## 1 Introduction
Considering these attributes, it is posited that the integration of HRMs as a stage of a launch vehicle can significantly enhance the overall performance of launch vehicles. The historical dominance of solid and liquid propulsion in space exploration has resulted in a wealth of knowledge and established technologies, creating a gravitational pull towards these proven systems. The preferences and investments of the space industry, including governmental space agencies and private companies, can strongly influence research priorities, with a bias towards traditional technologies. Additionally, the perception of higher risk and less maturity associated with hybrid rocket technology, especially in multi-stage setups, may make researchers and funding agencies more risk-averse. The limited operational examples of multi-stage launch vehicles utilizing hybrid rocket motors further contribute to a hesitancy among researchers and engineers to delve into this less-traveled domain. Despite these challenges, the dynamic nature of the rocket propulsion environment indicates that research interests are shifting and a focus on hybrid rocket mo \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Solid** & **Liquid** & **Hybrid** \\ \hline Simplicity & + & \(-\) & \(-\) \\ Reliability & + & N/A & + \\ Storage & + & \(-\) & N/A \\ Thrust & High & N/A & Limited \\ Operational Flexibility & + & \(-\) & + \\ Controllability & \(-\) & + & N/A \\ Restart Capability & \(-\) & + & + \\ Specific Impulse & Low & High & N/A \\ Safety & N/A & \(-\) & + \\ Cost-effectiveness & N/A & N/A & + \\ Grain Robustness & N/A & N/A & + \\ Eco-friendly & N/A & N/A & + \\ Performance & N/A & N/A & + \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of advantages and disadvantages of solid, liquid, and hybrid rocket motors. tors has begun, as technologies advance, the industry needs to evolve, and early adopters demonstrate success in the field of hybrid propulsion for multi-stage launch vehicles [11]. Therefore, the optimization studies conducted for multi-stage launch vehicles in the literature are quite limited due to the complexities they entail. Specifically, the number of publications on hybrid rocket motors used in multi-stage launch vehicles is significantly less compared to those on solid- and liquid-fueled rockets [11]. Among the challenges encountered in these studies, obtaining high \(I_{sp}\) over a minimum of 280 s is deemed most critical. Some examples of the optimization problem objectives studied are payload maximization, gross lift-off weight (GLOW) minimization, and \(F\) consumption minimization [12; 13; 14]. The studies discussed trajectory optimization and guidance methods for an advanced launch system, likely focusing on techniques to optimize the path and control of launch vehicle trajectories, as well as the optimization of launch vehicle ascent trajectories under path constraints and coast arcs, exploring methods to enhance the efficiency and performance of launch trajectories. In order to solve these optimization problems, there is a variety of solution methods. For example, genetic algorithms (GAs) have mostly been used to obtain orbit, vehicle weight, stage length (\(L^{*}\)), and total cost [15; 16]. This approach leverages computational methods to efficiently explore and refine design variables, contributing to the development of more efficient and cost-effective launch systems. Researchers continue to refine and combine optimization techniques like GA with other algorithms such as sequential quadratic programming (SQP) to further enhance trajectory design and performance across various phases of flight. The SQP method is used by many researchers to find the flight path, motor characteristics, and maximum angle of attack (\(\alpha\)) [17; 18; 19; 20]. These interdisciplinary approaches underscore the importance of computational optimization in advancing the precision and effectiveness of launch vehicle design and operations.