## 3 Trajectory Optimization Techniques Overview
### Autonomous Navigation Models
The following equations describe the inverse relationship amongst classical and modified equinoctial elements:  
\[\mathrm{a}=\frac{\mathrm{P}}{1-\mathrm{f}^{2}-\mathrm{g}^{2}} \tag{33}\]  
\[\mathrm{e}=\sqrt{\mathrm{f}^{2}+\mathrm{g}^{2}} \tag{34}\]  
\[\mathrm{V}_{i}^{k+1}=\mathrm{V}_{i}^{k}+\mathrm{c}_{1}\cdot\mathrm{r}_{1} \cdot\left(\mathrm{p}_{i}^{k}-\mathrm{x}_{i}^{k}\right)+\mathrm{c}_{2}\cdot \mathrm{r}_{2}\cdot\left(\mathrm{p}_{g}^{k}-\mathrm{x}_{i}^{k}\right) \tag{26}\]0  
\[\mathrm{V}_{i}^{k+1}=\mathrm{V}_{i}^{k}+\mathrm{c}_{1}\cdot\mathrm{r}_{1} \cdot\left(\mathrm{p}_{i}^{k}-\mathrm{x}_{i}^{k}\right)+\mathrm{c}_{2}\cdot \mathrm{r}_{2}\cdot\left(\mathrm{p}_{g}^{k}-\mathrm{x}_{i}^{k}\right) \tag{26}\]1  
\[\mathrm{V}_{i}^{k+1}=\mathrm{V}_{i}^{k}+\mathrm{c}_{1}\cdot\mathrm{r}_{1} \cdot\left(\mathrm{p}_{i}^{k}-\mathrm{x}_{i}^{k}\right)+\mathrm{c}_{2}\cdot \mathrm{r}_{2}\cdot\left(\mathrm{p}_{g}^{k}-\mathrm{x}_{i}^{k}\right) \tag{26}\]2  
\[\mathrm{V}_{i}^{k+1}=\mathrm{V}_{i}^{k}+\mathrm{c}_{1}\cdot\mathrm{r}_{1} \cdot\left(\mathrm{p}_{i}^{k}-\mathrm{x}_{i}^{k}\right)+\mathrm{c}_{2}\cdot \mathrm{r}_{2}\cdot\left(\mathrm{p}_{g}^{k}-\mathrm{x}_{i}^{k}\right) \tag{26}\]3  
The ECI state vector is expressed in terms of MEE using:  
\[\stackrel{{\rightarrow}}{{\mathrm{r}}}=\left\{ \begin{array}{c} \frac{\mathrm{r}}{\mathrm{s}^{2}}\bigl{[}\cos(\mathrm{L})+\alpha^{2}\cos( \mathrm{L})+2\mathrm{hksin}(\mathrm{L})\bigr{]}\\ \frac{\mathrm{r}}{\mathrm{s}^{2}}\bigl{[}\bigl{(}\sin(\mathrm{L})-\alpha^{2} \cos(\mathrm{L})+2\mathrm{hkcos}(\mathrm{L})\bigr{]}\\ \frac{2\mathrm{r}}{\mathrm{s}^{2}}\bigl{[}\mathrm{h}\mathrm{s}\mathrm{in}( \mathrm{L})-\mathrm{kcos}(\mathrm{L})\bigr{]}\end{array} \right\} \tag{39}\]  
\[\stackrel{{\rightarrow}}{{\mathrm{v}}}=\left\{ \begin{array}{c}- \frac{1}{\mathrm{s}^{2}}\sqrt{\frac{\mathrm{u}}{\mathrm{p}}}\bigl{[}\bigl{(} \sin(\mathrm{L})+\alpha^{2}\sin(\mathrm{L})-2\mathrm{hkcos}(\mathrm{L})+ \mathrm{g}-2\mathrm{fhk}+\alpha^{2}\mathrm{g})\bigr{]}\\ -\frac{1}{\mathrm{s}^{2}}\sqrt{\frac{\mathrm{u}}{\mathrm{p}}}\bigl{[}\bigl{(} -\cos(\mathrm{L})+\alpha^{2}\sin(\mathrm{L})+2\mathrm{hksin}(\mathrm{L})- \mathrm{f}+2\mathrm{ghk}+\alpha^{2}\mathrm{f})\bigr{]}\\ \frac{2}{\mathrm{s}^{2}}\sqrt{\frac{\mathrm{u}}{\mathrm{p}}}\bigl{[}\bigl{(} \mathrm{kcos}\mathrm{L}+\mathrm{ksin}(\mathrm{L})+\mathrm{fh}+\mathrm{gk}) \bigr{]}\end{array} \right\} \tag{40}\]  
where \(\alpha,\mathrm{s},\mathrm{r},\mathrm{q}\) can be expressed in terms of MEE as:  
\[\alpha^{2}=\;\mathrm{h}^{2}-\mathrm{k}^{2} \tag{41}\]  
\[\mathrm{s}^{2}=1+\mathrm{h}^{2}+\mathrm{k}^{2} \tag{42}\]  
\[\mathrm{r}\;=\;\frac{\mathrm{P}}{\mathrm{q}} \tag{43}\]  
\[\mathrm{q}=1+\mathrm{fcos}(\mathrm{L})+\mathrm{gsin}(\mathrm{L}) \tag{44}\]  
The motion of the spacecraft is modelled by the following second order differential equation:  
\[\frac{\mathrm{d}^{2}\stackrel{{\rightarrow}}{{\mathrm{r}}}}{ \mathrm{d}\mathrm{t}^{2}}+\frac{\stackrel{{\rightarrow}}{{ \mathrm{r}}}}{\mathrm{r}^{3}}=\stackrel{{\rightarrow}}{{ \mathrm{a}}}_{\mathrm{d}} \tag{45}\]where:  
r = magnitude of the inertial position,  
a\({}_{\rm d}\) = disturbing acceleration. The disturbing acceleration is replaced by \(\Delta\) when MEE are used. The perturbations will therefore be composed of the Earth effects and the applied thrust.