# Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control

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## Abstract

**:**

## 1. Introduction

## 2. Orbit Dynamics

#### 2.1. Reference Frames

#### 2.2. Perturbed Two-Body Problem

#### 2.3. Circular Restricted 3-Body Problem

- ${\sigma}_{1}{\tau}_{1}>0$ (solid black hyperbola), transit trajectories which cross the equilibrium region multiple times (i.e., evolve alternately around one of the primaries)
- ${\sigma}_{1}{\tau}_{1}<0$ (dashed black hyperbola), bouncing trajectories which may enter but never cross the equilibrium region (they evolve around only one of the two primaries)
- ${\sigma}_{1}={\tau}_{1}=0$, quasi-periodic orbits that evolve only within the equilibrium region
- ${\sigma}_{1}=0$ (blue line) or ${\tau}_{1}=0$ (red line), trajectories asymptotic to or from the quasiperiodic orbits.

## 3. Nonlinear Orbit Control

**Proposition**

**1.**

**ψ**and $\left(\partial \mathit{\psi}/\partial \mathit{z}\right)$ are continuous, $\left|\mathit{b}\right|>0$ unless $\mathit{\psi}=\mathbf{0}$, and ${u}_{T}^{(max)}\ge {x}_{7}\left|\mathit{b}+{\mathit{a}}_{\mathit{P}}\right|$, then the feedback control law

**Proposition**

**2.**

**ψ**and $\left(\partial \mathit{\psi}/\partial \mathit{z}\right)$ are continuous, $\left|\mathit{b}\right|>0$ unless $\mathit{\psi}=\mathbf{0}$, and ${u}_{T}^{(max)}<{x}_{7}\left|\mathit{b}+{\mathit{a}}_{\mathit{P}}\right|$ and ${\mathit{b}}^{T}{\mathit{a}}_{\mathit{P}}\le 0$, then the feedback control law

**Proposition**

**3.**

**ψ**and $\left(\partial \mathit{\psi}/\partial \mathit{z}\right)$ are continuous, $\left|\mathit{b}\right|>0$ unless $\mathit{\psi}=\mathbf{0}$, and ${x}_{7}\left|{\mathit{a}}_{\mathit{P}}\right|<{u}_{T}^{(max)}<{x}_{7}\left|\mathit{b}+{\mathit{a}}_{\mathit{P}}\right|$, then the feedback control law in Equation (24) leads a dynamical system governed by Equations (4) and (7) to converge asymptotically to the target set associated with Equation (20).

#### Nonlinear Control for Semi-Major Axis, Eccentricity, and Inclination

- 1.
- ${x}_{1}=0$ (rectilinear trajectories);
- 2.
- ${x}_{1}={p}_{d}$, ${x}_{2}^{2}+{x}_{3}^{2}={e}_{d}^{2}$, and ${x}_{4}={x}_{5}=0$ (equatorial elliptical orbits with semilatus rectum ${p}_{d}$ and eccentricity ${e}_{d}$);
- 3.
- ${x}_{1}={p}_{d}$, ${x}_{2}^{2}+{x}_{3}^{2}=0$, and ${x}_{4}^{2}+{x}_{5}^{2}={tan}^{2}{\displaystyle \frac{{i}_{d}}{2}}$ (circular orbits with radius ${p}_{d}$ and inclination ${i}_{d}$);
- 4.
- ${x}_{1}={p}_{d}$, ${x}_{2}^{2}+{x}_{3}^{2}=0$, and ${x}_{4}^{2}+{x}_{5}^{2}=0$ (circular equatorial orbits with radius ${p}_{d}$);
- 5.
- ${x}_{1}={p}_{d}$, ${x}_{2}^{2}+{x}_{3}^{2}={e}_{d}^{2}$, and ${x}_{4}^{2}+{x}_{5}^{2}={tan}^{2}{\displaystyle \frac{{i}_{d}}{2}}$ (target set).

## 4. Interplanetary Transfer

- 1.
- The first scenario, named “APO”, achieves this objective by first targeting a specific “optimal apocenter” altitude for the initial capture around Venus and then employing nonlinear low-thrust control to reach the desired target orbit.
- 2.
- The second scenario, labelled as “L1”, focuses on the use of low-energy captures designed in the equilibrium region about ${L}_{1}$.

#### 4.1. Impulsive Maneuvers in the Geocentric Arc and Propellant Consumption Analysis

#### 4.2. Comparative Analysis of Mission Profiles

- 1.
- Launch and insertion into a direct TLI trajectory on November 24, 2032, at 1:03 UTC.
- 2.
- Application of a three-dimensional impulsive maneuver $\Delta {\mathit{V}}_{\mathbf{1}}$ at apogee to lower the pericenter altitude and slightly change the inclination (magnitude of 7.7 m/s).
- 3.
- Application of an impulsive maneuver $\Delta {V}_{2}$ at perigee in the direction of velocity for insertion into the departure hyperbola (magnitude of 560 m/s).
- 4.
- Ballistic interplanetary arc until arrival at Venus with a pericenter altitude of 300 km and a velocity of 10.473 km/s.

- 1.
- Launch and insertion into a retrograde TLI trajectory on 23 November 2032, at 21:17 UTC.
- 2.
- Application of a three-dimensional impulsive maneuver $\Delta {\mathit{V}}_{\mathbf{1}}$ at apogee to lower the pericenter altitude and slightly change the inclination (magnitude of 24.9 m/s).
- 3.
- Application of an impulsive maneuver $\Delta {V}_{2}$ at perigee in the direction of velocity for insertion into the departure hyperbola (magnitude of 565.6 m/s).
- 4.
- Interplanetary arc with an intermediate corrective impulse $\Delta {\mathit{V}}_{\mathit{c}}$ (magnitude of 16.8 m/s).
- 5.
- Arrival at Venus with a pericenter altitude of 300 km and corresponding velocity of 10.473 km/s.

## 5. Capture and Orbit Injection at Venus

#### 5.1. Capture in Highly Elliptic Orbit and Low-Thrust Orbit Injection

- Determine the braking $\Delta {V}_{3}$ (applied along $-\widehat{v}$).
- Calculate ${x}_{{7}_{i}}={x}_{{7}_{0}}{e}^{-\Delta {V}_{3}/{c}_{HT}}$ (Tsiolkovsky’s law), with ${x}_{{7}_{0}}$ determined in Section 4 based on $\Delta {V}_{E}$. This value serves as the initial value for the mass ratio ${x}_{7}$ and helps establish the maximum thrust acceleration: ${u}_{T}^{(max)}=5\times {10}^{-5}\phantom{\rule{0.222222em}{0ex}}{g}_{0}/{x}_{{7}_{i}}$.
- Propagate the spacecraft’s motion while applying low-thrust control: numerically integrate the dynamic Equations (4)–(7) using the control law (25) to determine ${\mathit{u}}_{\mathit{T}}$, inserted in (8). Gravitational perturbations related to spherical harmonics (up to degree and order 15) and the third body (the Sun) are taken into account using the Expressions (9)–(11), respectively. The initial conditions utilize the equinoctial elements corresponding to the starting state (dependent on the control activation point based on the strategy) and ${x}_{{7}_{i}}$ for the mass ratio. The propagation continues until the desired target orbit is reached (although this may not always occur).
- Identify the final mass ratio ${x}_{{7}_{f}}$ after integration.

`fminsearch`routine. The optimization of weights is performed for each strategy using a reference case that applies $\Delta {V}_{SOI}$, and then these optimized weights are used for all considered $\Delta {V}_{3}$ values.

#### 5.2. Low-Energy Capture and Low-Thrust Orbit Injection

#### 5.3. Discussion of Results

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Sketch of the flow in the equilibrium region projected onto the $[{\sigma}_{1},{\tau}_{1}]$ plane: green hyperbolas and lines indicate the boundaries, black hyperbolas represent transit (solid) and bouncing (dashed) trajectories, straight lines represent trajectories asymptotic to (blue) and from (red) the quasi-periodic orbit.

**Figure 4.**Trajectory of the optimal solution for the year 2032 in the HCI reference system. The top view is shown in the ecliptic plane, while the bottom image provides an out-of-plane perspective (the z-axis is not to scale, emphasizing the out-of-plane component). The marker at the origin represents the Sun.

**Figure 5.**Magnitude of $\Delta {\mathit{V}}_{\mathit{c}}$ depending on the application time, backward propagated from Venus.

**Figure 6.**Geocentric arc trajectory, from the moment of spacecraft release into TLI orbit by the launcher until the TVI maneuver at perigee. The figure also illustrates a segment of the departure hyperbola.

**Figure 7.**Final mass ratio as a function of the applied $\Delta {V}_{3}$ for the APO.A strategy with an elliptical target orbit. The maximum value is highlighted in red. The gray dashed vertical line represents the position of $\Delta {V}_{SOI}$.

**Figure 8.**Trends of semi-latus rectum, eccentricity, inclination, and mass ratio for the APO.A strategy with an elliptical target orbit.

**Figure 9.**Trajectory for APO.C strategy with circular target orbit. The violet segment represents the ballistic trajectory from the first to the second pericenter, while the blue segment represents the portion under orbital control, starting at the red marker.

**Figure 10.**Trends of semi-latus rectum, eccentricity, inclination, and mass ratio for APO.C strategy with circular target orbit.

**Figure 11.**Low-energy trajectory with the application of $\Delta {V}_{{L}_{1}}$ in the synodic reference frame centered at ${L}_{1}$ (red dot). The solid curve represents the trajectory after the impulse, while the dashed curve is prior to it, starting from the arrival at Venus. The purple and yellow markers represent Venus and ${L}_{2}$.

**Figure 12.**Trajectory for L1.B strategy with elliptical target orbit (in orange). The red marker indicates the starting point of the orbital control, the yellow marker indicates Venus.

**Figure 13.**Successful application of low-thrust control along the trajectory radius for an elliptical target orbit. Bullets are used to depict the pericenters (red) and apocenters (blue). The values for the corresponding final mass ratios are represented by red (apocenter) and blue (pericenter) asterisks in the lower figure.

**Figure 14.**Percentage mass ratio variation (

**left**) and total flight time in days variation (

**right**) with ${N}_{1}$ (horizontal axis) and ${N}_{2}$ (vertical axis) for elliptical orbit.

**Figure 15.**Overlaid profiles of semilatus rectum, eccentricity, and inclination for the L1.B strategy with elliptical target orbit and temporary propulsion system failure (multiple graphs with varying ${N}_{1}$ and ${N}_{2}$, each with a different color).

Parameter | Symbol [Unit] | Value |
---|---|---|

Semi-major axis | a [km] | 216,500 |

Eccentricity | e | 0.969 |

Inclination | i [deg] | 57 or 123 |

Release Altitude | h [km] | 70,000 |

**Table 2.**Solutions minimizing ${v}_{\infty}^{\mathrm{TOT}}$ for the years 2029, 2031, and 2032. All times are given in UTC.

Departure | Arrival | ToF | ${\mathit{v}}_{{\mathit{\infty}}_{1}}$ | ${\mathit{v}}_{{\mathit{\infty}}_{2}}$ | ${\mathit{v}}_{\mathit{\infty}}^{\mathbf{TOT}}$ | ||
---|---|---|---|---|---|---|---|

Date | Date | (Days) | (km/s) | (km/s) | (km/s) | ||

25 October 2029 | 05:00 | 3 April 2030 | 19:24 | 160.6 | 2.8098 | 4.8299 | 7.6397 |

23 May 2031 | 16:00 | 26 October 2031 | 13:36 | 155.9 | 2.5632 | 3.8096 | 6.3728 |

6 December 2032 | 05:00 | 12 May 2033 | 17:00 | 157.5 | 3.1757 | 2.7201 | 5.8958 |

**Table 3.**Numerical results for the Hohmann transfer from Earth to Venus, using JPL data [52].

Parameter | Symbol [Unit] | Value |
---|---|---|

Departure v-infinity | ${v}_{{\infty}_{1}}$ [km/s] | 2.4991 |

Arrival v-infinity | ${v}_{{\infty}_{2}}$ [km/s] | 2.7109 |

Time of Flight | ToF [days] | 146.04 |

Synodic period | ${\tau}_{\mathrm{syn}}$ [days] | 584 |

Components (m/s) | Magnitude (m/s) | ||
---|---|---|---|

$\widehat{\mathit{v}}$ | $\widehat{\mathit{n}}$ | $\widehat{\mathit{b}}$ | |

$5.9$ | $14.8$ | $-5.4$ | $16.8$ |

**Table 5.**Orientation angles of the arrival hyperbola at Venus in the VCI reference frame for APO and L1 scenarios.

Scenario | i | $\mathbf{\Omega}$ | $\mathit{\omega}$ |
---|---|---|---|

APO | 67.3338${}^{\circ}$ | 352.4962${}^{\circ}$ | 342.7342${}^{\circ}$ |

L1 | 49.8824${}^{\circ}$ | 185.4746${}^{\circ}$ | 137.0611${}^{\circ}$ |

**Table 6.**Velocity components of $\Delta {\mathit{V}}_{\mathbf{1}}$ in the VNB reference frame for the APO and L1 scenarios.

Scenario | Components (m/s) | Magnitude (m/s) | ||
---|---|---|---|---|

$\widehat{\mathit{v}}$ | $\widehat{\mathit{n}}$ | $\widehat{\mathit{b}}$ | ||

APO | 7.6585 | $-0.0764$ | $-3.7229\times {10}^{-4}$ | 7.6589 |

L1 | 24.2227 | 5.5241 | 0.0307 | 24.8447 |

Target Orbit | Strategy | Scenario | |||
---|---|---|---|---|---|

APO | L1 | ||||

Mass Ratio | Time of Flight | Mass Ratio | Time of Flight | ||

Elliptical | A | 68.05% | 17.77 days | 64.65% | 30.14 days |

B | 68.04% | 16.87 days | 65.82% | 24.31 days | |

C | 68.12% | 10.31 days | 65.82% | 24.31 days | |

Circular | A | 55.40% | 83 days | 53.13% | 88 days |

B | 54.49% | 92 days | 55.63% | 75 days | |

C | 56.10% | 86 days | 55.71% | 75 days |

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## Share and Cite

**MDPI and ACS Style**

De Angelis, G.; Carletta, S.; Pontani, M.; Teofilatto, P.
Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control. *Aerospace* **2023**, *10*, 887.
https://doi.org/10.3390/aerospace10100887

**AMA Style**

De Angelis G, Carletta S, Pontani M, Teofilatto P.
Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control. *Aerospace*. 2023; 10(10):887.
https://doi.org/10.3390/aerospace10100887

**Chicago/Turabian Style**

De Angelis, Giulio, Stefano Carletta, Mauro Pontani, and Paolo Teofilatto.
2023. "Earth-Venus Mission Analysis via Weak Capture and Nonlinear Orbit Control" *Aerospace* 10, no. 10: 887.
https://doi.org/10.3390/aerospace10100887