Design optimization of a low-cost three-stage launch vehicle with modular hybrid rocket motors

This article investigates the impact of modular propulsion system design on the performance and cost of a three-stage hybrid rocket. Furthermore, it conducts a multi-objective optimization of unit payload cost, take-off mass, and payload mass ratio, considering factors such as the number of motors and layout considerations. The optimization design scheme for the three-stage hybrid rocket is divided into four cases. In the first case, each stage is equipped with a fixed single motor, and each stage is independently optimized without modular design. The second case considers the use of multiple motors in the first and second stages, still without modular design. The third case also involves multiple motors in the first and second stages, but all motors in each stage have identical parameters except for the nozzle expansion ratio, implementing a modular design. In the fourth case, the number and layout of the motor design method are the same as those in the third case, with independent optimization in the third stage using partial modular design. The results indicate that the unit payload cost of the multi-motor non-modular design case can be reduced by 13.12% compared to the single-motor non-modular design case. Within the modular case, the full modular design case is slightly inferior to the partial modular design case. Based on the above data, it can be concluded that the first and second stages of modular rockets offer the best performance and the lowest cost.


Introduction
Modular methods present substantial benefits in terms of reducing the development cycle and minimizing repetitive tasks in space mission development.Furthermore, modularity is extensively applied in aerospace transportation tools, including missiles and launch vehicles, where diverse module components can be pre-manufactured and stored in a module library, effectively reducing redundant development costs [1][2] .
The research focus of this article is a hybrid rocket motor, which emerges as a suitable choice for a modular spacecraft due to its simple structure and distinct modules.The hybrid rocket motor combines solid fuel with liquid oxidant [3][4] .Figure 1 depicts the typical structure of a hybrid rocket.Concerning the modular optimization design for hybrid rockets.Kanazaki et al. [5] compared non-cluster schemes for motor modules and concluded that the overall optimization scheme using clusters would diminish the rocket's launch performance.However, they did not take into account different structural layouts and cost factors.The article is structured as follows: The first section introduces the concept of modular ideals and explains that this article will focus on the optimization design of hybrid rockets.The second section applies multi-disciplinary optimization theories to develop models for propulsion, structure, aerodynamics, trajectory, and cost disciplines, and it presents the optimization methods employed in this study.The third section proposes an optimized modular rocket scheme.The fourth section presents the optimization results for the mathematical model introduced in the second section and the optimization plan proposed in the third section.Finally, the fifth section analyzes and summarizes the calculation results, concluding.

Description of the multidisciplinary problem
In this article, our focus is on a three-stage low earth orbit (LEO) launch vehicle with hybrid rocket motors, aimed at reaching an orbital height of 300 km.Subsequently, we develop multiple calculation models for various disciplines.

Propulsion design
The design section for hybrid rocket propulsion facilitates the determination of motor characteristic parameters such as mass, specific impulse, and thrust.To fulfill the modular manufacturing requirements, a circular-shaped grain is chosen.Furthermore, grain combustion follows the assumption of parallel layer combustion in solid rocket motors.Additionally, a mixture of 98% hydrogen peroxide (H 2 O 2 ) and hydroxyl-terminated polybutadiene (HTPB) is utilized as the oxidant and fuel, respectively.The initial input parameters consist of the initial thrust (F i ), initial combustion chamber pressure (P ci ), initial oxygen-to-fuel ratio (α i ), and nozzle expansion ratio (ε).Utilizing these parameters along with the chemical composition of the propellant, thermal calculations were conducted using RPA software to determine the specific impulse (I s ), characteristic velocity (c * ), adiabatic combustion temperature (T f ), and specific heat ratio (k).
The differential equation [4] used to calculate the pressure in the combustion chamber can be obtained from Equation (1), 0.2 *

( ) (
) Where P c is the combustion chamber pressure, V c is the inflation volume of the combustion chamber, which is the free space of the combustion chamber.R g is the universal gas constant; A b is the combustion area; A p is the area of the combustion channel.

Structure design
The overall structural layout of the launch vehicle follows the conventional three-stage rocket design.The structural calculation primarily involves determining parameters such as takeoff mass, structural dimensions, and layout based on the motor design outcomes.Furthermore, the overall layout of the launch vehicle consists of two main sections: the motor structure and other components.The propulsion system incorporates a pressurized feeding system, while the motor structure encompasses a combustion chamber, nozzle, oxidant tank, pipelines, gas bottles, motor casing, and various cables.Additionally, the launch vehicle comprises other structural elements, including interstage segments, flight control components, satellite rocket adapters, payloads, and a fairing.

Material selection.
To reduce the mass of the rocket while maintaining structural integrity, various components such as the gas bottle, tank, rocket skin, combustion chamber shell, and nozzle are constructed using carbon fiber composites.The tank is equipped with a rubber capsule lining, the thrust chamber insulation layer is composed of high-silicon oxygen material, the pipelines and valves are fabricated from aluminum alloy, and the catalytic bed consists of nickel-based silver-plated material, while its shell is made of stainless steel.

Mass estimation.
The calculation of motor mass is predicated on the following assumption: within a pressurized feeding system, the gas is assumed to be in a state of adiabatic expansion within the gas bottle, and the transportation of oxidant in the tank is considered to be an isobaric process.The mass of compressed gas and the volume of gas bottles and tanks are determined through the application of the first law of thermodynamics, the ideal gas state equation, and the law of mass conservation.To establish the diameter of the combustion chamber, the outer diameter of the propellant column is taken into account, accounting for the thickness of the insulation layer and shell.The thickness of all shells is computed using the third strength theory, followed by the derivation of mass parameters based on the chosen material density.Components not explicitly mentioned in the motor adhere to the same principles as other rocket components, and their estimation is derived from engineering experience.

Structural layout.
An optimized design scheme that applies the same structural layout to the third stage of the launch vehicle is accepted.This layout includes two oxidant tanks and two gas bottles, which are connected in parallel around a central thrust chamber, which is illustrated in Figure 2. The first and second stages employ a sequential structure consisting of a thrust chamber, an oxidant tank, and a gas bottle, according to the modular design requirements.The structural layout can range from a single thrust chamber to multiple thrust chambers, as depicted in Figure 3.For subsequent layouts with 5 or more thrust chambers, a central surround structure will be employed.

Aerodynamic design
The field of aerodynamics utilizes estimation methods, while the atmospheric environmental parameters are based on the 1976 U.S. standard atmospheric parameters [6] .Considering the structural similarity to the "Titan II," its lift coefficient and drag coefficient [7] are employed, which are specified in Table 1.

Aerodynamic coefficient Velocity range Formula
3.55

Trajectory design
A three-degree-of-freedom particle trajectory model that considers the Earth's rotation is employed.The integral equations governing the system are presented below.
Where g is the gravitational acceleration, a e is the inertial acceleration, and a c is the Coriolis acceleration.r 0 is the position of the launch point, V is the velocity vector in the inertial coordinate system of the launch point, a is the acceleration in the inertial coordinate system of the launch point, V 0 is the initial velocity of the carrier, m t is the real-time mass, P is the thrust (including control force) in the inertial coordinate system of the launch point, which can be obtained by the dynamics discipline through coordinate transformation of the thrust (including control force) in the projectile coordinate system without considering thrust eccentricity deviation, N F is the aerodynamic force in the inertial coordinate system of the launch point and obtained through the discipline of aerodynamics, and ṁ t is the real-time propellant mass flow rate.
The flight trajectory profile is illustrated in Figure 4.

Cost design
The cost calculation primarily takes into account the interstage structure, cables, payload attach fitting, head cones, and attitude control costs.The estimation of the cost model for these components is based on their mathematical relationship with mass.Due to the use of a hybrid rocket motor, the cost model is divided into solid and liquid parts [8][9] [10] , = The liquid portion of the cost (C liquid ) is composed of the cost of the oxidant tank (C tank ), the cost of the gas bottle (C gasbottle ), the cost of the propellant (C p ), and the cost of the pipeline and valve (C 1 ).The cost of the tank and gas bottle is calculated by considering the surface area (S t ).
For non-modular rocket motors, the cost of solid components is calculated using a component-wise approach, Where C p is the cost of solid grain, based on the mass of the solid grain and the complexity of the manufacturing of the grain; C c is the cost of the motor shell, which is related to the manufacturing process, materials, and shell diameter; C n is the cost of the nozzle, which is related to the selected nozzle type, material, nozzle length, and expansion ratio; C s represents the cost of motor skin, which is mainly related to the selected material and mass.
The solid part of the modular motor adopts an overall estimation method based on the mass of the solid rocket motor, as shown below, , , ln ln ln 2 ln 2 ( ) ( ) 0.623 0.377 -0.3387 0.5126 0.6167 ( )= 1.6680 1.3867 Where C s, dry is the net weight of the combustion chamber shell, C s,p is the cost of solid propellant, L C is the learning factor, C s1 is the coefficient, W sol is the total mass of the motor, and N N is the number of nozzles.The cost of modularization is adjusted based on expert experience, so the cost of modularized motors is calculated as follows, Where c model is the modular cost coefficient, and c 1 and c 2 are the shape and unit price factors of the grain, respectively.

Description of the optimization problem
This section presents a comprehensive introduction to the optimization design scheme, design variables, process constraints, and target indicators within the context of the previously mentioned multidisciplinary computing model.

Optimal design problem statement
The optimization design scheme for the three-stage hybrid rocket is divided into four cases.Case 1 involves a fixed single motor for each stage, with independent optimization for each stage and no modular design.Case 2 incorporates a single motor in the third stage, while the first and second stages utilize multiple motors.The number and layout of these motors are optimized as variables, with each stage independently optimized without modular design.Case 3 also includes a single motor for the third stage, with multiple motors in the first and second stages.The number and layout of motors are considered, and all motors in each stage have identical parameters except for the nozzle expansion ratio.A modular design is adopted in this case.In Case 4, the number and layout of motors are taken into account, and the parameters of each motor in the first and second stages are identical except for the nozzle expansion ratio.In this case, the third stage is independently optimized, and a partial modular design is implemented.
The detailed design variables and their corresponding ranges are displayed in Table 2.The same design variables and ranges are used in the modular design motor.The number of motors is treated as a discrete variable determined through traversal.

Optimization constraints and objectives
The optimization problem entails multi-objective optimization, to minimize the unit payload cost (C U ) and takeoff mass (M total ), while maximizing the payload mass (M payload ) ratio (η), which is obtained by (10).These objectives are utilized to holistically evaluate the performance characteristics and cost advantages of modular launch vehicles.

Optimization algorithm
In this article, we adopt the NSGA2 optimization algorithm, which treats the multi-objective problem as a frontier solution.The optimization equation is given by (11), which includes variables, constraints, and a target.In this equation, g1-g3 represent the axial overload constraint for the three stages, g4-g6 represent the thrust-to-weight ratio constraint for the three stages, g7 represents the interstage ratio constraint between the 1st stage and the 2nd stage, g8 and g9 represent the orbit height constraint, g10 represents the maximum dynamic pressure constraint, g11 represents the slenderness ratio constraint, and h represents the program turn end angle of attack constraint.

Pareto optimal front
The Pareto optimal front for the four cases is displayed in Figure 5.The upper left corner represents the optimal three-dimensional dominance relationship under the three objective functions.To facilitate analysis and data observation, three sets of two-dimensional projection solutions are provided for different combinations of objectives, as depicted in Figure 5 to Figure 8.The range of objective function solutions for the four cases can be found in Table 3.     [11] .This cost falls within the range of the unit payload costs presented in Table 3, indicating that hybrid rockets offer certain cost advantages.However, it should be pointed out that due to numerous factors such as market fluctuations, the more important aspect of this article is the relative cost comparison between the four optimization cases.
As this article specifically examines small launch vehicles in low Earth orbit, the payload mass ratio is considerably lower compared to heavy launch vehicles at the same orbital altitude.Nonetheless, for a three-stage solid launch vehicle of similar scale and target orbit, such as Kuaishou-1, the payload mass ratio is around 0.9%.While hybrid rockets require additional oxidizer tanks and pipelines, resulting in a slightly lower payload mass ratio than solid launch vehicles, the difference is relatively comparable.
Comparing Case 1 with Case 2 in Figure 6 to Figure 8 reveals that arranging discrete motor units can reduce takeoff mass, decrease unit payload cost, and improve the payload mass ratio.Analyzing the constraints, Case 1's slenderness ratio result closely approaches the constraint boundary, while Case 2 still possesses some margin.In contrast to using a single motor in each stage, this design reduces the overall slenderness ratio of the launch vehicle, resulting in a decrease in the overall mass of the shell and other components, ultimately delivering superior outcomes.
When contrasting modular Cases 3 and 4 with Case 2, it is apparent that they have lower unit payload costs.This cost efficiency is attributed to the significantly reduced manufacturing expenses associated with modular batch production.However, the simultaneous optimization using the same motors throughout all stages, or solely in the first or second stage, unavoidably compromises the individual capabilities of each stage.Consequently, the motors in each stage are not independently optimized, resulting in a slightly lower payload mass ratio when compared to Case 2.
Upon comparing modular Case 3 with Case 4, it can be observed that Case 4 exhibits a slight superiority in Table 3.This arises due to the limited thrust of the third-stage motor, employing the same motor design across all three stages leads to significant performance losses.

Optimal results
The primary objective of this study is to examine the overall impact of modular launch vehicles on enhancing carrying capacity and reducing costs.To achieve this, the cost-optimal solution is selected among the four cases of Pareto frontier solutions.The detailed design variables, constraints, and optimization solutions for these four cases are presented in Table 4 4, detailing the number of motors at each stage.Because the motors within the same stage of the rocket possess identical designs, the thrust curve corresponds to the combined thrust of all motors in that particular stage.This is demonstrated in Figure 9 to Figure 14.When comparing Case 1 and Case 2, disregarding modularity, it is evident that stages equipped with multiple motors have a higher thrust compared to those with a single motor.The operating duration of first-stage motors in Case 2, Case 3, and Case 4 accounts for 67.45%, 74.06%, and 69.95% of Case 1, respectively.Similarly, the operating duration of second-stage motors is 96.84%, 56.49%, and 53.39% of Case 1, respectively, showcasing the properties of high thrust and short operational time.
In the comparison between the non-modular Case 2 and the modular Cases 3 and 4, it is evident that the third-stage motor thrust in Case 3 is significantly higher than that in the other three cases.This can be attributed to the fact that all stages in Case 3 utilize the same motor, resulting in capacity limitations at each stage.Additionally, the second-stage motor thrust in Case 3 and Case 4 is also significantly higher than in the non-modular cases.Therefore, adopting a modular design involves making a tradeoff, where the thrust performance is slightly compromised in exchange for achieving a completely consistent design for all stages of the motors.

Trajectory characteristic.
The trajectory results of the four cases align with the motor thrust curve.Specifically, the modular design case exhibits significantly higher thrust during the second stage compared to the non-modular motor case.This is evident in Figure 10, which displays the altitude-time curve.Within the first 200 seconds, the climb height of the modular case (Case 3 and Case 4) is lower than that of the non-modular case (Case 1 and Case 2).However, during the operation of the secondstage and third-stage motors, the climb height of the modular cases surpasses that of the non-modular cases.This observation can be attributed to the synchronous optimization design of the three-stage rockets.
The velocity-time curve depicted in Figure 12 illustrates that Case 2 features the briefest free glide phase, lasting 116 seconds.In the modular setups, particularly Cases 3 and 4, synchronous optimization of motor parameters across the first and second stages is employed.As a result, these cases demonstrate decreased flight durations and quicker orbit entry times, amounting to 412.9 seconds and 417.4 seconds, respectively.A comparison between Case 3 and Case 4 illustrates that the modular advantage primarily arises from the synchronous motor design in the first and second stages.However, it is worth mentioning that Case 3, which involves a three-stage design, slightly trails behind Case 4 in terms of cost and performance.

Conclusion
This article investigates the modular optimization design of hybrid rocket motor-powered small LEO launch vehicles, focusing on unit payload cost, takeoff mass, and payload mass ratio as design objectives.
The optimization results demonstrate that employing discrete multiple motors in each stage can effectively reduce structural loads and lower flight costs.While the modular design exhibits significant advantages in terms of cost discipline, it does sacrifice overall performance due to the synchronous optimization of Case 3 and Case 4. A comparison between Case 3 and Case 4 reveals that the modular launch vehicles' first and second stages outperform the synchronous optimization of all three stages.Consequently, this article concludes that the optimal cost and performance balance can be achieved by focusing on the first and second stages of modular hybrid rocket-powered small launch vehicles.These findings provide valuable reference and design insights for future flights and modular rocket designs.

Figure 1 .
Figure 1.The typical layout structure of a hybrid rocket.The article is structured as follows: The first section introduces the concept of modular ideals and explains that this article will focus on the optimization design of hybrid rockets.The second section applies multi-disciplinary optimization theories to develop models for propulsion, structure, aerodynamics, trajectory, and cost disciplines, and it presents the optimization methods employed in this study.The third section proposes an optimized modular rocket scheme.The fourth section presents the optimization results for the mathematical model introduced in the second section and the optimization plan proposed in the third section.Finally, the fifth section analyzes and summarizes the calculation results, concluding.

Figure 2 .
Figure 2. The structural layout at the third stage.The first and second stages employ a sequential structure consisting of a thrust chamber, an oxidant tank, and a gas bottle, according to the modular design requirements.The structural layout can range from a single thrust chamber to multiple thrust chambers, as depicted in Figure3.For subsequent layouts with 5 or more thrust chambers, a central surround structure will be employed.

Figure 3 .
Figure 3.The structural layout at the first and second stages.
Where C is the component cost, M is the component mass, and a 1 and b 1 are regression coefficients.

Figure 9 .
Figure 9.The thrust time curve at the first stage.Figure 10.The trajectory time-height curve.

Figure 10 .
Figure 9.The thrust time curve at the first stage.Figure 10.The trajectory time-height curve.

11 Figure 11 .
Figure 11.The thrust time curve at the second stage.Figure 12.The trajectory time-velocity curve.

Figure 12 .
Figure 11.The thrust time curve at the second stage.Figure 12.The trajectory time-velocity curve.

Figure 13 .
Figure 13.The thrust time curve at the third stage.Figure 14.The trajectory time-flight path angle curve.

Figure 14 .
Figure 13.The thrust time curve at the third stage.Figure 14.The trajectory time-flight path angle curve.

Table 2 .
Description and ranges of variables.

Table 3 .
The multi-objective optimization solution range.Currently, the unit payload cost for SpaceX's Falcon 9 in low Earth orbit is approximately $2720 per kilogram

Table 4 .
and Table5.In comparison to the Case 2, Case 3, and Case 4 demonstrate reductions of 13.12%, 32.17%, and 35%, respectively.Furthermore, modular Case 3 and Case 4 exhibit reductions of 21.91% and 25.18% respectively, in comparison to Case 2. Cost optimal solution.

Table 5 .
Constraints and targets.Taking into account the discrete variable of the number of motors per stage and employing a traversal algorithm, it was determined that a well-designed structural layout is of utmost importance.The optimization outcomes for the various cases are presented in Table