. In the figure, KTre is the ratio of the maximum abort return time constraint to the minimum return time of the single-impulse abort. The required total abort impulse has the same variation tendency and decreases with the delay of the abort time. The larger the maximum abort return time constraint is, the smaller the required total abort impulse for two-impulse abort is. Compared with the single-impulse abort case of the minimum abort return time, the required total abort impulse for the two-impulse abort is obviously reduced, which can save 350~1100 m/s.Figure 8
Total abort impulse versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.Figure9 depicts the variation in the required abort return time corresponding to the abort time in the cases of different maximum abort return time constraints. The change tendency of the required abort return time is similar under different maximum abort return time constraints. With the delay of the abort time, the required abort return time is increasing and changes approximately linearly. The larger the maximum abort return time constraint is, the longer the required abort return time is.Figure 9
Abort return time versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.


## 4.2.1. Single-Impulse Abort Orbit Characteristic

First, the single-impulse abort orbit characteristics are analyzed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth and the flight-path angle of the reentry point are consistent with those in Section4.1. Figure 6 illustrates the curve of the required abort impulse changing with the abort time under different flight-path angle ratios. The figure indicates that the variation tendency of the required abort impulse is the same and that the required abort impulse decreases with the delay of the abort time under different flight-path angles. In addition, compared to the flight-path angle before applying the abort impulse, the larger the variation of the flight-path angle after applying the abort impulse is, the larger the required abort impulse is.Figure 6
Abort impulse versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.Figure7 presents the variation in the abort return time corresponding to the abort time in the cases of different flight-path angle ratios. The required abort return time has the same variation tendency in the cases of different flight-path angles. The change in the required abort return time corresponding to the abort time is approximately linear and increases with the delay of the abort time. In addition, the larger the change in flight-path angle after applying the abort impulse is, the smaller the required abort return time is. When the flight-path angle is 0, the required abort return time is the minimum. In this case, the single-impulse abort is regarded as the minimum return time abort.Figure 7
Abort return time versus the abort time of the single-impulse abort orbit under different flight-path angle ratios.


## 4.2.2. Two-Impulse Abort Orbit Characteristic

Second, the two-impulse abort orbit characteristics are discussed. In terms of simulation parameters, the hybrid orbit parameters, the distance with respect to the Earth, and the flight-path angle of the reentry point are consistent with those in Section4.1. The variation in the total abort impulse is depicted in Figure 8 corresponding to the abort time in the cases of different maximum abort return time constraints. In the figure, KTre is the ratio of the maximum abort return time constraint to the minimum return time of the single-impulse abort. The required total abort impulse has the same variation tendency and decreases with the delay of the abort time. The larger the maximum abort return time constraint is, the smaller the required total abort impulse for two-impulse abort is. Compared with the single-impulse abort case of the minimum abort return time, the required total abort impulse for the two-impulse abort is obviously reduced, which can save 350~1100 m/s.Figure 8
Total abort impulse versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.Figure9 depicts the variation in the required abort return time corresponding to the abort time in the cases of different maximum abort return time constraints. The change tendency of the required abort return time is similar under different maximum abort return time constraints. With the delay of the abort time, the required abort return time is increasing and changes approximately linearly. The larger the maximum abort return time constraint is, the longer the required abort return time is.Figure 9
Abort return time versus the abort time of the two-impulse abort orbit under different maximum abort return time constraints.


## 5. Conclusion

This paper proposes a direct abort orbit design method for direct abort missions in the Earth-Moon transfer phase of crewed lunar exploration. According to the requirement of emergency rescue in the Earth-Moon transfer phase, two kinds of direct abort orbit schemes are presented. A serial design method is adopted to generate a high-fidelity direct abort orbit. In terms of a single-impulse abort orbit, an analytical model is established for the initial value calculation, and the accurate solution is determined in the high-fidelity orbit model. In terms of a two-impulse abort orbit, a hybrid optimization design process is introduced, combining the DE_CMSBHS algorithm and the SQP_Snopt algorithm. The feasibility of this method is verified by the results of simulation examples. A number of simulation calculations adopting this method are performed to analyze the direct abort orbit characteristic. With the delay of the abort time, the required abort impulse gradually decreases, and the abort return time gradually increases. Compared to a single-impulse abort scheme, a two-impulse abort scheme can effectively reduce the fuel consumption and can reduce 350~1100 m/s. The research conclusions can provide an important reference for the design of a direct abort orbit scheme in the Earth-Moon transfer phase of future crewed lunar exploration missions.



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*Source: 7026823-2022-03-19.xml*