## 3 Trajectory Optimization Techniques Overview  
Spacecraft trajectory optimization is of paramount importance due to its direct influence on the system performance [54]. It is one of the highly studied problems in the field of aerospace engineering [55; 56]. An optimal trajectory for a spacecraft can be defined as the one that meets some criteria, including initial and final conditions. The first serious attempt to classify these approaches was conducted back in 1998 by [PERSON] [57], known as the direct and indirect methods. Both branches aim to minimize cost functions and constraint through discrete approximations. The direct methods transcribe the continuous optimal control problem into an optimization problem [58] adopting the state-space representation. The system equations are satisfied by integrating them implicitly or explicitly within a finite interval, thereby converting it into a non-linear programming problem (NLP) [59]. Although the solution obtained from direct methods is not necessarily optimal, its simple implementation, reduced problem size, and larger domain of convergence makes this method a widely used one [60]. One of the direct methods is referred to as the shape based method [61; 62]. In the method, the state variables are interpolated, and the control variables are taken in the objective function. This is followed by a gradient-based technique or metaheuristic to minimize the cost by varying the state variables. The most popular technique among the direct methods is the Fourier series, specifically for low-thrust trajectory optimization applications [63; 64; 65]. The solution obtained by the shape-based method satisfies the equation of motion and the boundary conditions. Moreover, the solution obtained from the shape-based method can serve as a very good guess for other optimization approaches. Gradient-based methods such as NLP appear to be the most popular computational techniques for addressing trajectory optimization problems. The NLP uses the gradient information and is often capable of quick convergence and accurate results, leading to its popularity. However, the most noticeable limitation of gradient-based methods is that the analyst must have some a priori knowledge of the optimal trajectory. In other words, the initial guess should be near to the global optimal solution. Failing to do so will lead to a non-global optimal solution or the optimization routine will become extremely slow. An entirely different class of optimization approaches are referred to as metaheuristics, an alternative approach that addresses the issues faced by gradient-based methods. Essentially, a metaheuristic is an iterative approach to exploring and exploiting the search space that intelligently combines different concepts that are inspired by phenomena occurring in nature for guiding a subordinate heuristic [66]. These algorithms belong to a special class of artificial intelligence (AI) techniques, which also include evolutionary algorithms such as genetic algorithms (GA), PSO, and other insect/swarm inspired algorithms [67]. The metaheuristic approaches are classified into two types, single solution algorithms and population based algorithms, the latter are often used for spacecraft trajectory optimization problems [68; 69]. The second branch of optimization approaches is that of indirect methods. Indirect methods consider the dualized form of the equations, including the states and co-states within the discretized time frame [65]. Although the indirect methods are better than direct methods in terms of accuracy, they have certain limitations, such as:  
* The necessary conditions, including the co-state differential equations, the Hamiltonian, and the optimality conditions, must be expressed analytically. * Due to discretization of co-states, the problem size becomes large. * The analyst must guess certain aspects of the solution, such as portions of the time domain containing constrained or unconstrained control arcs.