## 3 Trajectory Optimization Techniques Overview
### Autonomous Navigation Models  
As discussed, PSO algorithms were used for trajectory optimization. PSO was first introduced in 1995 by [PERSON] and [PERSON] [71]. Among several metaheuristic algorithms available, PSO is the most frequently used technique, as demonstrated by several works that exploit this approach [72; 73; 74]. PSO is a metaheuristic algorithm with an enhanced ability to perform global optimization. The PSO comprises a fixed-size population of particles, N, that are candidate solutions to the problem which move inside the search domain, modifying their position through appropriate perturbation called velocity, which allows the particle to perform a displacement. The particles move iteratively until they converge onto a global optimal solution considering the goal of optimality and minimizing the cost function, which describes the quality of the solution and the imposed constraints. The position of the particles iterates according to:  
\[\mathrm{X}_{1}^{k+1}=\mathrm{X}_{i}^{k}+\mathrm{V}_{i}^{k+1} \tag{25}\]  
\[\mathrm{V}_{i}^{k+1}=\mathrm{V}_{i}^{k}+\mathrm{c}_{1}\cdot\mathrm{r}_{1} \cdot\left(\mathrm{p}_{i}^{k}-\mathrm{x}_{i}^{k}\right)+\mathrm{c}_{2}\cdot \mathrm{r}_{2}\cdot\left(\mathrm{p}_{g}^{k}-\mathrm{x}_{i}^{k}\right) \tag{26}\]  
where \(\mathrm{V}_{i}^{(k+1)}\) is the velocity required to move from \(\mathrm{k}^{\mathrm{th}}\) iteration to \((\mathrm{k+1})^{\mathrm{th}}\) iteration. \(\mathrm{p}_{i}^{k}\) = best position of particle i at time k,  
\(\mathrm{p}_{g}^{k}\) = global best solution for all particles at time k,  
\(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) = random numbers between 0 and 1,  
\(\mathrm{c}_{1}\) = cognitive parameter assigned with a value 2. This is a hyperparameter that enables defining the ability of the swarm to be influenced by the best local solutions found over the iterations. \(\mathrm{c}_{2}\) = scaling parameter assigned with a value 2. This is a hyperparameter that enables defining the ability of the swarm to be influenced by the best global solutions during the iterations. In our case, the PSO implementation requires an appropriate spacecraft dynamics model, optimality criteria, and constraints that fulfil the collision avoidance requirements. Typically, satellite motion in an orbit can be modelled using the classical orbital elements based on Gaussian variational equations. However, using these equations will result in ambiguity, especially for orbits with low eccentricities or inclinations [75]. In order to avoid this ambiguity, a new model that employs a set of modified equinoctial elements (MEE) developed in [76; 77] is used to solve the low thrust transfer problem. The MEE parameters, in terms of the classical orbit parameters, are used as inputs for the PSO algorithm and can be written as follows:  
\[\mathrm{p}=\mathrm{a}\left(1-\mathrm{e}^{2}\right) \tag{27}\]  
\[\mathrm{f}=\mathrm{e}\cos(\omega+\Omega) \tag{28}\]  
\[\mathrm{g}=\mathrm{e}\sin(\omega+\Omega) \tag{29}\]  
\[\mathrm{h}=\tan(\frac{\mathrm{i}}{2})\cos(\Omega) \tag{30}\]\[\mathrm{k}=\tan\biggl{(}\frac{\mathrm{i}}{2}\biggr{)}\sin(\Omega) \tag{31}\]  
\[\mathrm{L}\;=\;\omega\;+\;\Omega\;+\;\theta \tag{32}\]  
where:  
p = semi-latus rectum,  
a = semimajor axis,  
e = orbital eccentricity,  
i = orbital inclination,  
\(\omega\) = argument of perigee,  
\(\Omega\) = RAAN,  
L = true longitude,  
f,g = x,y components of the eccentricity vector in the orbital frame,  
h,k = x,y components of the node vector in the orbital frame.