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Reachable set analysis of practical translunar orbit via a retrograde semianalytic model
Earth, Planets and Space volume 75, Article number: 34 (2023)
Abstract
Different from the Apollo flight mode, a safer translunar flight mode for the crews is preferred. The previousarrived lunar module rendezvouses with the crew exploration vehicle at a low lunar destination orbit, and then the crews ride the lunar module to descend the lunar surface sampling destination. The lunar module, which includes the descent and ascent stages, flies from a low Earth orbit to the low lunar destination orbit with two tangential impulses. The low lunar destination orbital reachable set of the practical translunar orbit limits the feasible lunar surface sampling region. Therefore, this paper addresses the low lunar destination orbital reachable set of the practical translunar orbit. A retrograde semianalytic model is proposed for rapidly computing the practical twoimpulse translunar orbit firstly, which refers the ephemeris table twice for more precision perilune orbital elements. The reachable set is generated using the multiplelevel traversal searching approach with this retrograde semianalytic mode. Its envelope is rechecked by the continuation theory with the highprecision orbital model. Besides, some factors that affect the reachable set are also measured. The results show that neither the Earth–Moon distance nor the translunar duration affects the reachable set. However, if the translunar injection inclination is smaller than the inclination of the moon’s path, the reachable set becomes smaller or even reduces into an empty set. In brief, the proposed retrograde semianalytic model for computing the reachable set provides a helpful and fast tool for selecting an applicable lunar surface sampling site for the manned lunar mission overall design.
Graphical Abstract
Introduction
The moon as the nearest celestial body is also the compulsory training camp for human’s interstellar travels. After the Apollo project, especially when the evidences of water in lunar south pole are discovered by many lunar polar orbiters (Colaprete et al. 2010), new plans for the manned lunar mission by USA and other countries have become more diverse. The National Aeronautics and Space Administration (NASA) have published three important reports on the Constellation Program after 2005(Stanley et al. 2005; Condon 2007; Garn et al. 2008). Although the Constellation Program was replaced later by the Lunar Orbital PlatformGateway Program (Davis 2018), the aims of lunar Global Access and Anytime Return remained unchanged. Different from the lunar low latitude area detection of the Apollo project, the lunar polar and highlatitude districts would be the next manned lunar mission destination for water and possible lunar life. In the Constellation Program, NASA proposed a new idea called crew and cargo separation that the lunar module (i.e., descend and ascend stages are merged) transferred to a low lunar orbit (LLO) by a practical translunar duration about 4–5 day, to wait for a crew exploration vehicle for rendezvous and assembling. Compared with the Apollo flight mode, this new flight mode is safer for crews (Stanley et al. 2005). When it comes to the mode of a 3day translunar orbit used in the Apollo Project, the new mode is also more fuelsaving for translunar injection (TLI) and low lunar orbit insertion (LOI) (He and Shen 2020). To save fuel, TLI and LOI are hoped to be tangential to the periapsis velocity vector. With the two constraints (i.e., the translunar duration and tangential velocity increments), the orbital elements reachable set of LLO for the lunar module rendezvous with the crew exploration vehicle becomes a significant factor to select a feasible lunar surface sampling site. Therefore, this paper studies the reachable set of the practical twoimpulse translunar orbits.
In February 1959, the Luna1 lunar impact probe launched by Soviet Union did not reach the goal of hitting the lunar surface because of its inaccurate translunar orbital dynamics model and designed method. The Luna2, launched in September 1959, successfully hit the lunar surface and became the first human lunar probe. After that, dozens of lunar probes including the unmanned and manned cases have had successfully missions to the Moon. However, the wellknown translunar orbital dynamics models could be divided into only four items, namely, the circular restricted threebody problem (CR3BP), the double twobody model, the pseudostate model, and the highprecision model. Issac Newton described the CR3BP in his Philosophiae Naturalis Principia Mathematica, which shows the mathematical formulas of the orbits in a threebody system. CR3BP still need numerical integration (Lee et al. 2014). The double twobody model was proposed based on the principle of computing the celestial body’s gravitational sphere in the Pierre Simon Laplace’s monumental work Celestial Mechanics (Egorov 1958). Wilson (1970) and Byrnes and Hooper (1970) proposed the pseudostate model around 1970s. It is also named the multiconic method. It calculates a translunar orbit using both the Earth’s and the Moon’s gravitational forces acting on the spacecraft gradually. Such calculation consumes about 1% of the numerical integration time, and its model errors are no more than 5% compared with the double twobody model. The perilune attitude error for a translunar orbit is about 20 km (Zhou et al. 2019). Apart from the above translunar orbital dynamical models, the highprecision model considers all the known perturbation forces acting on the spacecraft. Thus, it is the most accurate dynamic model that reflects the real scenario with a huge potential in the near future (Yan and Gong 2011). To design a translunar orbit with a highprecision model, a simplified orbital model is applied usually to provide an initial value of the design variable (Mohammad and Roberto 2019). The double twobody model is selected usually to play the role of the simplified orbital model because of its semianalytical feature. However, the selections of the parameters are different. Peng et al. (2011) and Gao et al. (2018) selected the entry point parameters on the lunar influence sphere to play the orbital patched conic parameters. Li et al. (2015) selected the exit point parameters to do this. The above orbital patched conic techniques that select the entry or exit point are difficult to compute the precision orbital position and velocity at the epochs of the perilune and perigee. Because they referred the Jet Propulsion Laboratory (JPL) ephemeris table for the Moon’s position and velocity at the epochs of the entry or exit point, but ignore the Moon’s position and velocity deviations when the spacecraft flies inside the Moon’s gravitational influence sphere. And the lambert iteration is needed to compute the orbital elements in the Earthcentric segment, so its calculation efficiency is low. A set of the perilune parameters selected in our previous works (He et al. 2019) is an effective measure for avoiding the orbital perilunestate deviation.
The reachable set of nonlinear differential equations refers to the dynamic system as Eq. (1). If the system is consecutive within \(t \in \left[ {t_{0} ,t_{{\text{f}}} } \right]\), and there is an initial state set \({\varvec{x}}\left( {t_{0} } \right) \in \mathbf{\Theta }^{{\text{n}}} \subseteq \mathbf{R}^{{\text{n}}}\) at the moment \(t_{{0}}\). There exists a control set \({\varvec{u}}\left( t \right) \in \mathbf{U}^{{\text{m}}} \subseteq \mathbf{R}^{{\text{n}}}\) that causes the final state \({\varvec{x}}\left( {t_{{\text{f}}} } \right) \in \mathbf{\Omega }^{{\text{n}}} \subseteq \mathbf{R}^{{\text{n}}}\) at the moment \(t_{{\text{f}}}\), then the \(\mathbf{\Omega }^{{\text{n}}}\) can be called the reachable set of the initial set \({{\varvec{\Theta}}}^{{\text{n}}}\)(Grantham. 1981).
Here, \({\varvec{x}}\left( t \right)\) denotes the state variables of the consecutive system; \(\dot{\user2{x}}\left( t \right)\) is its differential variable. \({\varvec{f}}\left[ {} \right]\) denotes their dynamic model function. \({\varvec{y}}\left( t \right)\) denotes the observation variables or the variables which are concerned. \({\varvec{c}}\left[ {} \right]\) denotes their function. In a physical dynamic system, all of the elements in the final reachable set are sometimes not intuitive or concerned. However, its partial elements are concerned. Therefore, a relevant elements’ equation \({\varvec{y}}\left( t \right) = {\varvec{c}}\left[ {t,{\varvec{x}}\left( t \right)} \right]\) is built, which can be used for calculating the final concerned parameters’ set \({\mathbf{Y}}^{{\text{k}}} \subseteq {\mathbf{R}}^{{\text{n}}}\). If there are some inequality or equality constraints in the control process as Eq. (2), then \({\mathbf{Y}}^{{\text{k}}}\) will become small, dimensionless, even an empty set \({\mathbf{Y}}^{{\text{k}}} \to {{\varvec{\Phi}}}\).
Here, \({\varvec{g}}\left[ {} \right]\) and \({\varvec{\zeta}}\left[ {} \right]\) denote the inequality or equality constraints. To review the approaches of computing the orbital reachable set, the previous literatures mainly focused on the following aspects, such as the spacecraft formation flight (Lee and Hwang 2018), collision probability calculation (Bai et al. 2012), geostationary satellites collocation (Li 2014), engagement analysis of exoatmospheric interceptor (Chai et al. 2014), Earth reentry (Lu and Xue 2010), and Moon and Mars entry landing footprints (Arslantas 2016; Benito and Mease 2012). Methods vary in different cases, but all of them could be categorized essentially into two types.
Type I:
When a spacecraft is dominated by one force and the orbital state transition matrix can be derived by the linearization hypothesis in some form as \(\dot{\user2{x}} = {\varvec{Ax}} + {\varvec{Bu}}\), and if there is no control force or uncertain process model error in the duration \(t \in \left[ {t_{0} ,t_{{\text{f}}} } \right]\), a state transition matrix \({\varvec{\varPhi}}_{{\left( {t_{0} ,t_{{\text{f}}} } \right)}} = e^{{{\varvec{A}}\left( {t_{{\text{f}}}  t_{0} } \right)}}\) can be utilized to compute the final state \({\varvec{x}}_{{t_{{\text{f}}} }} ={\varvec{\varPhi}}_{{\left( {t_{0} ,t_{{\text{f}}} } \right)}} {\varvec{x}}_{{t_{0} }}\). Therefore, the final state reachable set, affected by an initial state uncertainty or a small orbital maneuver (Zhang et al. 2013; Wen et al.2018; Yang et al. 2019), can be approximately calculated by the state transition matrix (i.e., \(\delta {\varvec{x}}_{{t_{{\text{f}}} }} ={\varvec{\varPhi}}_{{\left( {t_{0} ,t_{{\text{f}}} } \right)}} \delta {\varvec{x}}_{{t_{0} }}\)). If there are cases such as some perturbation forces action on the spacecraft (Li et al.2011), large uncertainty range of the initial state, long initial–final duration (He et al. 2013), or a complex flight process (Feng et al.2019), the calculation of the reachable set by the utilization of linearization hypothesis leads to an inaccuracy, or even a wrong result.
Type II:
When a spacecraft has some orbital control powers or there are some uncertain processes, such as the general entry or reentry reachable set problems with uncertain atmospheric model parameters, the final state becomes more complex as \({\varvec{x}}_{{t_{{\text{f}}} }} ={\varvec{\varPhi}}_{{\left( {t_{0} ,t_{{\text{f}}} } \right)}} {\varvec{x}}_{{t_{0} }} + \int_{{t_{0} }}^{{t_{{\text{f}}} }} {{\varvec{Bu}} \cdot {\text{d}}t}\). Because of the unknowable controlling or uncertain perturbation forces \(\int_{{t_{0} }}^{{t_{{\text{f}}} }} {{\varvec{Bu}} \cdot {\text{d}}t}\) during the process, the reachable set cannot be computed by the state transition matrix approach. Komendera et al. (2012) had suggested an intelligent feedbackadjust idea to search the initial parameters of a reachable set’s envelope, but it also needs more computations and is not practical for the complex engineering problems. The practical approach is the twostep method which avoids a huge amount of calculations. Take the Earth reentry for example (Lu and Xue 2010). The first step computes the maximum range, and the second step computes the maximum crossranges with some fixedranges which are between the minimum and maximum ranges. The highprecision dynamical model is adopted instead of a linearization assumption. By this way, the reachable set calculated by the highprecision method is undoubtedly more accurate than that calculated by the linearization hypothesis. This method is a calculation strategy operated at two levels. The outer layer gradually gives the virtual objective point, while the inner layer is a general singleobjective nonlinear constrained optimization problem. According to the specific conditions, the inner optimization problem can be solved by an indirect method or a direct optimization algorithm. The basic model is shown in Eq. (3).
Here, \(n,m,l,p\) denote the number of the optimization variables, optimization objectives, inequality constraints, and equality constraints, respectively. \(f,g_{j} ,h_{k}\) denote the objective function, inequality constraint function, and equality constraint function, respectively. The s.t. is the abbreviations for the subjected to some constraints. The optimization results of the inner layer calculation constitute the envelope parameters of the reachable set. As for the shortcoming of this methodology, it is impossible to prove whether the reachable set is continuous or not.
The structure of the remainder of the article is as follows. After reviewing the abovementioned translunar orbital dynamical models and different approaches of the orbital reachable set generation, details of problem formulation including the generation strategy proposed in this paper are presented in “Problem statement and strategy” section. The retrograde semianalytical model and the multiplelayer traversal searching approach are described in “Reachable set rapid generation” section. In “Precision envelopes generation” section, the retrograde semianalytical model and the envelope of the reachable set are rechecked by the highprecision dynamics model. The property of the reachable set changes due to some influence factors are exhibited in “reachable set property measurement” section. Finally, conclusions are drawn in “Conclusion” section.
Problem statement and strategy
Problem statement
The practical twoimpulse translunar orbit has duration about 4 ~ 5 days, projected orbital altitudes at the epochs of TLI and LOI, and tangential incremental velocities of \(\Delta v_{{{\text{TLI}}}}\) and \(\Delta v_{{{\text{LOI}}}}\) as shown in Fig. 1.
Tangential impulses are reasonable for computing a translunar orbit. It is uneconomical that the translunar burn from a low earth orbit (LEO) with a nontangential thruster vector. It is same uneconomical for lunar orbit insertion with a nontangential thruster vector. The TLI burn duration of Apollo11 is 320 s, it is an impulse compared to the translunar duration about 4–5 days (Berry 1970). To describe the orbital reachable set of the practical twoimpulse translunar orbit via the mathematical formula in Eq. (1), \(t_{0}\) and \(t_{{\text{f}}}\) represent the epochs of TLI and LOI, respectively. The orbital initial state set is described as Eq. (4).
Here, \({\kappa }\) represents the radius of periapsis; subscript ‘EJ2’ represents the J2000.0 earthcentric coordinate system; \(h_{{{\text{LEO}}}}\) represents the altitude of LEO; \(R_{{\text{E}}}\) represents the radius of Earth; \(e\), \(i\), and \(f\) represent the eccentricity, inclination, and the true anomaly of the translunar orbit, respectively. The superscript of ‘lb’ and ‘ub’ represent the low and upper boundaries. The expression of \(e_{{{\text{EJ2}}}} < 1\) implies the translunar orbit is an elliptical orbit, which orbital energy is lower than the parabolic and hyperbolic orbits. The interval of \(\left[ {i_{{{\text{EJ2}}}}^{{{\text{lb}}}} ,i_{{{\text{EJ2}}}}^{{{\text{ub}}}} } \right]\) depends on the launching angle \(A_{0}\) and the latitude of launch of launch site of \(B_{0}\), i.e., \(\cos i_{{\text{E}}} = \sin A_{0} \cos B_{0}\). The equation of \(f_{{{\text{EJ2}}}} = 0\) is equivalent to that \(\Delta v_{{{\text{TLI}}}}\) is tangential.
The nominal orbit does not need any midcourse corrections. The reachable set of the orbital elements at \(t_{{\text{f}}}\) can be described as Eq. (5).
Here, the subscript of ‘MJ2’ represents the J2000.0 mooncentric coordinate system; \(h_{{{\text{LLO}}}}\) represents the altitude of LLO; and \(R_{{\text{M}}}\) represents the radius of Moon. The expression of \(e_{{{\text{MJ2}}}} > 1\) implies that the translunar orbit is a hyperbolic orbit when it arrives at the perilune point. The equation of \(f_{{{\text{MJ2}}}} = 0\) is equivalent to that \(\Delta v_{{{\text{LOI}}}}\) acts on the perilune point. The interval of \(\left[ {\Delta t^{{{\text{lb}}}} ,\Delta t^{{{\text{ub}}}} } \right]\) implies that the acceptable translunar duration is practical and useful in an engineering task.
Obviously, the reachable set as described in Eq. (5) is a multidimensional element set, which is difficult to present clearly and simply understand. However, only the inclination and the longitude of ascending node (LAN) in the lunar centricfixed coordinate system (i.e., the subscript of ‘LCF’) in Eq. (6) are the elements concerned, which affect the ground track of satellite (GTS) after lunar orbit insertion as shown in Fig. 2.
Twostep strategy
It implies that there are countless translunar orbits with constraints as Eq. (7) to be calculated for generating a valid \({\mathbf{Y}}_{{t_{{\text{f}}} }}^{{\text{k}}}\). The difficulty lies in two aspects, one is the orbital dynamics model and another is the optimization algorithm. If the highprecision orbital mode is applied, its computingtime for a large number of the orbits is unaffordable. And if the evolutionary algorithm is applied, it leads to the same problem of the cost of computing.
Reference Type II. mentioned in the bottom of the Introduction, a strategy is suggested, which contains two steps as shown in Fig. 3. The simple explanation is as follows:
Step 1:
A retrograde semianalytical model is established based on the double twobody concept, wherein the orbital elements at the perilune point are selected to play the role of the traversal searching variables. After traversal searching, the orbital elements, which satisfy the constraints, are recorded. Then, the topological structure and the influence relations of the elements in the reachable set are analyzed.
Step 2:
The orbital design variable, which affects the reachable set obviously, is selected to play the role of the numerical continuation element. Then, the precision envelops of the reachable set are rechecked with the highprecision orbital dynamical model under the continuation frame. Every point of the envelope implies a constrained optimal problem. Its solution is solved via the initial values from Step 1 via SQP optimization algorithm (Gill et al. 2005).
Step 1 not only provides the topological structure information and the influence relation of the design variables in the reachable set but also provides the initial value of the orbit design variables for Step 2. Step 2 checks the reachable set solved by Step 1 using the proven highprecision orbital dynamical model.
Reachable set rapid generation
A retrograde semianalytical model
The latitude and longitude on the Moon’s influence sphere, and flight duration from TLI to the moment when the lunar probe passes into the Moon’s influence sphere are selected as the translunar orbital design variables (Peng et al. 2011; Gao et al.2018). The Newton iteration is applied to calculate the Earthcentered true of anomaly of the entrance point on the Moon’s influence sphere, and then the translunar orbit before the moment when the lunar probe passes into the Moon’s influence sphere can be designed. The orbital design variables in the traditional application of the double twobody model cannot obtain a constant orbital altitude at the epoch of perilune point, and the design variables have no physical significance. Moreover, they are not easily translated into the reachable set elements in Eq. (6). In our previous work, a set of orbital elements at the epoch of the perilune suggested as the design variable show a well convergence performance to design the adaptive LEOphase and the fixedthrust circumlunar freereturn orbits (He et al. 2019). The latitude and longitude \(\left( {\lambda ,\varphi } \right)\) in the lunarcentric local vertical and local horizontal coordinate system (LVLH) at the epoch of perilune, the velocity vector azimuth angle \(i_{{{\text{prl}}}} ,\) and the value of the velocity \(v_{{{\text{prl}}}}\) before LOI are selected as the translunar orbital design variables as shown in Fig. 4.
The subscript of ‘prl’ represents the epoch of the perilune. The translunar orbital position and velocity vectors at this epoch before LOI can be described as Eq. (8).
Here, \(r_{{{\text{prl}}}}\) is equals to \(\kappa_{{{\text{MJ2}}}}\); \({\varvec{M}}_{{\text{y}}}\),\({\varvec{M}}_{{\text{z}}}\), and unused \({\varvec{M}}_{{\text{x}}}\) are the basic rotation matrix of the coordinate systems. If \(t_{{\text{f}}}\) (i.e., the epoch of LOI) and \(r_{{{\text{prl}}}}\) (i.e., \(r_{{{\text{prl}}}} = R_{{\text{M}}} + h_{{{\text{LLO}}}}\)) are set to constants, the instantaneous lunarcentered LVLH coordinate system can be treated as an inertial system, the position, and velocity vectors of the translunar orbit at this epoch before LOI in J2000.0 mooncentered coordinate system can be computed by Eq. (9).
Here, \({\varvec{M}}_{{{\text{LVLH2MJ2}}}}\) is the rotation matrix from the lunarcentered LVLH coordinate system to the J2000.0 lunarcentered coordinate system which is expressed of \({\varvec{M}}_{{{\text{LVLH2MJ2}}}} = {\varvec{M}}_{{\text{z}}} \left( {  {\Omega }_{\text{M}} } \right){\varvec{M}}_{{\text{x}}} \left( {  i_{\text{M}} } \right){\varvec{M}}_{{\text{z}}} \left( {  u_{\text{M}} } \right)\). The expressions of \(\left( {u_{{\text{M}}} ,i_{{\text{M}}} ,{\Omega }_{{\text{M}}} } \right)\) are the Moon’s argument of latitude, inclination, and right ascension of ascending node (RAAN), respectively, in the J2000.0 lunarcentered coordinate system. Therefore, the translunar orbit can be only computed by \(\left( {\lambda ,\varphi ,i_{{{\text{prl}}}} ,v_{{{\text{prl}}}} } \right)\). When \(\left[ {{\varvec{r}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} ,{\varvec{v}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} } \right]\) is translated into the modified classical orbital elements \(\left( {\kappa_{{{\text{MJ2}}}} ,e_{{{\text{MJ2}}}} ,i_{{{\text{MJ2}}}} ,{\Omega }_{{{\text{MJ2}}}} ,\omega_{{{\text{MJ2}}}} ,f_{{{\text{MJ2}}}}^{{{\text{prl}}}} } \right)\), the radius of the Laplace influence sphere \(\rho\) is 66200 km. Hence, the true anomaly \(f_{{{\text{in}}}}^{{{\text{MJ2}}}}\) at the moment of \(t_{{{\text{in}}}}\) can be computed by Eq. (10).
Here, the subscript of ‘in’ represents the epoch when the lunar module enters the Laplace influence sphere, and \(p_{{{\text{prl}}}} = \kappa_{{{\text{prl}}}} \left( {1 + e_{{{\text{prl}}}} } \right)\). Then, the position and velocity vectors of the translunar orbit \(\left[ {{\varvec{r}}_{{{\text{in}}}}^{{{\text{MJ2}}}} ,{\varvec{v}}_{{{\text{in}}}}^{{{\text{MJ2}}}} } \right]\) at this epoch in J2000.0 lunarcentered coordinate system can be computed by the principle of the twobody orbital state transfer matrix described as Eq. (11).
Here, \(\mu_{{\text{M}}}\) represents the lunar gravitational constant, and \(\sigma_{{{\text{prl}}}}^{{{\text{MJ2}}}} = {{\left( {{\varvec{r}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} \cdot {\varvec{v}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} } \right)} \mathord{\left/ {\vphantom {{\left( {{\varvec{r}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} \cdot {\varvec{v}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} } \right)} {\sqrt {u_{{\text{M}}} } }}} \right. \kern0pt} {\sqrt {u_{{\text{M}}} } }}\).
According to Gudermann Christoph’s transformation principle, the lunarcentered hyperbolic anomaly \(H\) at this epoch can be computed by Eq. (13).
The flight duration from \(t_{{{\text{in}}}}\) to \(t_{{\text{f}}}\) can be computed by Eq. (14).
Therefore, \(t_{{{\text{in}}}} = t_{{{\text{prl}}}}  \Delta t_{{{\text{in2prl}}}}\). The Moon’s position and velocity vectors \(\left[ {{\varvec{r}}_{{\text{M}}}^{{{\text{in}}}} ,{\varvec{v}}_{{\text{M}}}^{{{\text{in}}}} } \right]\) are obtained by the JPL ephemeris table. The position and velocity vectors of the translunar orbit \(\left[ {{\varvec{r}}_{{{\text{in}}}}^{{{\text{MJ2}}}} ,{\varvec{v}}_{{{\text{in}}}}^{{{\text{MJ2}}}} } \right]\) at the epoch of \(t_{{{\text{in}}}}\) in J2000.0 Earthcentered coordinate system can be computed by Eq. (15).
When \(\left[ {{\varvec{r}}_{{{\text{in}}}}^{{{\text{EJ2}}}} ,{\varvec{v}}_{{{\text{in}}}}^{{{\text{EJ2}}}} } \right]\) is translated into the modified classical orbital elements \(\left( {\kappa_{{{\text{EJ2}}}} ,e_{{{\text{EJ2}}}} ,i_{{{\text{EJ2}}}} ,{\Omega }_{{{\text{EJ2}}}} ,\omega_{{{\text{EJ2}}}} ,f_{{{\text{EJ2}}}}^{{{\text{in}}}} } \right)\), the eccentricity of \(e_{{{\text{EJ2}}}}\) is smaller than 1, and \(f_{{{\text{EJ2}}}}^{{{\text{in}}}}\) is the true anomaly of the translunar orbit at the epoch of \(t_{{{\text{in}}}}\). According to the Kepler's transformation principle, the flight duration from the moment of TLI to the moment \(t_{{{\text{in}}}}\) can be computed by Eq. (16).
Here, \(\tan \frac{{E_{{{\text{EJ2}}}}^{{{\text{in}}}} }}{2} = \sqrt {\frac{{1  e_{{{\text{EJ2}}}} }}{{1 + e_{{{\text{EJ2}}}} }}} \tan \frac{{f_{{{\text{EJ2}}}}^{{{\text{in}}}} }}{2}\), and the epoch of TLI (i.e., \(t_{0}\)) is \(t_{0} = t_{{{\text{in}}}}  \Delta t_{{{\text{TLI2in}}}}\). The total flight duration of the translunar orbit is \(\Delta t = \left( {t_{{\text{f}}}  t_{{0}} } \right) = \Delta t_{{{\text{TLI2in}}}} + \Delta t_{{{\text{in2prl}}}}\). The expressions in Eqs. (8 , 9, 10, 11, 12, 13, 14, 15, 16) are the socalled retrograde semianalytical model, which is proposed to compute the translunar orbit rapidly.
Multiplelayer traversal search
Based on the retrograde semianalytical model, a multilayer traversal search method is utilized to generate the reachable set of the translunar orbits rapidly. The flow chart is shown in Fig. 5.
\(\left( {\lambda ,\varphi ,i_{{{\text{prl}}}} ,v_{{{\text{prl}}}} } \right)\) are selected as the traversal searching elements in a fourlayer frame respectively. Then, \(\kappa_{{{\text{EJ2}}}}\), \(e_{{{\text{EJ2}}}}\), \(i_{{{\text{EJ2}}}}\), and, \(f_{{{\text{EJ2}}}}^{{{\text{in}}}}\) are tested by the constraints. If any constraint is not satisfied, the next loop will be restarted. Considering the error of the double twobody model, an error of the radius of perigee \(\Delta \kappa_{{{\text{EJ2}}}}\) is allowable as Eq. (17).
\(\left( {\lambda ,\varphi ,i_{{{\text{prl}}}} ,v_{{{\text{prl}}}} } \right)\) refresh its values by add themselves on \(\left( {\Delta \lambda ,\Delta \varphi ,\Delta v_{{{\text{prl}}}} ,\Delta i_{{{\text{prl}}}} } \right)\) in everyone’s layer, respectively. Until the searching process of the four layers is accomplished, the set of the orbital elements at the epoch of perilune constitutes the reachable set.
The reachable set based on the retrograde semianalytical model
To verify the effectiveness of the proposed retrograde semianalytical model and the multilayer traversal search frame, an example is tested. The useful orbital elements and constraints are optioned as follows:

A)
The moment of perilune \(t_{{\text{f}}}\) is set as 1 Jan 2025 0:00:00.000 UTCG (UTC, in Gregorian Calendar format).

B)
Refer to Apollo11 mission (Berry 1970), set \(h_{{{\text{LLO}}}}\) = 111 km and \(h_{{{\text{LEO}}}}\) = 185.2 km.

C)
Set \(\left[ {\lambda_{\min } ,\lambda_{\max } } \right]\) = [−1 80, 180] deg, \(\left[ {\varphi_{\min } ,\varphi_{\max } } \right]\) = [− 90, 90] deg, \(\Delta \lambda = \Delta \varphi\) = 2 deg. \(v_{{{\text{prl}}}}\) is set to ensure that the translunar orbit is a lunarcentered hyperbolic orbit. Therefore, its minimum value is \(v_{{{\text{prl}}}}^{{{\text{min}}}} = \sqrt {{{2\mu_{{\text{M}}} } \mathord{\left/ {\vphantom {{2\mu_{{\text{M}}} } {\left( {h_{{{\text{LLO}}}} + R_{{\text{M}}} } \right)}}} \right. \kern0pt} {\left( {h_{{{\text{LLO}}}} + R_{{\text{M}}} } \right)}}}\). The allowable \(\Delta v_{{{\text{LOI}}}}^{{{\text{allow}}}}\) is smaller than 1 km/s [14]. Its maximum value \(v_{{{\text{prl}}}}^{{{\text{max}}}} = \sqrt {{{\mu_{{\text{M}}} } \mathord{\left/ {\vphantom {{\mu_{{\text{M}}} } {\left( {h_{{{\text{LLO}}}} + R_{{\text{M}}} } \right)}}} \right. \kern0pt} {\left( {h_{{{\text{LLO}}}} + R_{{\text{M}}} } \right)}}} + \Delta v_{{{\text{LOI}}}}^{{{\text{allow}}}}\). Therefore, \(\left[ {v_{{{\text{prl}}}}^{{{\text{min}}}} ,v_{{{\text{prl}}}}^{{{\text{max}}}} } \right]\) = [2302.7, 2628.3] m/s, and set \(\Delta v_{{{\text{prl}}}}\) = 1 m/s. A circumlunar freereturn orbit or a hybrid orbit is used to being the translunar orbit of the crew exploration vehicle. Both of them have a retrograde LLO after lunar orbit insertion by a tangential maneuver at perilune. The LLO is destination orbit for the lunar module to rendezvous with the crew exploration vehicle, Therefore, set \(\left[ {i_{{{\text{prl}}}}^{{{\text{min}}}} ,i_{{{\text{prl}}}}^{{{\text{max}}}} } \right]\) = [90, 270] deg, and set \(\Delta i_{{{\text{prl}}}}\) = 2 deg.

D)
According to the large position error of the semianalytical model, set \(\Delta \kappa_{{{\text{EJ2}}}}\) = 1000 km. And to analyze the influence of the flight duration on the reachable set, set \(\left[ {\Delta t_{\min } ,\Delta t_{\max } } \right]\) = [3, 6] day.

E)
The inclination of the moon’s path plane varies between 18 to 28 deg. Considering an orbital inclination error of 2 deg, set \(\left[ {i_{{{\text{EJ2}}}}^{{{\text{min}}}} ,\,i_{{{\text{EJ2}}}}^{{{\text{max}}}} } \right]\) = [16, 30] deg.
After a pure analytical searching process, a mass of the constrained translunar orbits is obtained. The distributions of the design variables are shown in Fig. 6a, b, c.
The longitude of \(\lambda\) is scattered in [− 100 to 0] W deg while \(\varphi\) symmetrically is scattered in [− 50 to 50] deg. The value of the velocity at \(v_{{{\text{prl}}}}\) is scattered in [2410, 2540] m/s. It becomes larger with \(\lambda\) and \(\varphi\) become larger to a certain extent. The inclination of \(i_{{{\text{prl}}}}\) is scattered in [90–180] deg (\(\varphi\) < 0) and [180–270] deg (\(\varphi\) > 0), and its value tends to 180 deg from 90 and 270 deg in both directions as \(\lambda\) becomes larger.
The orbital elements concerned as shown in Fig. 6d are the inclination and the LAN in the lunarcentric LVLH and the lunar fixed coordinate systems. The inclination in the lunarcentric LVLH coordinate system is scattered in [90–180] deg. However, when the inclination is less than about 167 deg, the LAN only distributes between about [80–140] deg and [260–320] deg. The case in the lunar fixed coordinate system is similar, except for minor differences caused by the Moon’s rotation and libration.
Precision envelopes generation
Optimize a precise translunar orbit
The position and velocity vectors of the translunar orbit at the moment of perilune before LOI in J2000.0 lunarcentered coordinate system \(\left[ {{\varvec{r}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} ,{\varvec{v}}_{{{\text{prl}}}}^{{{\text{MJ2}}}} } \right]\) can be computed by Eq. (8) and (9). Then, the Moon’s position and velocity vectors \(\left[ {{\varvec{r}}_{{\text{M}}}^{{{\text{prl}}}} ,{\varvec{v}}_{{\text{M}}}^{{{\text{prl}}}} } \right]\) in the J2000.0 Earthcentered coordinate system at this epoch are computed by JPL ephemeris table. The position and velocity vectors at this epoch in the J2000.0 Earthcentered coordinate system are computed by Eq. (18).
The translunar orbit is computed via a 6day retrogradetime numerical integration with the initial states of \(\left[ {{\varvec{r}}_{{{\text{prl}}}}^{{{\text{EJ2}}}} ,\,{\varvec{v}}_{{{\text{prl}}}}^{{{\text{EJ2}}}} } \right]\) using the Runge Kutta 7–8 integrator. The epoch of the perigee is found via the interpolation. Then, the position and velocity are translated into the modified classical orbital elements \(\left( {\kappa_{{{\text{EJ2}}}} ,e_{{{\text{EJ2}}}} ,i_{{{\text{EJ2}}}} ,{\Omega }_{{{\text{EJ2}}}} ,\omega_{{{\text{EJ2}}}} ,f_{{{\text{EJ2}}}}^{{{\text{TLI}}}} } \right)\). Wherein, \(\left {\sin \left( {f_{{{\text{EJ2}}}}^{{{\text{TLI}}}} } \right)} \right < {\upvarepsilon }\), which is the determined by the interpolation accuracy. \(\kappa_{{{\text{EJ2}}}}\) and \(i_{{{\text{EJ2}}}}\) are set as the equality and the inequality constraints due to the rocket. Therefore, the inclination and the RAAN at the epoch of LOI are the same with the target values. The model of searching a precise translunar orbit via an optimal algorithm is shown in Eq. (19).
Here, the superscript ‘tar’ represents the target values of the orbital elements. To verify the model above, the same constraints and the interval of the design variables are set as that in section (the reachable set based on the retrograde semianalytical model). The difference is the consideration of the precision perturbation orbital model. It contains the Sun’s and the Moon’s perturbation, and the Earth's nonspherical perturbation of 6 × 6 order of in WGS84 table. The solar radiation pressure and the atmosphere perturbation are ignored. An LLO with \(i_{{{\text{MJ2}}}}^{{{\text{tgt}}}}\) = 150 deg and \({\Omega }_{{{\text{MJ2}}}}^{{{\text{tgt}}}}\) = 50 deg is selected as the destination orbit in an optional manner.
A set of proximate design variables and the lunarcentered modified classical orbital elements in different lunarcentered coordinate systems are listed in Table 1 and Table 2. The Earthcentered modified classical orbital elements at the moment of TLI computed by the retrograde semianalytical model are listed in Table 3.
The values in Table 1 are selected to play the role of the initial values of the design variables in Eq. (19). The optimal values of the design variables, the lunarcentered modified classical orbital elements in different lunarcentered coordinate systems, and the Earthcentered modified classical orbital elements computed via Eq. (19) with the highprecision numerical integration are listed in Tables 4, 5, and 6, respectively.
The iterative process of the optimal objective function and the constraints’ fitness are shown in Fig. 7. The iteration converges quickly within 10 steps. It has two major reasons: One is the utilization of the SQP_snopt optimization kit (Gill et al. 2005), which has good ability to solve nonlinear programming problems (Song et al. 2020). Another is the design variables obtained from the retrograde semianalytical model, which provides an effective initial values.
Recheck the envelope of the reachable set
The reachable set is generated rapidly and clearly based on the retrograde semianalytical model, and the accuracy of its envelope still has errors. According to the methodology in Eq. (3) and Fig. 6(c), \(i_{{{\text{prl}}}}\) is selected to play the role of the continuation element. In the continuation frame, there are three design variables left. \(\left[ {i_{{{\text{EJ2}}}}^{{{\text{min}}}} ,\,\,i_{{{\text{EJ2}}}}^{{{\text{max}}}} } \right]\) is still set as an inequality constraint, and the translunar orbit is numerical integrated via a timeretrograde manner with a fixed \(\Delta t\). The error absolute value between the perigee radius and the radius of LEO is set as the optimal objective function. The frame is shown in Eq. (20).
The initial values \({\varvec{x}}^{0}\) of the design variables are found from the reachable set in section (the reachable set based on the retrograde semianalytical model). \({\varvec{x}}_{{n{  1}}}^{{{\text{opt}}}}\) plays the role of the initial values for the iteration optimization that \(n \ge 2\) as shown in Eq. (21). The orbital elements are solved quickly and easily using this continuation theory, the envelope of the reachable set are constituted with them.
Corresponding to the timeretrograde integrate fixed time \(\Delta t \in \left[ {3,6} \right]\) day, respectively, the envelope of the reachable set in the lunarcentered LVLH and the lunar fixed coordinate systems is shown in Fig. 8.
It shows that the model error of the retrograde semianalytical model for the practical translunar orbit with 6day duration is larger than that with 3day flight duration. When the flight duration is 3 days, the envelope solution of the reachable set is more consistent with the solution computed with the highprecision model. The inclinations in the lunarcentered LVLH and the lunarcentered fixed coordinate systems both approach to 180 deg. When the inclination is close to 180 deg, the LAN has a large error than other cases due to its numerical singularity.
Reachable set property measurement
The property of the reachable set changes due to some influence factors. It is useful to understand the relation of them for designing a manned lunar task in the overall design phase. In consideration of the effectiveness of the retrograde semianalytical model, the property of the reachable set is measured. The distance of the Earth–Moon, the transfer duration, and the declination of the Moon are tested as follows, respectively.
The distance of the Earth–Moon
The eccentricity of the Moon’s path changes from 1/23 to 1/5. It leads a result that the distance of the Earth–Moon changes from about \(3.6 \times 10^{8}\) to \(4.1 \times 10^{8}\) m in a lunation. After 1 Jan 2025 0:000.000 UTCG, 8 Jan 2025 0:000.000 UTCG, and 21 Jan 2025 5:000.000 UTCG in this lunation are selected as the epochs for testing, which correspond the nearest and the farthest Earth–Moon distances, respectively.
The results are shown in Fig. 9a, b. It shows outwardly that the Earth–Moon distance leads nonobvious influence to the reachable set except for the orbital inclination is approaching to 180 deg in the lunarcentered LVLH frame. In practice, however, when the orbital inclination is approaching to 180 deg in the lunarcentered LVLH frame, the singularities of LAN emerge. Arbitrary value of LAN does not have significance. In a word, the Earth–Moon distance leads nonobvious influence to the reachable set.
The transfer duration
The reachable sets with the translunar duration of 3 days, 4 days, 5 days, and 6 days are shown in Fig. 10, respectively. All of them arrive at the epoch of 1 Jan 2025 0:00:00.000 UTCG as perilune.
It shows that the LANs in the lunarcentered LVLH and the lunarcentered fixed coordinate systems have distinguishing features. Both of them has two manners (i.e., ascent orbit and descent orbit) arriving their perilune. When the translunar duration is longer than 4 days, the perilune point is on the front of the Moon (i.e., the side is directly visible from the Earth). This property is significant for directly tracking and controlling from the Earth’s surface tracking stations.
The declination of the Moon
The Moon’s position in the J2000 Earthcentric coordinate system is described via the right ascension and declination. The declination of the Moon describes the Moon’s position along the perpendicular direction of the J2000.0 mean equatorial plane. After 1 Jan 2025 0:000.000 UTCG, 7 Jan 2025 00.00.00.000 UTCG and 12 Jan 2025 00.00.00.000 UTCG in this lunation are selected as the epochs, which correspond the smallest and the biggest cases of the moon’s declinations. The values of the Moon’s declinations are 1.359 deg and 28.447 deg, respectively. The reachable sets are shown, respectively, in Fig. 11a, b. Refer to the discussion of the singularities of LAN when the orbital inclination is approaching to 180 deg, when we compare the difference between Fig. 9a, b. The reachable sets have nonobvious difference if just looking at these two pictures. However, the distributions of the LEO inclination as shown in Fig. 12 have a significant difference. It implies that if the inclination of LEO is less than the declination of the Moon, no translunar orbit exists. In other words, the reachable set is an empty set. The long period of the declination of the Moon is the Metonic cycle (i.e., 18.6 years), which periodically changes from 18.3 to 28.6 deg. The maximum declination of the Moon will be 28.6 deg in 2025 and the minimum is 18.3 deg in 2034. As long as the inclination of LEO is greater than 18.3 deg in 2034, the reachable set of the translunar orbit will not be an empty set. Moreover, the reachable set of the translunar orbit does not become a nonempty set at any time in 2025, and the inclination of the LEO must be greater than 28.6 deg.
Conclusion
A retrograde semianalytical model is suggested to generate the reachable set of the practical twoimpulse translunar orbit with two tangential maneuvers. Compared with the traditional twobody conic method in the literatures, this model does not have internal iteration, and its perilune point altitude can be an accurate constant. The multilayer traversal searching frame and the precision envelops recheck have exhibited its rapidity and effectiveness. The classical influence factor test exhibits the changeable property of the reachable set. Some conclusions can be drawn as follows:

A)
The distance of the Earth–Moon leads nonobvious influence to the reachable set.

B)
The translunar duration leads to different perilune point on the Moon’s surface. If there are only tracking stations on the Earth’s surface, and a necessary support of the tracking and control for the lunar module’s LOI, the translunar duration is suggested to be more than 4 days.

C)
It is a necessary condition that the inclination of LEO is greater than the declination of the Moon at the epoch of perilune. Otherwise, the reachable set of the translunar orbits becomes an empty set.
Availability of data and materials
Not applicable.
Abbreviations
 NASA:

National Aeronautics and Space Administration
 LLO:

Low lunar orbit
 LEO:

Low earth orbit
 TLI:

Translunar injection
 LOI:

Lunar orbit insertion
 CR3BP:

Circular restricted threebody problem
 JPL:

Jet Propulsion Laboratory
 LAN:

Longitude of ascending node
 LCF:

Lunar centric fixed
 GTS:

Ground track of satellite
 LVLH:

Local vertical and local horizontal
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Funding
This work was supported by the National Natural Science Foundation of China (No. 11902362) and the Young Elite Scientists Sponsorship Program by CAST(2021JCJQQT027).
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The idea of rapidly calculating the reachable set of the lunar module’s practical translunar orbit using the lunar perilune elements was put forward by Mr. BYH. BYH and SGW completed the derivation of retrograde semianalytic model. BYH completed the coding, figures plotting, and manuscript writing work. Pro. HNL proofread the manuscript. All the authors read and approved the final manuscript.
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Boyong He, Associate Prof. received his Ph.D. degree in aeronautical and astronautical science and technology from National University of Defense Technology, China, in 2017. After graduation from doctoral, he joined State Key Laboratory of Astronautic Dynamics (ADL), Xi’an Satellite Control Center as an assistant professor, Xi’an, China in 2018, and in Northwestern Polytechnical University, China, as a postdoctoral fellow (2019–2021). Currently, he implements 1 project from National Natural Science Foundation of China, 1 project from the China Postdoctoral Science Foundation, and 1 project from Young Elite Scientists Sponsorship Program by CAST. His area of expertise is in space mission planning and astrodynamics.
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He, By., Wu, Sg. & Li, Hn. Reachable set analysis of practical translunar orbit via a retrograde semianalytic model. Earth Planets Space 75, 34 (2023). https://doi.org/10.1186/s4062302301782y
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DOI: https://doi.org/10.1186/s4062302301782y