## 3 Trajectory Optimization Techniques Overview
### Autonomous Navigation Models
The disturbing acceleration is expressed in the rotating RSW frame whose principal axes are defined by:  
\[{\rm Q_{T}}=\left[\frac{\stackrel{{\rightarrow}}{{\rm r}}}{ \stackrel{{\rightarrow}}{{\rm r}}}\left[\frac{\stackrel{{ \rightarrow}}{{\rm r}}}{\stackrel{{\rightarrow}}{{\rm r}}} \times\stackrel{{\rightarrow}}{{\rm v}}\right)\times\stackrel{{ \rightarrow}}{{\rm r}}\right]\left[\stackrel{{ \rightarrow}}{{\rm r}}\times\stackrel{{\rightarrow}}{{\rm v}} \right] \tag{46}\]  
The J\({}_{2}\) perturbations are modelled using the following equations:  
\[{\rm f_{r}}=-\frac{3\mu{\rm J_{2}R^{2}}}{2r^{4}}\Bigg{[}1-12\frac{\left(\rm h \sin L-k\cos L\right)^{2}}{s^{4}}\Bigg{]} \tag{47}\]  
\[{\rm f_{s}}=-\frac{12\mu{\rm J_{2}R^{2}}}{2r^{4}}\bigg{[}\frac{\left(\rm h \sin L-k\cos L\right)\left(\rm h\cos L-k\sin L\right)}{s^{4}}\bigg{]} \tag{48}\]  
\[{\rm f_{w}}=-\frac{6\mu{\rm J_{2}R^{2}}}{2r^{4}}\Bigg{[}\frac{\left(\rm h \sin L-k\cos L\right)\left(1-k^{2}-h^{2}\right)}{s^{4}}\Bigg{]} \tag{49}\]  
The total perturbation, if the thrust applied in the low thrust manoeuvre is very small, can be expressed as:  
\[\stackrel{{\rightarrow}}{{\Delta}}=\stackrel{{ \rightarrow}}{{\Delta}}_{\rm pert}+\stackrel{{\rightarrow}}{{ \Delta}}_{\rm T} \tag{50}\]  
where \({\Delta}_{\rm pert}\) corresponds to J\({}_{2}\) and \({\Delta}_{\rm T}\) is thrust. The thrust acceleration \(\stackrel{{\rightarrow}}{{\rm a}}_{\rm T}\) can be expressed as:  
\[\stackrel{{\rightarrow}}{{\rm a}}_{\rm T}=\frac{\rm T}{\rm m} \stackrel{{\rightarrow}}{{\rm u}}_{\rm T}=\frac{\rm cn_{0}}{c-n_ {0}t}u \begin{bmatrix}\sin\alpha\cos\beta\\ \cos\beta\cos\alpha\\ \sin\beta\end{bmatrix}  \tag{51}\]  
where:  
\(\stackrel{{\rightarrow}}{{\rm u}}_{\rm T}=\) thrust control vector,  
T = thrust,  
c = effective exhaust gas velocity,  
n\({}_{0}\) = thrust to mass ratio at the initial time. The thrust control magnitude is expressed in the form of a square wave with frequency \({\rm f_{T}}\) and phase \({\rm k_{T}}\) to reduce the infinitely dimensional optimal control problem into a finite non-linear programming problem. \[{\rm u}=\mbox{ square }\left(\pi{\rm f_{T}}\frac{t}{{\rm t_{f}}}+{\rm k_{T}}\right) \tag{52}\]  
where:  
t\({}_{\rm f}\) = time duration of the total manoeuvre, which is determined by the optimization algorithm. The thrust vector angles are expressed using the following equations:  
\[\alpha(t)=\pi\alpha_{\rm M}\sin\frac{\pi}{2}\bigg{[}\bigg{(}\frac{t}{{\rm t_{f }}}\bigg{)}^{{\rm P_{1}}}+\bigg{(}\frac{t}{{\rm t_{f}}}\bigg{)}^{{\rm P_{2}}} \bigg{]} \tag{53}\]\[\beta(t)=\pi\beta_{M}\sin\frac{\pi}{2}\Bigg{[}\bigg{(}\frac{t}{t_{f}}\bigg{)}^{P1}+ \bigg{(}\frac{t}{t_{f}}\bigg{)}^{P2}\Bigg{]} \tag{54}\]  
where \(\alpha_{M}\), \(\beta_{M}\), \(P_{1}\), \(P_{2}\) are the control parameters for the thrust orientation control. The thrust acceleration vector in the RSW frame is expressed as:  
\[\overset{\simeq}{a}_{T}^{RSW}=\frac{T}{m} \begin{bmatrix}\sin\alpha\cos\beta \\ \cos\beta\cos\alpha\\ \sin\beta\end{bmatrix}  \tag{55}\]  
The thrust control vector can be transformed to the ECI frame as follows:  
\[\overset{\simeq}{u}_{T}^{ECI}=Q_{T}\overset{\simeq}{u}_{T}^{RSW} \tag{56}\]  
Since the satellite operates in low earth orbit, the effects of aerodynamic drag must be considered. MSIS90 atmospheric model was employed in the PSO algorithm [78].The drag in RSW frame is expressed as:  
\[ \begin{bmatrix}\Delta_{D_{R}}\\ \Delta_{D_{S}}\\ \Delta_{DW}\end{bmatrix} =\frac{1}{2}\rho SC_{D}V_{R} \begin{bmatrix}v_{r}\\ v_{s}\\ 0\end{bmatrix}  \tag{57}\]  
where:  
\(\rho\) = atmospheric density,  
S = aerodynamic reference area,  
\(C_{D}\) = drag coefficient,  
\(V_{R}\) = velocity magnitude. The velocity components \(v_{r}\), \(v_{s}\) are expressed as:  
\[v_{r}=\sqrt{\frac{\mu}{P}}(f\sin L-g\cos L) \tag{58}\]  
\[v_{s}=\sqrt{\frac{\mu}{P}}(1+f\cos L-g\sin L) \tag{59}\]