# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Orbit Dynamics

#### 2.1. Modified Equinoctial Elements

**a**represents the $\left(3\times 1\right)$-vector of the non-Keplerian acceleration that affects the spacecraft’s motion. Its components, denoted with $\left({a}_{r},{a}_{\theta},{a}_{h}\right)$, are the projections of

**a**into the local vertical local horizontal (LVLH) rotating frame aligned with $\left(\widehat{r},\widehat{\theta},\widehat{h}\right)$, where unit vector $\widehat{r}$ is directed toward the instantaneous position vector

**r**(taken from the Mars center), whereas $\widehat{h}$ is aligned with the spacecraft orbital angular momentum. Vector

**a**includes both the thrust acceleration and the perturbing acceleration generated by the space environment. These two contributions can be distinguished, i.e., $a={a}_{T}+{a}_{P}$, where subscripts T and P refer respectively to thrust and perturbations. The perturbing acceleration is due to harmonics of the areopotential, third body gravitational attraction, and solar radiation pressure and must be projected in the LVLH-frame.

#### 2.2. Orbit Perturbations

- (a)
- harmonics of the areopotential, with coefficient $\left|{J}_{l,m}\right|\ge {10}^{-5}$ (i.e., ${J}_{2}$, ${J}_{3}$, ${J}_{4}$, ${J}_{2,2}$ and ${J}_{3,1}$),
- (b)
- third body gravitational pull due to the Sun, and
- (c)
- solar radiation pressure.

#### 2.3. Operational Orbit and Coverage Analysis

## 3. Nonlinear Orbit Control

#### 3.1. Operational Orbit

#### 3.2. Feedback Law and Related Stability Analysis

**0**. However, to be a Lyapunov function, $V$ must have a nonpositive time derivative, and this can be ensured through proper selection of the control action ${u}_{T}$. The $(3\times 1)$-vectors $b$ and $d$ are introduced as

**Proposition**

**1.**

**Proposition**

**2.**

- rectilinear trajectories ${x}_{1}=0$
- target set, i.e., $\psi =\mathbf{0}$

**d**, is to be analyzed, in order to verify the existence of possible singularity issues. Using the definitions of $\psi $ and G, the limit of ${\left[\left(\mathsf{\partial}\psi /\mathsf{\partial}z\right)\mathrm{G}\right]}^{-1}\left(\mathsf{\partial}\psi /\mathsf{\partial}t\right)$ as $\left\{i\to {i}_{d},\mathsf{\Omega}\to {\mathsf{\Omega}}_{d},e\to {e}_{d},p\to {p}_{d}\right\}$ yields three closed-form analytical expressions for the three components. The second and the third component of the vector ${\left[\left(\mathsf{\partial}\psi /\mathsf{\partial}z\right)\mathrm{G}\right]}^{-1}\left(\mathsf{\partial}\psi /\mathsf{\partial}t\right)$ turn out to tend to zero, while the first component tends to

## 4. Numerical Simulations

- $c=30$ km/s, effective exhaust velocity
- ${u}_{T}{}^{(max)}=5\xb7{10}^{-5}{g}_{0}({g}_{0}=9.8{\mathrm{m}/\mathrm{s}}^{2})$

#### 4.1. Orbit Injection with Temporary Propulsion Failure

#### 4.2. Orbit Injection with Dispersed Initial Conditions and Errors in Perturbation Estimation

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | reference surface |

${C}_{R}$ | solar radiation pressure coefficient |

${D}_{n}$ | nodal day |

${J}_{l,m}$ | coefficients of gravitational harmonics |

$M$ | mean anomaly |

${P}_{sr}$ | solar radiation pressure |

${R}_{M}$ | Martian equatorial radius |

$T$ | thrust |

${T}_{max}$ | maximum available thrust |

${T}_{n}$ | nodal orbital period |

$U$ | Martian gravitational potential |

$V$ | Lyapunov function |

$\mathbf{a}$ | non-Keplerian acceleration vector |

${a}_{d}$ | desired semimajor axis |

${\mathbf{a}}_{\mathbf{P}}$ | perturbation acceleration |

${\mathbf{a}}_{T}$ | thrust acceleration |

$e$ | osculating eccentricity |

${e}_{d}$ | desired eccentricity |

$f$ | true anomaly |

$i$ | osculating inclination |

${i}_{d}$ | desired inclination |

$l,m,n,s,q$ | modified equinoctial elements (MEE) |

$m$ | instantaneous mass of the vehicle |

${m}_{0}$ | initial spacecraft mass |

$\dot{m}$ | mass time rate |

$p$ | osculating semilatus rectum |

${p}_{d}$ | desired semilatus rectum |

$\mathbf{r}$ | position vector |

$r$ | radius |

${r}_{A}$ | apoapse radius |

${r}_{P}$ | periapse radius |

$t$ | actual time |

${t}_{f}$ | time of flight |

$\mathsf{\Omega}$ | osculating RAAN |

${\mathsf{\Omega}}_{d}$ | desired RAAN |

$\zeta $ | heading angle |

$\mu $ | Martian gravitational parameter |

$\nu $ | shadow function |

$\xi $ | absolute longitude |

$\varphi $ | latitude |

$\mathit{\psi}$ | vector of final conditions |

$\omega $ | osculating argument of periapse |

${\omega}_{d}$ | desired argument of perigee |

${\omega}_{M}$ | Martian angular rate |

## Appendix A. Planetary Parameters of Mars

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**Figure 10.**Maintenance phase: time histories of the thrust acceleration magnitude (

**top**) and the true anomaly (

**bottom**).

**Figure 14.**Monte Carlo analysis 1: radius (

**a**) and related zoom (

**b**); the dotted line represents the Martian radius.

Latitude (Deg) | 50 | 60 | 70 | 80 | 90 |
---|---|---|---|---|---|

$\mathrm{min}\left\{{\epsilon}_{min}\right\}$ (deg) | 0.4 | 9.7 | 19.2 | 29.0 | 38.7 |

$\mathrm{max}\left\{{\epsilon}_{min}\right\}$ (deg) | 1.0 | 10.3 | 19.8 | 29.4 | 38.7 |

Latitude (Deg) | 0 | 10 | 20 | 30 | 40 | 50 |
---|---|---|---|---|---|---|

${\epsilon}_{min}=0\mathrm{deg}$ | 902.6 | 902.9 | 595.4 | 566.3 | 311.5 | / |

${\epsilon}_{min}=5\mathrm{deg}$ | 909.6 | 908.1 | 699.1 | 592.8 | 394.6 | 71.6 |

Mean Value | Standard Deviation | |
---|---|---|

Time of convergence [days] | 87.53 | 1.57 |

Mass Ratio | 0.894 | 1.063 × 10^{−2} |

**Table 4.**Monte Carlo 2: mean value, standard deviations, and bounds for the stochastic initial orbit elements.

Parameter | Unit | Mean Value | Std Deviation | Lower Bound | Upper Bound |
---|---|---|---|---|---|

Periares radius | [km] | - | - | 3596 | 10,000 |

Apoares radius | [km] | - | - | 80,000 | 150,000 |

Inclination | [deg] | 92.29 | 5.00 | 87.29 | 97.29 |

RAAN | [deg] | 64.70 | 5.00 | 59.70 | 69.70 |

Argument of periares | [deg] | 342.39 | 5.00 | 337.39 | 437.39 |

True anomaly | [deg] | 180.00 | 5.00 | 175.00 | 185.00 |

Mean Value | Standard Deviation | |
---|---|---|

Time of convergence [days] | 77.10 | 4.46 |

Mass Ratio | 0.898 | 5.813 × 10^{−2} |

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**MDPI and ACS Style**

Santoro, R.; Pustorino, M.; Pontani, M.
Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control. *Aerospace* **2023**, *10*, 670.
https://doi.org/10.3390/aerospace10080670

**AMA Style**

Santoro R, Pustorino M, Pontani M.
Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control. *Aerospace*. 2023; 10(8):670.
https://doi.org/10.3390/aerospace10080670

**Chicago/Turabian Style**

Santoro, Riccardo, Marco Pustorino, and Mauro Pontani.
2023. "Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control" *Aerospace* 10, no. 8: 670.
https://doi.org/10.3390/aerospace10080670