# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 2 Orbit Dynamics
### Modified Equinoctial Elements  
Modified equinoctial elements are used to describe the dynamics of the spacecraft, modeled as a point mass. The orbital motion of the vehicle is mainly affected by the Mars gravitational field. Therefore, the spacecraft dynamics can be investigated by employing a perturbed two-body problem model, with the inclusion of the most relevant orbit perturbations. Fundamental planetary parameters are reported in Appendix A. The space vehicle is assumed to be equipped with a steerable, throttleable low-thrust propulsion system to perform the orbit injection maneuvers. Nonlinear orbit control using low-thrust propulsion is being employed for the purpose of driving a spacecraft toward some operational conditions and maintaining it in the desired orbit, compensating for perturbations. The operational conditions can be expressed in terms of osculating orbit elements. Orbit dynamics can be described in terms of osculating orbit elements, i.e., semimajor axis \(a\), eccentricity \(e\), inclination \(i\), right ascension of the ascending node (RAAN) \(\Omega\), argument of periapsis \(\omega\), and true anomaly \(f\). However, the Gauss equations [16; 17], which govern the time evolution of the classical orbit elements, become singular in the presence of a circular or equatorial orbit (and when an elliptic orbit transitions to a hyperbola). For these reasons, the modified equinoctial elements [18; 19] are chosen, in conjunction with the semilatus rectum (parameter), used in place of \(a\). The six elements are defined as  
\[ \begin{array}{ll}x_{1}=a\big{(}1-e^{2}\big{)}&x_{2}=e\cos(\Omega+\omega)&x_{3 }=e\sin(\Omega+\omega)\\ x_{4}=\tan\frac{i}{2}\cos\Omega&x_{5}=\tan\frac{i}{2}\sin\Omega&x_{6}=\Omega+ \omega+f\end{array}  \tag{1}\]  
These elements are nonsingular for all Keplerian trajectories, with the only exception of equatorial retrograde orbits (\(i=\pi\)). If \(\eta:=1+x_{2}\cos x_{6}+x_{3}\sin x_{6}\), the instantaneous radius is \(r=x_{1}/\eta\). Letting \(z:=\big{[}x_{1}\quad x_{2}\quad x_{3}\quad x_{4}\quad x_{5}\big{]}^{T}\), the governing equations for the modified equinoctial elements can be written as  
\[\dot{z}=\mathrm{G}(z,x_{6})\mathbf{a} \tag{2}\]  
\[\dot{x}_{6}=\sqrt{\frac{\mu}{x_{1}^{3}}}\eta^{2}+\sqrt{\frac{x_{1}}{\mu}}\frac {x_{4}\sin x_{6}-x_{5}\cos x_{6}}{\eta}a_{h} \tag{3}\]  
where \(\mu\) represents the Martian gravitational parameter, whereas the expression of \(\mathrm{G}(z,x_{6})\) is [5; 14]  
\[\mathrm{G}=\sqrt{\frac{x_{1}}{\mu}} \begin{bmatrix}0&\frac{2x_{1}}{\eta}&0\\ \sin x_{6}&\frac{(\eta+1)\cos x_{6}+x_{2}}{\eta}&-\frac{x_{4}\sin x_{6}-x_{5} \cos x_{6}}{\eta}x_{3}\\ -\cos x_{6}&\frac{(\eta+1)\sin x_{6}+x_{3}}{\eta}&\frac{x_{4}\sin x_{6}-x_{5} \cos x_{6}}{\eta}x_{2}\\ 0&0&\frac{1+x_{4}^{2}+x_{5}^{2}}{2\eta}\cos x_{6}\\ 0&0&\frac{1+x_{4}^{2}+x_{5}^{2}}{2\eta}\sin x_{6}\end{bmatrix}  \tag{4}\]  
Vector \(\mathbf{a}\) represents the \((3\times 1)\)-vector of the non-Keplerian acceleration that affects the spacecraft's motion. Its components, denoted with \((a_{r},a_{\theta},a_{h})\), are the projections of \(\mathbf{a}\) into the local vertical local horizontal (LVLH) rotating frame aligned with \(\left(\hat{r},\hat{\theta},\hat{h}\right)\), where unit vector \(\hat{r}\) is directed toward the instantaneous position vector \(\mathbf{r}\) (taken from the Mars center), whereas \(\hat{h}\) is aligned with the spacecraft orbital angular momentum. Vector \(\mathbf{a}\) includes both the thrust acceleration and the perturbing acceleration generated by the space environment. These two contributions can be distinguished, i.e., \(\mathbf{a}=\mathbf{a}_{T}+\mathbf{a}_{P}\), where subscripts T and P refer respectively to thrust and perturbations. The perturbing acceleration is due to harmonics of the neopotential, third body gravitational attraction, and solar radiation pressure and must be projected in the LVLH-frame. Let \(T_{max}\) and \(\widetilde{m}_{0}\) represent the maximum available thrust magnitude and the initial mass.