# Optimum, Suboptimal and Solar Sailing Orbital Maneuvers for a Spacecraft Orbiting the Earth

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## Abstract

**:**

## 1. Introduction

## 2. Model Used

- Objective Function: T,where T is the time of transfer, the variable to be minimized with respect to the control;
- Subject to: Equations of motion, constraints in the state and control.

## 3. Optimal Solution

- (a)
- The differential equations that give the Lagrange multipliers, called “adjoint equations”. When we combine them with the equations of motion, we have a complete set of differential equations that solves the problem;
- (b)
- The “Transversality Conditions”, which are the conditions that the Lagrange multipliers must achieve at the end of the numerical integration. They complete the conditions to be satisfied by the solution of the problem when combined to the fact that the maneuver starts at a given initial orbit and ends in a different final orbit;
- (c)
- The “Maximum Principle of Pontryagin”, which states that it is necessary to maximize the magnitude of the scalar product of the Lagrange multipliers with the right-hand side of the equations of motion. This point gives the condition to find the optimal angles of “pitch” and “yaw” at each instant of time.

- (i)
- Give estimates for the initial and final times of the maneuver and the initial values of the Lagrange multipliers;
- (ii)
- Make a numerical integration of the adjoint equations and the equations of motion, using the values for the “pitch” and “yaw” angles obtained from the Maximum Principle of Pontryagin;
- (iii)

- Extremize $\sum _{i=1}^{9}}{p}_{i}{f}_{i$ with respect to A.

## 4. Numerical Method

- (i)
- First, we need to make the system satisfy the constraints given by the problem (usually some orbital elements of the final orbit). For this step, we use the update:$${\mathbf{X}}_{i+1}={\mathbf{X}}_{i}-\nabla {\mathbf{f}}^{T}\xb7{\left[\nabla \mathbf{f}\xb7\nabla {\mathbf{f}}^{T}\right]}^{-1}\mathbf{f}$$
- (ii)
- After the constraints are satisfied, we take some steps to minimize the fuel consumed. For this phase, the updates are given by:$${\mathbf{X}}_{i+1}={\mathbf{X}}_{i}+\left(\gamma \frac{J\left(\mathbf{X}\right)}{\nabla J\left(\mathbf{X}\right)\xb7\mathbf{d}}\right)\frac{\mathbf{d}}{\left|\mathbf{d}\right|}$$$$d=-\left(\mathbf{I}-\nabla {\mathbf{f}}^{T}{\left[\nabla \mathbf{f}\xb7\nabla {\mathbf{f}}^{T}\right]}^{-1}\mathbf{f}\right)\xb7\nabla J\left(\mathbf{X}\right)$$

## 5. Solar Sailing

#### Sail Attitude Strategy

- (i)
- Increase the spacecraft orbital energy, where the resulting SRP force is directed to the direction of the spacecraft geocentric velocity (as illustrated in Figure 1);
- (ii)
- Decrease the spacecraft orbital energy, where the resulting SRP force is directed in the opposite direction of the spacecraft geocentric velocity (as illustrated in Figure 2).

- Semi-major axis $\to {a}_{ref}$:
- -
- If $a\left(t\right)<{a}_{ref}$: implement Strategy (i);
- -
- If $a\left(t\right)>{a}_{ref}$: implement Strategy (ii);

- Eccentricity $\to {e}_{ref}$:
- -
- If $e\left(t\right)<{e}_{ref}$: implement Strategy (i) if the spacecraft geocentric eccentricity vector $\mathbf{e}$ points in the opposite direction of the Earth velocity and Strategy (ii) if otherwise;
- -
- If $e\left(t\right)>{e}_{ref}$: implement Strategy (i) if $\mathbf{e}$ points in the direction of the Earth velocity and Strategy (ii) if otherwise;

- Inclination $\to {i}_{ref}$:
- -
- If $i\left(t\right)<{i}_{ref}$: ${\delta}_{geo}=+{35}^{\circ}$ if the spacecraft geocentric true longitude is $\nu \approx {0}^{\circ}$ and ${\delta}_{geo}=-{35}^{\circ}$ if $\nu \approx {180}^{\circ}$;
- -
- If $i\left(t\right)>{i}_{ref}$: ${\delta}_{geo}=-{35}^{\circ}$ if $\nu \approx {0}^{\circ}$ and ${\delta}_{geo}=+{35}^{\circ}$ if $\nu \approx {180}^{\circ}$,

## 6. Results

- Initial orbit:
- -
- Semi-major axis: 41,904.1$\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$;
- -
- Eccentricity: $0.018$;
- -
- Inclination: $0.{688}^{\circ}$;
- -
- Ascending node: $-29.{8}^{\circ}$;
- -
- Argument of perigee: $7.{0}^{\circ}$.

- Initial data:
- -
- Total mass (vehicle + fuel): $300\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}$;
- -
- Magnitude of thrust available: $2\phantom{\rule{0.166667em}{0ex}}\mathrm{N}$.

- Imposed conditions of final (geostationary) orbit:
- -
- Semi-major axis: 42,164.2$\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$;
- -
- Eccentricity: $0.0$;
- -
- Inclination: $0.{0}^{\circ}$.

#### 6.1. Optimal Maneuver

- Constraint satisfaction tolerance: $0.03$;
- Initial guess:
- -
- Propulsion start: $100.{0}^{\circ}$;
- -
- Propulsion end: $110.{0}^{\circ}$;
- -
- Initial pitch angle: $180.{0}^{\circ}$;
- -
- Initial yaw angle: $-45.{0}^{\circ}$;
- -
- Initial pitch rate of change: $0.5$;
- -
- Initial yaw rate of change: $0.0$.

- Effectively achieved orbit:
- -
- Semi-major axis: 42,161.23$\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$;
- -
- Eccentricity: $0.000$;
- -
- Inclination: $0.{0}^{\circ}$;
- -
- Ascending node: $265.{5}^{\circ}$;
- -
- Argument of perigee: $93.{2}^{\circ}$;
- -
- True anomaly: $171.{8}^{\circ}$.

- Duration of the maneuver: 15,237.6$\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$.

#### 6.2. Suboptimal Maneuvers

#### 6.3. Solar Sail Spacecraft

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Orbital elements of the spacecraft over time. (Red line: initial value; Blue line: final value).

Maneuver | ${\mathit{s}}_{\mathit{s}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{s}}_{\mathit{e}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{A}}_{0}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{B}}_{0}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{A}}^{\prime}$ | ${\mathit{B}}^{\prime}$ | Duration (s) |
---|---|---|---|---|---|---|---|

Optimal | 249 | 290 | $-26.6$ | $55.7$ | - | - | 15,237.6 |

Linear | 235 | 282 | $-22$ | 45 | $0.687$ | $0.067$ | 17,121.3 |

Constant | 221 | 271 | $-12$ | 55 | 0 | 0 | 18,305.4 |

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## Share and Cite

**MDPI and ACS Style**

Meireles, L.G.; Gomes, V.M.; Prado, A.F.B.d.A.; Melo, C.F.d.
Optimum, Suboptimal and Solar Sailing Orbital Maneuvers for a Spacecraft Orbiting the Earth. *Symmetry* **2023**, *15*, 512.
https://doi.org/10.3390/sym15020512

**AMA Style**

Meireles LG, Gomes VM, Prado AFBdA, Melo CFd.
Optimum, Suboptimal and Solar Sailing Orbital Maneuvers for a Spacecraft Orbiting the Earth. *Symmetry*. 2023; 15(2):512.
https://doi.org/10.3390/sym15020512

**Chicago/Turabian Style**

Meireles, Lucas Gouvêa, Vivian Martins Gomes, Antônio Fernando Bertachini de Almeida Prado, and Cristiano Fiorilo de Melo.
2023. "Optimum, Suboptimal and Solar Sailing Orbital Maneuvers for a Spacecraft Orbiting the Earth" *Symmetry* 15, no. 2: 512.
https://doi.org/10.3390/sym15020512