# Periodic Solutions of Nonlinear Relative Motion Satellites

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## Abstract

**:**

## 1. Introduction

#### 1.1. Importance of Relative Motion Satellites

- The space rendezvous problem.
- The formation flight problem.

#### 1.2. Mathematical Models of Relative Motion Satellites

## 2. Formulation of Motion Model

- Phasing of maneuvers to accomplish a rendezvous in the timing sequence, which will bring the two satellites into close proximity.
- Maneuvers of finalizing rendezvous that involve the relative motion between the two satellites for rendezvous and docking.

#### 2.1. General Circular Relative Motion Satellites

#### 2.2. Linear Circular Relative Motion Satellites

## 3. Solutions of HCW Equations

#### Nonlinear Circular Relative Motion Satellites

## 4. Periodic Solution of Nonlinear Relative Motion Satellites

#### 4.1. Legitimacy of Perturbation Techniques

#### 4.2. Legitimacy of Relative Motion Equations

## 5. Periodic Solutions by the Lindstedt–Poincar é Technique

**Zero order solutions**

**First order solutions**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Solutions of linear relative motion in two dimensions. (

**a**) Analytical solution of linear relative motion at different phases. (

**b**) Numerical solutions by using numerical integration of explicit Euler method.

**Figure 3.**Solutions of linear relative motion in three dimensions (

**a**) Analytical solutions at different phases when ${\phi}_{0}=\pi /4$. (

**b**) Numerical solutions by using numerical integration of the explicit Euler method.

**Figure 4.**Solution of nonlinear relative motion in two dimensions. (

**a**) Solutions of the Lindstedt–Poincaré technique at different phases. (

**b**) Numerical solutions by integration of the explicit Euler method.

**Figure 5.**Solution of nonlinear relative motion in three dimensions; (

**a**) Solutions of the Lindstedt–Poincaré technique at different phases when ${\phi}_{0}=\pi /4$. (

**b**) Numerical solutions by integration of the explicit Euler method.

**Figure 6.**Comparison between analytical solutions of linear and nonlinear relative motion in two dimensions at different phases.

**Figure 7.**Comparison between analytical solutions of linear and nonlinear relative motion in three dimensions at different phases when ${\phi}_{0}=\pi /4$.

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**MDPI and ACS Style**

Pal, A.K.; Abouelmagd, E.I.; García Guirao, J.L.; Brzeziński, D.W.
Periodic Solutions of Nonlinear Relative Motion Satellites. *Symmetry* **2021**, *13*, 595.
https://doi.org/10.3390/sym13040595

**AMA Style**

Pal AK, Abouelmagd EI, García Guirao JL, Brzeziński DW.
Periodic Solutions of Nonlinear Relative Motion Satellites. *Symmetry*. 2021; 13(4):595.
https://doi.org/10.3390/sym13040595

**Chicago/Turabian Style**

Pal, Ashok Kumar, Elbaz I. Abouelmagd, Juan Luis García Guirao, and Dariusz W. Brzeziński.
2021. "Periodic Solutions of Nonlinear Relative Motion Satellites" *Symmetry* 13, no. 4: 595.
https://doi.org/10.3390/sym13040595