# Phasing Maneuver Analysis from a Low Lunar Orbit to a Near Rectilinear Halo Orbit

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mission Scenario and Dynamic Model

#### 2.1. Reference Frames

**Inertial Reference Frame**: all inertial coordinate frames have their axes aligned with the axis as defined for the Jet Propulsion Laboratory (JPL) DE2000 Ephemeris files. The x-axis points in the direction of the vernal equinox at midday of the 1st of January 2000. The z-axis points in the direction normal to the mean equatorial plane, in the direction of the North pole, while the y-axis supplements the set to be a right-hand coordinate system. The center of the inertial frame can be located in the center of a celestial body or in the barycenter of a system of celestial bodies.**Synodic (Co-rotating) Reference Frame**: very suitable coordinate frames for the illustration and treatment of libration point orbits are co-rotating coordinate frames. The x-axis points from the primary body to the secondary body, the z-axis is the axis of the angular velocity of the bodies and is therefore normal to the orbital plane of the two primaries and the y-axis supplements the coordinate system to be a right-hand system, it can be centered on the secondary center or on the system’s barycenter.**Local Vertical Local Horizon Reference Frame**: The LVLH frame is defined with respect to the primary body around which the target is orbiting. Denoting with ${\mathit{r}}_{it}$ the target position with respect to the primary i, with ${\left[\phantom{\rule{0.166667em}{0ex}}{\dot{\mathit{r}}}_{it}\phantom{\rule{0.166667em}{0ex}}\right]}_{{\mathcal{M}}_{i}}$ the target velocity as seen from the primary, and with ${\mathit{h}}_{it}={\mathit{r}}_{it}\times {\left[\phantom{\rule{0.166667em}{0ex}}{\dot{\mathit{r}}}_{it}\phantom{\rule{0.166667em}{0ex}}\right]}_{{\mathcal{M}}_{i}}$ the target specific angular momentum with respect to the primary, the LVLH frame unit vectors are defined and named as follows,- ●
- $\widehat{\mathit{k}}=-{\mathit{r}}_{it}/\parallel {\mathit{r}}_{it}\parallel $ points to the primary and is called R-bar;
- ●
- $\widehat{\mathit{\u0237}}=-{\mathit{h}}_{it}/\parallel {\mathit{h}}_{it}\parallel $, is perpendicular to the target instantaneous orbital plane and is called H-bar;
- ●
- $\widehat{\mathit{\u0131}}=\widehat{\mathit{\u0237}}\times \widehat{\mathit{k}}$ completes the right-handed reference frame, and is called V-bar.

The above definition of the LVLH frame is consistent with the one given by Fehse in its classical reference book for spacecraft rendezvous and docking [13]. With a little abuse of notation, the name of the unit vectors are maintained, as standard practice in the community.

#### 2.2. Departure and Arrival Boundary Conditions

#### 2.3. Equations of Motion

#### 2.4. Boundary Conditions

## 3. Phasing Trajectory Computation

- Determine a first guess on the departure velocity from the lunar orbit.
- Apply a differential correction propagation to correct the initial velocity, based on the boundary conditions at the final time, so that the spacecraft arrives at the final point with an acceptable error, along an acceptable manifold.

#### 3.1. Phasing Using Lambert’s Initial Conditions

**V**for the transfers shown above is performed assuming the following vector relationship:

#### 3.2. Phasing Using Hohmann Initial Conditions

#### 3.3. Numerical Optimization

**e**is the error between final state and desired state,

## 4. Validation Using the Elliptic Restricted Three-Body Problem

#### Comments on $\mathrm{\Delta}V$ Expenditure

## 5. Discussion

#### 5.1. Lambert, Two-Impulse

- The boundary conditions needed to solve the problem are computed using Lambert’s two-body problem method.
- The best simulation results are obtained with a stable manifold computed via propagation from an 80 km perturbation and it is met at the aposelene. The final position error is zero, since we are on the desired manifold.
- The time of flight of the phasing trajectory is 78.5 h (half period of the target orbit).
- The total expenditure is $\mathrm{\Delta}V=660$ m/s.
- The phasing trajectory is shown in Figure 10, and it is a full out-of-plane path. This is why this approach may result in an increased fuel expenditure. To be noted is the fact that the orientation of the LLO orbit was not specified, thus the amount of out-of-plane imposed by Lambert could change if different LLO orientations are chosen.

#### 5.2. Hohmann, Two-Impulse

- The boundary conditions needed to solve the problem are computed using a planar two-body Hohmann approach.
- The stable manifold is computed via propagation from an 80 km perturbation and it is met at the aposelene. The final position error is zero, since we are on the desired manifold.
- The time of flight is similar to the previous case (half period of the target orbit).
- The total expenditure is $\mathrm{\Delta}V=671$ m/s.
- The phasing trajectory is shown in Figure 11.

#### 5.3. Multiple Impulse Optimization

- The optimization procedure does not use differential correction, and it produces position errors at the final time, which depend on propagation equations, relative weight between error and fuel expenditure, optimization soft stopping conditions on the final state.
- The optimization used soft constraints on the simplified dynamics, in order to evaluate the relative influence between accuracy and fuel expenditure. Although hard constraints could have produced different results, the choice appears justified by the fact that further iterations are necessary in the mission design phase.
- A different number of impulses was evaluated, with the best results obtained for a 4-impulse sequence (see Table 5), although almost all $\mathrm{\Delta}V$ was used with the first two impulses, making it similar to the other two solutions.
- Different stable manifolds were evaluated in terms of $\mathrm{\Delta}V$ expenditure (see Table 7). The computed $\mathrm{\Delta}V$ are 688 m/s and 687 m/s for a manifold propagation of one orbital period (80 km and 100 km perturbations).
- The phasing trajectory time of flight is half orbital period and it is shown in Figure 13.
- Improvement in the final position error and $\mathrm{\Delta}V$ can be obtained by longer manifold propagation, and longer time of flight of phasing trajectory. In particular, a five-period manifold propagation, and a phasing trajectory time of flight of 701 h yields a position error within the requirements and a $\mathrm{\Delta}V$ = 678.4 m/s, as shown in Figure 16.

#### 5.4. ER3BP Propagation

- Phasing trajectory time of flight of half period, $\mathrm{\Delta}V$ = 703 m/s, see Figure 18. The figure also shows the loss of accuracy at the Periselene, as the propagation time increases.
- The validity of CR3BP is maintained only if the time of flight is much lower than the propagation time as expected.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

LAE | Lunar Ascent Element |

LOP-G | Lunar Orbital Platform–Gateway |

NRHO | Near Rectilinear Halo Orbit |

LLO | Low Lunar Orbit |

TOF | Time of Flight |

LVLH | Local Vertical Local Horizon |

ER3BP | Elliptic Restricted Three-Body Problem |

CR3BP | Circular Restricted Three-Body Problem |

CLTV | Cis-Lunar Transfer Vehicle |

## Appendix A. Differential Correction

**x**= [x, y, z, $\dot{x}$, $\dot{y}$, $\dot{z}$] ${}^{T}$ is the state vector describing the motion of the vehicle. Let $\mathbf{x}\left(t\right)$ be some trajectory going from ${\mathbf{x}}_{0}$ to ${\mathbf{x}}_{1}$ under motion dynamics described by Equation (1). For any given initial condition ${\mathbf{x}}_{0}$, the resulting trajectory at time t is defined by a

**flow map**$\varphi (t,{t}_{0})$ as:

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Information | Data |
---|---|

Comment | LOP-G Southern NRHO 9:2 Resonance |

Originator | FreeFlyer (AI-SOLUTION) |

Object Name | LOP-G |

Center Name | Moon |

Start Time | 2020-01-09T00:21:00.000 |

Stop Time | 2020-02-07T03:21:00.000 |

Period | 6.56 days |

Periselene | 1500 km |

Aposelene | 70,000 km |

Initial Orbit | Insertion Point |
---|---|

LLO Polar | near NRHO Aposelene |

100 km altitude | ~50 km below (R-Bar), ~86 km behind (V-Bar) |

Propagation Time | True Anomaly |
---|---|

1.4T | 152 deg. |

1.35T | 159 deg. |

1.3T | 163 deg. |

1.25T | 167 deg. |

1.2T | 170 deg. |

1.15T | 172 deg. |

1.1T | 175 deg. |

1.05T | 177 deg. |

1.0T | 180 deg. |

TOF | $\mathbf{\Delta}\mathit{V}$ [m/s] |
---|---|

0.5 NRHO T | 671 |

0.55 NRHO T | 688 |

0.7 NRHO T | 766 |

Weight | Value |
---|---|

${q}_{1}$ | $9\times {10}^{4}$ |

${q}_{2}$ | 10 |

Number of Impulses | Total $\mathbf{\Delta}\mathit{V}$ |
---|---|

2 | 661 m/s |

3 | 709 m/s |

4 | 688 m/s |

5 | 717 m/s |

Perturbation (km) | Total $\mathbf{\Delta}\mathit{V}$ |
---|---|

80 (original) | 688 m/s |

90 | 687 m/s |

100 | 687 m/s |

150 | 687 m/s |

200 | 686 m/s |

No. Impulses | 100 km, 5T | 100 km, 5.2T | 100 km, 5.4T |
---|---|---|---|

3 | 676 m/s | 671 m/s | >700 m/s |

4 | 671 m/s | 754 m/s | >700 m/s |

5 | 677 m/s | 1045 m/s | >700 m/s |

Perturbation (km) | Total $\mathbf{\Delta}\mathit{V}$ |
---|---|

80 (original) | 673 m/s |

90 | 672 m/s |

100 | 671 m/s |

150 | 674 m/s |

200 | 682 m/s |

Time of Flight $\mathbf{\Delta}\mathit{V}$ | |
---|---|

T = 0 | 639 m/s |

T = 1.19 day | 32 m/s |

T = 1.28 day | 0.01 m/s |

T = 2.2 day | 0.05 m/s |

Time of Flight $\mathbf{\Delta}\mathit{V}$ | |
---|---|

T = 0 | 639 m/s |

T = 0.91 day | 39.4 m/s |

T = 1.638 day | 0.001 m/s |

T = 0.67 day | 0.001 m/s |

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**MDPI and ACS Style**

Bucchioni, G.; Innocenti, M.
Phasing Maneuver Analysis from a Low Lunar Orbit to a Near Rectilinear Halo Orbit. *Aerospace* **2021**, *8*, 70.
https://doi.org/10.3390/aerospace8030070

**AMA Style**

Bucchioni G, Innocenti M.
Phasing Maneuver Analysis from a Low Lunar Orbit to a Near Rectilinear Halo Orbit. *Aerospace*. 2021; 8(3):70.
https://doi.org/10.3390/aerospace8030070

**Chicago/Turabian Style**

Bucchioni, Giordana, and Mario Innocenti.
2021. "Phasing Maneuver Analysis from a Low Lunar Orbit to a Near Rectilinear Halo Orbit" *Aerospace* 8, no. 3: 70.
https://doi.org/10.3390/aerospace8030070