## 3 Optimization of the Hybrid Last-Stage Launch Vehicle  
Launch vehicle optimization is the process that enables the efficient and effective design and operation of launch vehicle systems used in space exploration and payload launches. It involves adjustments and improvements to ensure that the launch vehicle systems function in the most efficient manner possible during space research and payload launches. The main concept behind optimization problems is to determine optimization design parameters until the objective function reaches the desired minimum or maximum values. In the specific case of the hybrid last stage, the objective of the optimization is to determine the characteristics of the hybrid rocket motor such as throat diameter and certain flight path parameters, such as the TVC angle, to achieve a target trajectory for a specified PM. The PSO algorithm was selected as the optimization method. The PSO algorithm falls under the category of swarm intelligence methods [50]. The use of SAs is motivated by their simplicity, robustness, and accuracy [51]. They are reported to be more efficient than GAs in terms of function evaluation and ease of implementation, while PSO is computationally more expensive compared to gradient-based methods [46; 47]. However, there is a research gap in the application of PSO, particularly with a coupling strategy, in the field of hybrid small satellite launch vehicles. Therefore, the PSO algorithm is considered a promising and improvable optimization method. PSO is a metaheuristic optimization algorithm inspired by the social behavior of birds and fish. In PSO, a population of potential solutions, called particles, moves through the solution space to find the optimal solution. Each particle represents a potential solution and is characterized by its position and velocity in the search space. The PSO algorithm operates through a dynamic iterative process. It begins with the initialization of a population of particles, each representing a potential solution to the optimization problem, and random assignments of positions and velocities. Parameters such as the number of particles, maximum iterations, and control factors are defined to guide the algorithm's behavior. In each iteration, the particles update their velocities based on a formula considering their past velocities, personal best positions, and the best position found by any particle in the swarm. These updated velocities are then used to adjust the particles' positions. The algorithm continually evaluates the fitness of these new positions, updating personal bests and the global best solution if improvements are found. The process iterates for a predefined number of iterations or until convergence is achieved, ultimately producing the best solution found as the output [23]. The hyperparameter settings for PSO include a swarm size of 100 particles to balance exploration and exploitation. The inertia weight starts around 0.9 and linearly decreases to 0.4 over iterations, regulating the balance between the particle's current and historical velocities. Acceleration coefficients are set to 2.0, controlling the influence of the particle's best known position and the global best known position on its movement. The optimization problem for launch vehicles, specifically those with a modified last stage featuring hybrid propulsion, falls into the category of multi-objective design optimization. The subsystems within the launch vehicle are interrelated and exhibit coupling effects. For instance, the center of gravity location and inertia properties are calculated based on the new geometry obtained from CATIA, using the optimization design parameters. To ensure effective and realistic applications, the use of multi-objective multidisciplinary design optimization (MDO) algorithms is crucial for achieving more efficient and accurate optimization [52; 53; 54].