## 1 Introduction
The uniform trigonometric method (UTM) has been used to optimize the re-entry trajectory of reusable launch vehicles, and SRM characteristics and trajectory are optimized using a hybrid method that combines GA and SQP methods [21; 22]. In the last decades, the particle swarm optimization (PSO) technique, which is also a class of heuristic methodology, has become prominent. Simplicity is the main reason for using a swarm algorithm (SA). Because the basic version of the PSO appears very intuitive, it is considerably easier to program and implement. Although it is computationally expensive in comparison with gradient-based methods, in the literature it is reported as being more efficient than GAs because of the decreased number of function evaluations [23; 24]. The adoption and further exploration of the PSO in various fields highlight its potential for addressing complex optimization challenges. As researchers continue to refine and adapt PSO techniques to specific application domains, its versatility and effectiveness are likely to be more widely recognized and integrated into diverse optimization problems in the future. Despite its promising features, few researchers have concentrated on PSO to use it in launch vehicle design optimization problems [25; 26; 27]. These studies collectively investigate the application of the PSO to optimize ascent trajectories of multi-stage launch vehicles, including addressing ascent phase trajectory optimization problems and enhancing trajectories for vehicles equipped with multi-combined cycle engines. Each of these studies delves into the application of the PSO in optimizing launch vehicle trajectories for different purposes. In this study, the optimization objective is the determination of hybrid motor characteristics and some flight path parameters such as thrust vector control (TVC) angle to obtain a target trajectory for a specified payload mass (PM). To accomplish this task, the nonlinear coupling between aerodynamics, propulsion, structure, and trajectory subsystems is considered in the six-degrees-of-freedom (6 DOF) simulation. Aerodynamic coefficients and structural properties are modeled using Data Compendium (DATCOM) v2011 and Computer-Aided Three-Dimensional Interactive Application (CATIA) V5R19, respectively. The method was applied to the Minotaur-I launch vehicle. Reference is made to the Minotaur-I launch vehicle, which has a total of four stages as solid, e.g., M55A1 and SR19 as the first and second stages used in Minuteman missiles, Orion 50 XL and Orion 38 as the third and fourth stages taken from the Pegasus rocket. The hybrid rocket motors considered in the present analysis use cryogenic LOX, which requires low values of the mixture ratio to give high \(l_{sp}\) when burning with fuels containing hydrocarbons, and paraffin-based solid \(F\) as a propellant combination, which presents a good \(l_{sp}\), a large regression rate coefficient (\(n\)), higher performance, and has a blowdown feed system [10]. The subsystems are updated in each iteration step based on the hybrid rocket propulsion characteristics and geometry, according to the optimization steps, leading to the attainment of the optimum solution. The literature gap in the launch vehicle design and optimization can be filled by modeling the interaction between aerodynamics, propulsion, structure, and trajectory subsystems in 6 DOF simulation. Additionally, the performance of launch vehicles can be increased by using hybrid propulsion systems, which have lower production and manufacturing costs than other types of propulsion systems. In the literature, there are articles in which simulations are conducted using 6 DOF equations of motion. However, in recent literature studies involving optimization with the use of a 6 DOF trajectory model, it is assumed that aerodynamic coefficients, center of gravity, and mass or moment of inertia values remain constant throughout the optimization process.