# End-to-End Optimization of Power-Limited Earth–Moon Trajectories

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{−3}m/s).

_{1}of the Earth–Moon system (intermediate EML

_{1}rendezvous) [8]. The need to pass through the vicinity of the libration point EML

_{1}is a well-known property of low-thrust transfer between an Earth orbit and a lunar orbit. However, the degree of non-optimality of the solution using EML

_{1}as the junction point of geocentric and selenocentric segments of trajectory has not been sufficiently studied so far.

_{1}rendezvous. Concluding remarks are presented in Section 6.

## 2. End-to-End Optimization of the Perturbed Power-Limited Trajectories

_{K}= L − K is the deviation of true longitude $L=\nu +\omega +\mathsf{\Omega}$ from auxiliary longitude K, e is the eccentricity, ω is the argument of perigee, Ω is the right ascension of ascending node, i is the inclination, ν is the true anomaly, ${s}^{2}=1+{i}_{x}^{2}+{i}_{y}^{2}$, $q=1+{e}_{x}\mathrm{cos}L+{e}_{y}\mathrm{sin}L$, $\xi ={i}_{x}\mathrm{sin}L-{i}_{y}\mathrm{cos}L$, μ is the gravitational parameter of the central celestial body, a

_{LPr}, a

_{LPt}, a

_{LPn}are the circumferential, radial, and binormal components of the thrust acceleration, respectively, and a

_{pt}, a

_{pr}, and a

_{pn}are the components of the perturbing acceleration.

_{t}/dK ≡ 0. However, during the process of designing multi-revolution trajectories to the Moon, it is necessary to take into account perturbing accelerations from the gravity of other celestial bodies, which explicitly depend on time. Therefore, the differential equation for t must be included in system (1).

_{K}varies very little in typical multi-revolution optimal trajectories. For example, in the 500-revolution optimal perturbed orbit transfer presented in [23], the value of L

_{K}varies from 0 to −0.092 degrees. Therefore, in this article, we will limit ourselves to considering the problems of transfer with a fixed angular distance along the auxiliary longitude ΔK instead of ΔL, and we will optimize the values of L

_{K}at the left or right ends of the trajectory. In this case, the final value of the auxiliary longitude will be fixed: K

_{f}= K

_{0}+ ΔK.

**a**

_{pgc}and selenocentric

**a**

_{psc}segments has the form:

**r**is the vector of the spacecraft’s position with respect to the central celestial body,

**r**

_{M}is the vector of the geocentric position of the Moon,

**r**

_{E}is the vector of the selenocentric position of the Earth,

**r**

_{S}is the vector of the geocentric or the selenocentric position of the Sun, and μ

_{E}, μ

_{M}, and μ

_{S}are the gravitational parameters of the Earth, the Moon, and the Sun.

_{b}= Tc/2 is given, but within this limitation, the thrust magnitude T and the exhaust velocity magnitude c can be varied arbitrarily. It is known that the differential equations of the optimal motion of a spacecraft for the LP-problem are divided into dynamic and parametric parts [24]. The dynamic part (1) does not depend on the spacecraft mass m. The dependence of the spacecraft mass on time t is calculated by the relation $m\left(t\right)={m}_{0}{P}_{b}/\left[{P}_{b}+{m}_{0}{J}_{LP}\left(t\right)\right]$, where ${J}_{LP}\left(t\right)=\frac{1}{2}{\displaystyle {\int}_{{t}_{0}}^{t}{a}_{LP}^{2}\left(t\right)dt}$ and ${a}_{LP}\left(t\right)=\sqrt{{a}_{LPr}^{2}\left(t\right)+{a}_{LPt}^{2}\left(t\right)+{a}_{LPn}^{2}\left(t\right)}$. Therefore, in the case under consideration, the problem of minimizing fuel consumption m

_{p}= m

_{0}− m

_{f}is equivalent to the problem of minimizing the cost function J

_{LP}[10]. The cost function of the form J

_{LP}is often called the “energy” criteria. The choice of this cost function facilitates the numerical solution of the problem. However, the main reason for choosing such a criterion is the possibility of the continuation of the obtained solutions to optimal trajectories with a constant exhaust velocity and finite thrust [11,20,22].

_{1}is the time of passing the junction point of the geocentric and selenocentric segments, and K

_{1}is the intermediate value of the auxiliary longitude value K

_{0}< K

_{1}< K

_{f}. The value of ${K}_{1}^{-}$ is calculated by the relation of ${K}_{1}^{-}={K}_{0}+\Delta {K}_{gc}$, and the final auxiliary longitude is ${K}_{f}={K}_{1}^{+}+\Delta {K}_{sc}$. As was shown in [10,11], the zero value of K

_{0}and ${K}_{1}^{+}={K}_{1}^{-}={K}_{1}$ can be used without a loss of generality.

**a**

_{LP}, the perturbing accelerations, and the time costate variable,

**a**

_{LP}= [a

_{LPt}, a

_{LPr}, a

_{LPn}]

^{T}, a

_{LP}= |

**a**

_{LP}|,

**A**

^{T}= (A

_{t}, A

_{r}, A

_{n}),

_{p}, p

_{ex}, p

_{ey}, p

_{ix}, p

_{iy}, p

_{LK}, p

_{m}, and p

_{t}are the costate variables to the corresponding state variables of the system p, e

_{x}, e

_{y}, i

_{x}, i

_{y}, L

_{K}, m, t, ${k}_{1}=\frac{1}{{q}^{2}}\sqrt{\frac{{p}^{3}}{\mu}}$, ${k}_{2}=\frac{1}{q}\sqrt{\frac{p}{\mu}}$. From the maximum condition for the Pontryagin function (3) with respect to the control

**a**

_{LP}, it is easy to obtain an expression for the optimal control:

_{t}/dK ≠ 0 and p

_{t}(K) ≠ 0. The equations of optimal motion have the form:

**x**

^{T}= (p, e

_{x}, e

_{y}, i

_{x}, i

_{y}),

**p**

_{x}^{T}= (p

_{p}, p

_{ex}, p

_{ey}, p

_{ix}, p

_{iy}).

_{1}of passing the junction point (optimal junction point or libration point EML

_{1}), is fixed and the time of departure from the initial Earth orbit and the time of insertion into the final lunar orbit must satisfy the necessary optimality conditions.

_{0}= 0 without loss of generality [10], can be written as:

_{1}have the following form:

**p**(0), initial values of L

_{x}_{K}and t for the geocentric segment, as well as five components of the vector

**p**(K

_{x}_{1}

^{+}), the initial value of p

_{t}, and the final value of L

_{K}for the selenocentric segment (a total of 14 decision variables). In addition, the six elements of the geocentric orbit

**x**(K

_{1}

^{+}) and L

_{K}(K

_{1}

^{+}) at the junction point at the beginning of the selenocentric segment are unknown. Thus, the problem under consideration contains 14 + 6 = 20 decision variables.

_{1}*, which is equal to the selenocentric distance of the EML

_{1}point at the junction moment t

_{1}. This choice allows us to use a smooth continuation from the trajectory with intermediate EML

_{1}rendezvous to the trajectory with an optimal junction point.

_{1}can be rewritten using variables in Cartesian coordinates:

**r**

^{−}and

**v**

^{−}are the position and velocity vectors of the spacecraft in the geocentric coordinate system at the final moment t

^{−}of the geocentric segment,

**r**

^{+}and

**v**

^{+}are the position and velocity vectors of the spacecraft in the selenocentric coordinate system at the initial moment t

^{+}of the selenocentric segment, and

**r**

_{M}and

**v**

_{M}are the position and velocity vectors of the Moon in the geocentric coordinate system at a fixed moment of junction t

_{1}, $t\left({K}_{1}^{-}\right)={t}^{-}$ and $t\left({K}_{1}^{+}\right)={t}^{+}$.

^{+}= |

**r**

^{+}|.

**p**is continuous at the junction point, the vector

_{v}**p**undergoes a discontinuity along the selenocentric radius of the junction point, and the time costate variable p

_{r}_{t}also undergoes a discontinuity due to the fixed value of t

_{1}.

- the values of five components of the vector
**p**(0);_{x} - the initial values of L
_{K}and t for the geocentric segment; - the values of five components of the vector
**p**(K_{x}_{1}^{+}); - the initial value of p
_{t}(K_{1}^{+}) and the final value of L_{K}(K_{f}) for the selenocentric segment; - the values of six osculating elements of the orbit (
**x**(K_{1}), L_{K}(K_{1})) at the junction point; - the value of Lagrange multiplier λ
_{1}.

## 3. Canonical Transformation of Costate Variables

**x**= (x

_{1}, x

_{2}, …, x

_{n})

^{T},

**p**= (p

_{1}, p

_{2}, …, p

_{n})

^{T}represent the (old) system of state and costate variables and

**x***= (x*

_{1}, x*

_{2}, …, x*

_{n})

^{T},

**p***= (p*

_{1}, p*

_{2}, …, p*

_{n})

^{T}represent another (new) system. The transformation between the coordinates

**x**and

**x***can be expressed in terms of the transformation function of the state variable

**F**:

**S**is the transformation function of the state variables from

**x*** to

**x**.

**p***, it is necessary to calculate the Jacobian matrix ∂

**S**(

**x***)/∂

**x***. If the vector function

**S**has a complex analytical representation that makes it difficult to calculate its Jacobian, and the vector function of the direct transformation of state variables

**F**is simpler, then the second part of the Equation (18) can be used, for which the matrix ∂

**F**(

**x**)/∂

**x**must be calculated. The high-precision calculation of the matrices ∂

**S**(

**x***)/∂

**x*** and ∂

**F**(

**x**)/∂

**x**is possible via the complex step differentiation [26] or via automatic differentiation using the algebra of dual numbers [27] if necessary.

**Q**and

**G**are used:

**Q**is much simpler than that of

**G**, so it is reasonable to use the second equation in (20) for the canonical transformation.

## 4. General Scheme for The Solution of End-to-End Trajectory Optimization

**f**

_{gc}, at the junction point

**f**

_{junc}, and on the selenocentric segment of trajectory

**f**

_{sc}:

**f**= [

**f**

_{gc}

^{T},

**f**

_{junc}

^{T},

**f**

_{sc}

^{T}]:

**G**

_{gc}is the transformation function from

**x**

_{cc}to

**x**

_{mee}in geocentric motion (using the gravitational parameter of the Earth),

**Q**

_{sc}is the transformation function from

**x**

_{mee}to

**x**

_{cc}in selenocentric motion (using the gravitational parameter of the Moon), and ${x}_{M}={\left({r}_{M}^{\mathrm{T}},{v}_{M}^{\mathrm{T}}\right)}^{\mathrm{T}}$ is the vector of geocentric position and the velocity of the Moon. To solve systems (22), (23), and (24), it is necessary to calculate the decision vector

**z**= [

**z**

_{gc}

^{T},

**z**

_{junc}

^{T},

**z**

_{sc}

^{T}], which includes eight decision variables of the geocentric segment

**z**

_{gc}= [

**p**

_{x}^{T}(0), L

_{K}(0), t (0), λ

_{1}]

^{T}, six decision variables of the junction zone ${z}_{junc}={x}_{mee}^{+}$, and seven decision variables of the selenocentric segment

**z**

_{sc}= [

**p**

_{r}^{+T},

**p**

_{v}^{+T}, p

_{t}

^{+}]

^{T}. A canonical transformation, based on function

**Q**, is used to calculate the Cartesian coordinates, the components of velocity, and their costate variables at the junction point.

**f**to variations of the decision vector

**z**in the process of solving the boundary value problem.

**f**

_{gc}/∂

**z**

_{sc}= 0.

- The input data are set: the time t
_{1}of passing the junction point (in the case of solving the optimization problem of trajectories with intermediate EML_{1}rendezvous, t_{1}is equal to the time t_{EML}_{1}of passing the libration point), the elements of the initial Earth orbit**x**_{0gc}= (p_{0gc}, e_{x}_{0gc}, e_{y}_{0gc}, i_{x}_{0gc}, i_{y}_{0gc})^{T}, the elements of the final lunar orbit**x**_{fsc}= (p_{fsc}, e_{xfsc}, e_{yfsc}, i_{xfsc}, i_{yfsc})^{T}, the angular distance of geocentric segment of trajectory ΔK_{gc}, and the angular distance of selenocentric segment of trajectory ΔK_{sc}; - The continuation method is used to solve a boundary value problem for calculating the optimal perturbed LP-trajectory with a junction of the geocentric and selenocentric segments at the EML
_{1}point. The continuation parameter τ is introduced into the right-hand sides of the differential equations for the state and costate variables in such a way that for τ = 0, the differential equations coincide with the equations of the unperturbed LP-problem, and for τ = 1, they coincide with the equations of the perturbed LP-problem, including the perturbed part of the Hamiltonian H_{p}and the p_{t}-dependent part of Hamiltonian H_{t}(with additional equations for the time variable and its costate variable). To use the Hamiltonian form of writing these equations, the Hamiltonian must have the form H_{LP}+ τ∙(H_{p}+ H_{t}). As an initial guess for the decision vector, the zero values of costate variables are used in both segments of the trajectory: p_{p}(0) = p_{ex}(0) = p_{ey}(0) = p_{ix}(0) = p_{iy}(0) = p_{LK}(0) = 0 (which corresponds to the coasting motion of the spacecraft along the initial orbit of each segment of trajectory). At first, the geocentric segment of the LP-trajectory is calculated, and then, the selenocentric segment of the LP-trajectory is sequentially calculated; - The continuation method is used to solve a boundary value problem for calculating the optimal perturbed LP-trajectory with an optimal junction point. The values obtained for the perturbed LP-trajectory with intermediate EML
_{1}rendezvous are used as an initial guess for the decision vector of the end-to-end optimization problem. After setting the initial conditions, the system of differential equations of geocentric motion is numerically integrated from K_{0}to ${K}_{1}^{-}$. The residual vector of the boundary value problem of geocentric segment**f**_{gc}is calculated after integrating the differential equations of the geocentric motion using the technique specified in Section 3. The vector of residuals**f**_{junc}is calculated using (23). The initial conditions of the selenocentric segment are set at $K={K}_{1}^{+}$, and the numerical integration of the system of differential equations of selenocentric motion from ${K}_{1}^{+}$ to K_{f}is carried out. The vector of residuals of the boundary value problem of selenocentric segment**f**_{sc}is calculated. The matrix ∂**f**/∂**z**, which is required to calculate the right-hand sides of the differential equations of the continuation method, is calculated based on the use of automatic differentiation using CDNAD [22].

_{1}point is the magnitude of the velocity when passing through the junction point of two segments. To calculate the velocity of a spacecraft in a rotating coordinate system at the time of passing the optimal junction point in the framework of the perturbed ephemeris model, it is necessary to transform from the inertial coordinate system J2000 to the rotating coordinate system.

## 5. Numerical Examples

_{1}for a given date of passing the junction point. The problem of optimizing the argument of perigee and the RAAN of the initial orbit was not considered in this study.

^{3}/s

^{2}. For the selenocentric segment of trajectory, all altitudes are given relative to the mean radius of the Moon, 1738 km, and the gravitational parameter of the Moon is taken equal to 4902.799 km

^{3}/s

^{2}. The gravitational parameter of the Sun is taken equal to 132,712,440,018 km

^{3}/s

^{2}. The date of passing the junction point is 25 December 2023, 00:00:00 UTC. On the corresponding fixed date of the passage of the junction point, the radius of the Hill sphere of the Moon in the selenocentric inertial coordinate system is 58,082.52 km.

_{1}point for the identically given orbital elements of the boundary orbits and time of passing the optimal junction point and the libration point EML

_{1}. The angular distances of the transfer are fixed from 4 to 28 revolutions for the geocentric segment and from 1 to 7 revolutions for the selenocentric segment. The total value of the angular distances for transfer to the Moon is calculated as the sum of these values in the geocentric and selenocentric segments: ΔK

_{Σ}= ΔK

_{gc}+ ΔK

_{sc}. In this study, trajectories with a ratio of the number of revolutions in the geocentric and selenocentric segments of 4:1 are considered, and the total number of revolutions will be varied in the range of 5 to 35 revolutions. The main criterion to choose the angular distance of trajectory segments is the level of thrust acceleration of EPS. The ratio of the number of revolutions in the geo- and selenocentric segments was selected from the approximate equality condition of the average values of thrust acceleration in these segments.

_{1}rendezvous for the different angular distances of transfer.

_{ch}is the characteristic velocity, J

_{LP}is the cost function of the considered problem, ${\tilde{a}}_{gc}$ and ${\tilde{a}}_{sc}$ are the mean values of the thrust acceleration for the geocentric and selenocentric segments of the LP-trajectory, and ΔL

_{gc}and ΔL

_{sc}are the increments of the true longitudes on the geocentric and selenocentric segments of trajectory.

_{1}point in the inertial coordinate system J2000. In Figure 1 and everywhere else in this article, the position of the EML

_{1}libration point is indicated by the red marker “×”, and the position of the optimal junction point is indicated by the red circle. The motion of the Moon is indicated by the dashed gray line. Everywhere else in this article, the solution associated with the trajectory with an optimal junction point is indicated by a blue line, and the trajectory with an intermediate EML

_{1}rendezvous by an orange line.

_{1}point, the radius of perigee increases compared to trajectories with an optimal junction point, and this is necessary to ensure the approach to the libration point with zero relative velocity. Figure 2 shows the projections of the same optimal trajectories as in Figure 1 onto the XY plane of the rotating coordinate system.

_{1}point in the rotating coordinate system. On the right side of Figure 2, a part of the trajectories near the junction point are shown on an enlarged scale.

_{1}point. This is due to the fact that the smaller the thrust acceleration magnitude, the narrower the allowable size of the opening in the vicinity of the libration point EML

_{1}when spacecraft enters into the Hill sphere of the Moon and passes through the opening in such a way that it makes it possible to capture spacecrafts in the orbit around the Moon.

_{1}rendezvous, on the contrary, have fractures and discontinuities in the control program at the EML

_{1}point (Figure 2 and Figure 3).

**r**and

**v**on the 4-revolution LP-trajectory to the Moon. For the trajectory with an intermediate EML

_{1}rendezvous (orange line), there is a discontinuity in all six costate variables, since two separate optimization problems for the geocentric and selenocentric segments of trajectory are being solved. In the case of the trajectory with an optimal junction of two segments (blue line), the dependences of the costate variables to the components of the velocity vector

**p**are continuous, but the costate variables to the components of the position vector

_{v}**p**have a discontinuity λ

_{r}_{1}·

**r**

^{+}/r

^{+}at the junction point.

**r**at the junction moment are ${p}_{rx}^{+}-{p}_{rx}^{-}$ = −2.0424·10

^{−5}, ${p}_{ry}^{+}-{p}_{ry}^{-}$ = −2.18182·10

^{−5}, ${p}_{rz}^{+}-{p}_{rz}^{-}$ = −1.25671·10

^{−5}for the 4-revolution trajectory, ${p}_{rx}^{+}-{p}_{rx}^{-}$ = 1.18001·10

^{−6}, ${p}_{ry}^{+}-{p}_{ry}^{-}$ = 2.16396·10

^{−6}, ${p}_{rz}^{+}-{p}_{rz}^{-}$ = 1.22789·10

^{−6}for the 20-revolution trajectory, and ${p}_{rx}^{+}-{p}_{rx}^{-}$ = 6.48896·10

^{−7}, ${p}_{ry}^{+}-{p}_{ry}^{-}$ = 1.31404·10

^{−6}, ${p}_{rz}^{+}-{p}_{rz}^{-}$ = 7.40291·10

^{−7}for the 35-revolution trajectory. Therefore, the magnitude of the discontinuity in

**p**decreases as the number of revolutions increases.

_{r}_{1}is equal to (0.849073, 0, 0), and the velocity vector of the EML

_{1}point is equal to (32.18877, 0, 0) m/s. The nonzero value of the first component of the velocity vector of the libration point is due to the use of the ephemeris model of motion, in which the radial velocity of the Moon is not equal to zero. Table 3 shows that the trajectory with the optimal junction point has large values of the velocity components in the rotating coordinate system when passing through the junction point of two segments compared to the trajectories with an intermediate EML

_{1}rendezvous.

_{1}rendezvous are presented in Figure 5. The deviations of the velocity magnitude Δv

_{rotating}in the rotating coordinate system from the velocity magnitude of the libration point EML

_{1}at the junction point of two segments of trajectory are 266.7827 m/s for a 5-revolution trajectory, 224.6906 m/s for a 20- revolution trajectory, and 218.0903 m/s for a 35-revolution trajectory

_{1}as the total number of revolutions increases (dashed trend lines in Figure 6).

_{junction point}between the optimal junction point and the libration point EML

_{1}on the total number of revolutions of trajectories in the inertial coordinate system. In this Figure, the obtained values of distance between two points are indicated by the black squares. The graph shows the dependence of d

_{junction point}on the total increment value of the angular variable K with extrapolation to the region of large values of the angular distance (dashed line).

_{1}point. According to the prediction of our study, this distance will decrease to 893 km on a 100-revolution trajectory and to 449 km on a 200-revolution trajectory.

_{p}of 300 km and the apogee altitude h

_{a}of 35,793 km to an elliptical lunar orbit having the perilune altitude h

_{p}of 1000 km and an apolune altitude h

_{a}of 10,000 km. The full set of elements of the initial and final orbits used in this example is presented in Table 4.

_{1}rendezvous, and the blue line corresponds to the trajectory with an optimal junction point. The main parameters of these trajectories are presented in Table 5.

## 6. Conclusions

_{1}of the Earth–Moon system was used. The numerical results of the end-to-end optimization of power-limited trajectories to the Moon were presented. The comparison was made between the obtained trajectories with the optimal junction point and with the intermediate EML

_{1}rendezvous.

_{1}was shown. The optimal junction point can be quite far from the libration point EML

_{1}(by 2560–17,760 km in the considered numerical examples) and the velocity of the spacecraft at this point can differ significantly from the velocity of the libration point (by 218–267 m/s). It was shown that with an increase in the total number of revolutions, the optimal junction point approaches EML

_{1}, and the velocity of the spacecraft at the optimal junction point relative to the velocity of EML

_{1}asymptotically tends to zero with increasing angular distance of transfer.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**x**

_{cc}can be expressed in terms of the modified equinoctial elements

**x**

_{mee}with the following expressions (here, true longitude can be represented as L = L

_{K}+ K):

_{x}cosL + e

_{y}sinL), α

^{2}= i

_{x}

^{2}− i

_{y}

^{2}.

**x**

_{mee}can be expressed in terms of the Cartesian coordinates

**x**

_{cc}with the following expressions:

## References

- Herman, A.L.; Conway, B.A. Optimal, Low-Thrust, Earth–Moon Orbit Transfer. J. Guid. Control. Dyn.
**1998**, 21, 141–147. [Google Scholar] [CrossRef] - Pérez-Palau, D.; Epenoy, R. Fuel optimization for low-thrust Earth–Moon transfer via indirect optimal control. Celest. Mech. Dyn. Astr.
**2018**, 130, 21. [Google Scholar] [CrossRef] - Carletta, S. Design of fuel-saving lunar captures using finite thrust and gravity-braking. Acta Astronaut.
**2021**, 181, 190–200. [Google Scholar] [CrossRef] - Taheri, E.; Abdelkhalik, O. Fast initial trajectory design for low-thrust Restricted Three-Body Problems. J. Guid. Control. Dyn.
**2015**, 38, 2146–2160. [Google Scholar] [CrossRef] - Yang, G. Earth-moon Trajectory Optimization Using Solar Electric Propulsion. Chin. J. Aeronaut.
**2007**, 20, 452–463. [Google Scholar] [CrossRef][Green Version] - Kluever, C.A.; Pierson, B.L. Optimal earth-moon trajectories using nuclear electric propulsion. J. Guid. Control. Dyn.
**1997**, 20, 239–245. [Google Scholar] [CrossRef] - Gao, Y.; Wang, Z.; Zhang, Y. Low thrust Earth–Moon transfer trajectories via lunar capture set. Astrophys. Space Sci.
**2019**, 364, 219. [Google Scholar] [CrossRef] - Yoon, S.W.; Petukhov, V.G.; Ivanyukhin, A.V. Minimum-Thrust Lunar Trajectories. In Proceedings of the 72nd International Astronautical Congress (IAC), Dubai, United Arab Emirates, 25–29 October 2021. [Google Scholar]
- Walker, M.; Ireland, B.; Owens, J. A set of modified equinoctial elements. Celest. Mech.
**1985**, 36, 409–419, Erratum in**1986**, 38, 391–392. [Google Scholar] [CrossRef] - Ivanyukhin, A.V.; Petukhov, V.G. Optimization of Multi-revolution Limited Power Trajectories Using Angular Independent Variable. J. Optim. Theory Appl.
**2021**, 191, 575–599. [Google Scholar] [CrossRef] - Petukhov, V.; Ivanyukhin, A.; Popov, G.; Testoyedov, N.; Yoon, S.W. Optimization of finite-thrust trajectories with fixed angular distance. Acta Astronaut.
**2022**, 197, 354–367. [Google Scholar] [CrossRef] - Ranieri, C.L.; Ocampo, C.A. Indirect optimization of low Earth orbit to low lunar orbit transfers. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, American Institute of Aeronautics and Astronautics, Honolulu, HI, USA, 18–21 August 2008. [Google Scholar] [CrossRef]
- Haissig, C.M.; Mease, K.D.; Vinh, N.X. Canonical transformations for space trajectory optimization. In Proceedings of the AIAA/AAS Astrodynamics Conference, Hilton Head, SC, USA, 10–12 August 1992. [Google Scholar]
- Taheri, E.; Arya, V.; Junkins, J.L. Costate mapping for indirect trajectory optimization. Astrodynamics
**2021**, 5, 359–371. [Google Scholar] [CrossRef] - Junkins, J.L.; Taheri, E. State vector representations for low-thrust trajectory optimization. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Snowbird, UT, USA, 19–23 August 2018. [Google Scholar]
- Ivashkin, V.V. Optimization of Space Maneuvers at Limited Distances from the Planets; Nauka Press: Moscow, Russia, 1975; p. 392. (In Russian) [Google Scholar]
- Morante, D.; Rivo, M.S.; Soler, M. A Survey on Low-Thrust Trajectory Optimization Approaches. Aerospace
**2021**, 8, 88. [Google Scholar] [CrossRef] - Golan, O.M.; Breakwell, J.V. Minimum fuel lunar trajectories for a low-thrust power-limited spacecraft. Dyn. Control.
**1994**, 4, 383–394. [Google Scholar] [CrossRef] - Ozimek, M.T.; Howell, K.C. Low-thrust transfers in the Earth-Moon system, including applications to libration point orbits. J. Guid. Control Dyn.
**2010**, 33, 533–549. [Google Scholar] [CrossRef] - Petukhov, V.G. Method of continuation for optimization of interplanetary low-thrust trajectories. Cosm. Res.
**2012**, 50, 249–261. [Google Scholar] [CrossRef] - Standish, E.M.; Newhall, X.X.; Williams, J.G.; Folkner, W.M. JPL Planetary and Lunar Ephemerides, DE403/LE403. Interoff. Memo.
**1995**, 314.10-127. [Google Scholar] - Petukhov, V.G.; Yoon, S.W. Optimization of perturbed spacecraft trajectories using complex dual numbers. Part 1: Theory and method. Cosmic Res.
**2021**, 59, 401–413. [Google Scholar] [CrossRef] - Petukhov, V.G.; Yoon, S.W. Optimization of perturbed spacecraft trajectories using complex dual numbers. Part 2: Numerical Results. Cosmic Res.
**2021**, 59, 517–528. [Google Scholar] [CrossRef] - Irving, J.H. Low Thrust Flight: Variable Exhaust Velocity in Gravitational Fields. In Space Technology; Seifert, H.S., Ed.; Wiley and Sons Inc.: New York, NY, USA, 1959. [Google Scholar]
- Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies; Academic Press: Cambridge, MA, USA, 1967. [Google Scholar]
- Martins, J.R.R.A.; Sturdza, P.; Alonso, J.J. The Connection Between the Complex-Step Derivative Approximation and Algorithmic Differentiation. In Proceedings of the 39th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 8–11 January 2001. [Google Scholar] [CrossRef][Green Version]
- Yu, W.; Blair, M. DNAD, a Simple Tool for Automatic Differentiation of Fortran Codes Using Dual Numbers. Computer Physics Communications
**2013**, 184, 1446–1452. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**The projection of the optimal 5-(

**upper**row), 20-(

**middle**row), and 35-(

**lower**row) revolution trajectories with an optimal junction point and with an intermediate EML

_{1}rendezvous in the inertial coordinate system J2000.

**Figure 2.**The projection of optimal 5-(

**upper**row), 20-(

**middle**row), and 35-(

**lower**row) revolution trajectories with an optimal junction point and with an intermediate EML

_{1}rendezvous onto the XY plane in the rotating coordinate system.

**Figure 4.**Variation of the adimensional values of costate variables on the 4-revolution LP-trajectory to the Moon.

**Figure 5.**The time dependences of the velocity in the rotating coordinate system on the 5-(on the

**left**), 20-(in the

**middle**), and 35-revolution (on the

**right**) trajectories with an optimal junction point and with an intermediate EML

_{1}rendezvous.

**Figure 6.**The positions of the optimal junction points in the rotating coordinate system onto the XY (on the

**left**) and XZ (on the

**right**) planes.

**Figure 7.**The dependence of distance between the optimal junction point and the libration point EML

_{1}on the total value of angular variable K in the inertial coordinate system.

**Figure 8.**The projection of optimal 18-revolution trajectories from GTO with low perigee altitude of 300 km to elliptical lunar orbit onto the XY plane in the rotating coordinate system.

ΔK_{Σ} | Δt, Days | Δv_{ch}, m/s | J_{LP}, m^{2}/s^{3} | ${\tilde{\mathit{a}}}_{\mathit{g}\mathit{c}}$, mm/s^{2} | ${\tilde{\mathit{a}}}_{\mathit{s}\mathit{c}}$, mm/s^{2} | ΔL_{gc}, Orbits | ΔL_{sc}, Orbits |
---|---|---|---|---|---|---|---|

5 | 12.5019 | 2426.431 | 4.166911 | 5.019797 | 4.437600 | 4.000847 | 1.001065 |

10 | 23.2934 | 2423.336 | 2.061159 | 2.488547 | 2.153214 | 8.000450 | 2.000285 |

15 | 34.1184 | 2414.030 | 1.367589 | 1.654349 | 1.415061 | 12.00030 | 2.999980 |

20 | 44.9189 | 2407.796 | 1.023116 | 1.240632 | 1.056037 | 16.00022 | 3.999832 |

25 | 55.7171 | 2403.652 | 0.817266 | 0.992601 | 0.843267 | 20.00017 | 4.999750 |

30 | 66.5044 | 2400.837 | 0.680367 | 0.826451 | 0.702236 | 24.00014 | 5.999701 |

35 | 77.3294 | 2398.331 | 0.582603 | 0.708573 | 0.601601 | 28.00011 | 6.999663 |

ΔK_{Σ} | Δt, Days | Δv_{ch}, m/s | J_{LP}, m^{2}/s^{3} | ${\tilde{\mathit{a}}}_{\mathit{g}\mathit{c}}$, mm/s^{2} | ${\tilde{\mathit{a}}}_{\mathit{s}\mathit{c}}$, mm/s^{2} | ΔL_{gc}, Orbits | ΔL_{sc}, Orbits |
---|---|---|---|---|---|---|---|

5 | 16.3599 | 2666.178 | 4.512647 | 4.897396 | 4.544912 | 4.000075 | 0.999946 |

10 | 27.9986 | 2671.468 | 2.239662 | 2.426028 | 2.176598 | 8.000047 | 1.999964 |

15 | 39.6325 | 2672.903 | 1.498561 | 1.611979 | 1.448160 | 12.00002 | 2.999969 |

20 | 51.1954 | 2663.745 | 1.128752 | 1.205753 | 1.091558 | 16.00001 | 3.999971 |

25 | 62.8006 | 2649.619 | 0.905442 | 0.962620 | 0.878230 | 20.00000 | 4.999972 |

30 | 74.6966 | 2637.367 | 0.75464 | 0.801968 | 0.734214 | 24.00000 | 5.999972 |

35 | 86.8793 | 2623.987 | 0.645747 | 0.687248 | 0.625917 | 27.99999 | 6.999971 |

**Table 3.**The magnitudes of adimensional position and dimensional velocity vectors of the optimal junction point in the rotating coordinate system.

ΔK_{Σ} | x_{rotating} | y_{rotating} | z_{rotating} | v_{x rotating}, m/s | v_{y rotating}, m/s | v_{z rotating}, m/s |
---|---|---|---|---|---|---|

5 | 0.856126 | 0.045084 | −0.006894 | 237.8436 | −180.8093 | 11.06208 |

10 | 0.850736 | 0.021426 | −0.006398 | 219.0976 | −152.6332 | 3.993939 |

15 | 0.849788 | 0.013438 | −0.005965 | 213.9416 | −147.4695 | 1.357233 |

20 | 0.849493 | 0.009780 | −0.005628 | 211.8339 | −145.3045 | 0.075297 |

25 | 0.849353 | 0.007504 | −0.005384 | 210.4100 | −143.3563 | −0.630621 |

30 | 0.849268 | 0.005700 | −0.005218 | 209.0966 | −141.3347 | −1.062691 |

35 | 0.849217 | 0.004278 | −0.005094 | 207.7519 | −139.5594 | −1.392600 |

Boundary Orbit | h_{p}, km | h_{a}, km | i, Degrees | Ω, Degrees | ω, Degrees |
---|---|---|---|---|---|

Initial Earth orbit | 300 | 35793 | 25 | 4 | 248 |

Final lunar orbit | 1000 | 10000 | 30 | 4 | 248 |

**Table 5.**The results of the optimal 18-revolution trajectories from GTO with perigee altitude of 300 km to elliptical lunar orbit.

Type of Junction | Δt, Days | Δv_{ch}, m/s | J_{LP}, m^{2}/s^{3} | ${\tilde{\mathit{a}}}_{\mathit{g}\mathit{c}}$, mm/s^{2} | ${\tilde{\mathit{a}}}_{\mathit{s}\mathit{c}}$, mm/s^{2} |
---|---|---|---|---|---|

Optimal junction | 32.9135 | 2935.234 | 2.347619 | 2.631531 | 2.368071 |

Intermediate EML_{1} rendezvous | 39.45431 | 3327.815 | 2.533057 | 2.550099 | 2.366935 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Petukhov, V.; Yoon, S.W.
End-to-End Optimization of Power-Limited Earth–Moon Trajectories. *Aerospace* **2023**, *10*, 231.
https://doi.org/10.3390/aerospace10030231

**AMA Style**

Petukhov V, Yoon SW.
End-to-End Optimization of Power-Limited Earth–Moon Trajectories. *Aerospace*. 2023; 10(3):231.
https://doi.org/10.3390/aerospace10030231

**Chicago/Turabian Style**

Petukhov, Viacheslav, and Sung Wook Yoon.
2023. "End-to-End Optimization of Power-Limited Earth–Moon Trajectories" *Aerospace* 10, no. 3: 231.
https://doi.org/10.3390/aerospace10030231