Multidisciplinary Design Optimization of a Two-Stage LOX/Methane Partially Reusable Microlauncher

Abstract

With support from the Romanian Nucleu Program, INCAS has taken the initiative to develop a multidisciplinary optimization (MDO) environment capable of generating reusable microlauncher concepts that could be used as the first building blocks in the development and production of a locally based launch vehicle. This paper will present the main work carried out towards the accomplishment of this task, with several mathematical models being proposed to fill in the gaps associated with microlauncher reusability. Towards the end of the paper, a partially reusable microlauncher concept is proposed, which is capable of successfully accomplishing both the main mission, that of inserting a 100 kg satellite into a 400 km altitude, circular polar orbit, and the recovery mission, where the first-stage assembly (including the interstage) is recovered from a secondary location. Preliminary cost estimates are also given for the entire lifespan of the microlauncher, focusing on the economic feasibility of a reusable concept for small launch vehicles.

1. Introduction

Worldwide, an increase can be seen in the quantity of resources allocated to space programs, mainly for the development of space launchers, which are designed to transport a payload (satellite) safely from the ground into the desired orbit. A niche in the fleet of space launchers currently on the market is represented by launchers dedicated to satellites with small masses and dimensions. Small space launchers (known as microlaunchers) are designed to transport satellites whose masses are not large enough to justify launching them with medium-sized or even large launchers as secondary payloads.

Due to the increased competitiveness in the space launcher sector, the need to design a very efficient launcher in the preliminary phase is indisputable. In the case of small launchers, increased operational safety is desired, which correlates with a cost and a lift-off mass that are as low as possible. Traditionally, small satellites, whose masses are below 100 kg, are transported as secondary payloads alongside the main satellite, for which the exploitation mission has been defined (“piggyride” missions [1,2,3,4]). Due to their low priority, small satellites must be re-adapted to the main mission, with there being cases in which the desired orbit cannot be achieved and the secondary satellite is forced to be inserted into an inferior orbit. This drawback is the main reason for the need to design, manufacture and operate microlaunchers, vehicles that are already in various stages of development worldwide (mostly expendable).

In the last ten years, significant progress has been made by research institutes and private companies in the recovery of the first stages of launchers, which has led to a reduction in launch costs. We consider vertical-landing recovery technology to be the basis of future space launchers, whether they are of medium/large or small dimensions. In recent years, private American companies (SpaceX, Blue Origin and Rocket Lab) have made significant progress in the field of reusability, which has led to a reduction in launch costs.

With the introduction of the concept of reusability in the context of microlaunchers, there is a need to assess new research directions to be followed in order to achieve the safe recovery of the major components of vehicles (usually the first stages). Although some knowledge has been acquired in this field over time, it is clear that at the European level, there is a knowledge gap associated with space applications of the reusable launcher type.

One of the projects incorporated in the National Romanian Nucleu Program is intended to provide insight into the generation of constructive solutions for a family of reusable microlaunchers. Possible future operations with such vehicles to be conducted from Romania have also been considered, which would increase the visibility of the National Institute for Aerospace Research “Elie Carafoli” (INCAS), as well as that of the entire national research and development field. The results obtained in this project will have a high impact on defining future space strategies and technologies and could form the basis of the next research projects launched by INCAS, together with the European Space Agency (ESA) and other European partners.

Internationally, numerous studies are underway to investigate the feasibility of building and commercially operating a small, reusable launcher, the most advanced level of maturity being represented by the Electron vehicle, developed by Rocket Lab [5], an American company operating in New Zealand.

At the European level, the level of maturity reached by reusable launchers is low compared to that in the USA. However, there are ongoing projects for the design and development of similar constructive solutions. One of the projects with the highest chances of success in Europe is the reusable launcher MIURA 5 [6], developed by the Spanish company PLD Space. In this project, the use of a parachute and subsequent recovery of the lower stage from the water is envisioned.

Another massive project underway at the European level is the design and development of the CALLISTO lower-stage demonstrator [7], carried out by the DLR, CNES and JAXA consortium. For this vehicle, the recovery mode is a return to the launch area and a possible landing on a barge in the ocean.

The future vision of some of the major players in the field of European space launchers (CNES, ArianeGroup and ONERA), supported by funding from the European Space Agency, is the development of the reusable lower-stage demonstrator THEMIS [8]. Speaking strictly of reusable small launchers, the following space vehicles are in various stages of design and development at the European level: Orbex Prime, RFA One, Skykora XL and Maia [9].

A common trend among all of the reusable microlauncher studies underway at a European level is that of reducing the complexity of the launch vehicle. In the case of a microlauncher, this corresponds to using a constant-diameter architecture, the number of stages being as low as possible. As a single stage-to-orbit vehicle is not quite yet possible from a structural point of view, a two-stage architecture will be used as a baseline in this study. Following the global trend in favor of liquid propellant rocket engines (high specific impulse), this paper analyzes configurations based on these engines, a propellant pair of LOX/methane being used as reference.

2. Multidisciplinary Design Optimization Approach

Maybe one of the quickest and most efficient ways of obtaining a partially reusable microlauncher concept is by using a multidisciplinary design optimization approach (MDO). Several papers from recent literature have also analyzed the possibility of using an MDO approach with regard to the optimization of a reusable launch vehicle, the results being favorable [10,11,12].

Previous work performed at the National Institute for Aerospace Research “Elie Carafoli” (INCAS) has investigated the optimization of expendable microlaunchers, the most recent paper being [13]. Of interest is now, of course, to expand the applicability of the MDO approach to reusable microlaunchers; in this paper, we study the possibility of optimizing a partially reusable microlauncher, where only the first stage is recovered by means of autonomous landing.

The block scheme of the MDO algorithm proposed in the current paper is presented in Figure 1. In addition to the generation of a microlauncher preliminary concept that minimizes a set objective function (for the current case it will be related to mass minimization while successfully accomplishing both imposed missions), the MDO algorithm also generates reference trajectories for the launcher and recoverable first stage to follow, together with a cost estimative for the entire lifespan of the launcher.

A short explanation of how the developed MDO tool works will now be given to provide the reader a better understanding of the optimization process that occurs over time. The most important mathematical models used in each of the MDO main modules will be detailed in the following sections.

The optimization of the partially reusable microlauncher is performed by obtaining an optimization variable vector, following the use of the solution selection and advancement algorithm based on the evaluation of an imposed objective function. The solution is considered optimal upon convergence of the MDO algorithm, when the objective function has not improved after a specified number of iterations. It is important to state the fact that no optimization method can find the “optimal” result for complex problems such as the one analyzed in this paper (microlauncher optimization), the realistic objective of the MDO tool being to generate a microlauncher with very high performances which are close to the optimal ones (but which cannot be rigorously demonstrated mathematically).

After the final optimization variable vector has been obtained, the output data of the developed program (written in Matlab version R2024a [14]) are saved, the preliminary graphical concept of the launcher is generated, the numerical data representing the microlauncher performance databases are post-processed, the graphs of interest regarding the reference trajectories are realized and the cost estimation database is generated.

The choice of optimization variables is made in accordance with the mission requirements and its imposed launcher architecture. The launcher and its performances can be completely defined with the aid of the optimization variables and the global input data with the help of the mathematical models integrated in the main MDO modules. The disciplinary analysis is performed in a cascade sequence as seen in Figure 1, the core of the program being made up of five main modules:

• Preliminary design;
• Propulsion;
• Aerodynamics;
• Trajectory;
• Cost estimation.

Within the Preliminary design module, the microlauncher is sized both globally and at the level of major assemblies and subassemblies. At the same time, its mass estimate is also made, based on the breakdown scheme implemented. Within the Propulsion module, the propulsive performances of the liquid propellant rocket engines are determined. Within the Aerodynamics module, the aerodynamic characteristics of the small launcher and the recoverable first stage configurations are determined.

Within the Trajectory module, the evolution of the microlauncher and its lower stage is simulated. The flight simulator is based on a simplified three degrees of freedom dynamic model (3DOF), the developed Matlab code being additionally capable of optimizing the launcher’s ascent trajectory and the lower stage’s recovery trajectory. Within the Cost Estimation module, a list of preliminary costs is generated for the development, production and operation of the small space launch vehicle, ending with obtaining a total cost per launch and a total price per launch valid for the considered operational lifetime.

To reduce the computational effort, the Cost Estimation module is called only after the convergence of the MDO algorithm, the effective optimization loop containing only the main modules of Preliminary design, Propulsion, Aerodynamics and Trajectory, as shown in Figure 1. This is due to the lack of any output data of the Cost Estimation module that is needed later within the MDO algorithm (inside the Objective function module).

If integrating the impact of the launcher cost within the objective function is desired, then the Cost Estimation module can be easily introduced within the optimization loop of the MDO algorithm, but, for the current paper, this aspect is being envisioned as not necessary, a direction of mass optimization of the generated constructive solution being preferred such as in papers [15,16].

The mathematical models developed for each main module are independent; thus, five individual computational codes are developed (one for each main module of the MDO), which are then integrated within the final MDO tool. The order in which the modules are assessed is very important, mainly because input data for latter modules are derived as output data from previous modules. Along with the five main modules listed above, within the architecture of the developed multidisciplinary optimization algorithm, the following secondary modules are also needed:

• Requirements and input data;
• Optimization variables;
• Objective function;
• Selection and advancement algorithm.

2.1. Preliminary Design Module

The estimation of the dimensions and masses of the main components of the microlauncher starting from a reduced number of input data and optimization variables is performed in the first main module of the MDO algorithm (block scheme described in Figure 1). In this module, a bottom-up strategy is implemented, the masses and dimensions of the major components of the launcher being individually computed, so that by summing all of the subassemblies, one can obtain the mass and dimensions of each stage, of the upper structure and finally of the entire microlauncher.

The breakdown scheme of an n-stage launcher can be observed in Figure 2 [16,17]. Here, the existence of two independent structures is observed, namely a lower structure made up of n stages and an upper structure dedicated to the satellite and avionics.

As mentioned in the Introduction Section of this paper, of great interest are microlauncher configurations with a two-stage architecture, and thus n = 2 for the scheme presented in Figure 2. In this case, the first stage can be referred to as the lower stage (which will be recovered), while the second stage can be referred to as the upper stage (which will not be recovered).

To be implemented in the MDO algorithm, the mathematical model used for the launcher preliminary design (sizing and weight assessments) must be

• Robust, so that it can be used regardless of the selected optimization variables;
• Fast, so that it does not require a high computational time;
• Accurate, so that the results correspond to a correct estimate of the size and mass of the launcher constructive solution.

Therefore, analytical and semi-empirical, closed-relation models are preferred, which offer a high-accuracy first estimation after a very low computational time.

The upper structure is mandatory regardless of the type of microlauncher (reusable or expendable) or stage number and consists of the following components [16,17,18]:

• Payload (can be one or more satellites);
• Payload adapter;
• Vehicle Equipment Bay (VEB) that includes avionics and additional electrical systems required for the main mission;
• Fairing.

The payload mass and its dimensions are considered input data, being defined before running the multidisciplinary optimization algorithm. Thus, only the last three components need to be sized accordingly, with the aid of simplified mathematical models.

If the specifications of the adapter used are not known prior to the microlauncher design, a semi-empirical mathematical model is implemented, where the payload adapter mass is dependent on the payload mass [18] and its height is based on the maximum diameter of the payload and a structural complexity factor [16,17].

The avionics and additional electrical power systems (EPS) are located in the VEB area, which is integrated either inside the payload adapter or inside the upper stage. For microlaunchers, using an architecture where the payload adapter houses the VEB is feasible, the additional length of the VEB area being this way negligible. However, the VEB mass cannot be neglected and is approximated according to [19] based on a formulation dependent on the dry mass value of the launch vehicle.

The fairing geometry is defined based on several predefined input data (such as fineness ratio, preferred fairing profile, tip bluntness ratio) taking also into consideration the interior space necessary to safely house the payload. Based on the results of works [16,17], one can use a simplified relation to estimate the fairing mass, which is computed based on the lateral surface area of the fairing.

For the classical, expendable launcher concept, both stages will have the same type of internal components (more details in paper [17]), while for the case of a partially reusable microlauncher, things are completely different, the two stages having distinct architectures. First, the upper stage is not recovered, arriving together with the satellite in the target orbit and then being de-orbited, disintegrating during the destructive reentry into the atmosphere [20]. Thus, the number of internal components of the expendable stage is reduced. For the case of the lower stage, since it is subject to the recovery process, its internal architecture is different compared to an expendable stage, requiring new critical assemblies without which the stage recovery could not be carried out safely [10].

In the case of the lower structure, the dimensions and masses of each stage are computed individually, their contributions being summed up at the end to obtain the final constructive solution of the microlauncher. For an expendable stage which incorporates a liquid propellant rocket engine, the breakdown scheme used in the Preliminary design module is presented in Figure 3a. Additionally, a simplified graphical representation of the main components modeled in the MDO tool is presented in Figure 3b.

Many mathematical models from the literature are implemented in this subpart of the Preliminary design module to asses each individual component depicted in Figure 3a. The propellant (fuel and oxidizer) masses are computed using the following relations:

$$M_{oxidizer}={\displaystyle \frac{R_m}{R_m+1}}{M}_{propellant}; M_{fuel}={\displaystyle \frac{1}{R_m+1}}{M}_{propellant}.$$

(1)

where $M_{propellant}$ is the total stage propellant mass (considered an optimization variable) and the optimal mixture ratio between oxidizer and fuel $R_m$ is estimated according to the model detailed in Section 2.2 through the following approximation:

$$R_m=a+b\cdot {P_c}^c+d\cdot {P_e}^e$$

(2)

where $P_c$ is the combustion chamber pressure (considered an optimization variable), $P_e$ is the exhaust pressure (considered an optimization variable) and ($a$,$b$,$c$,$d$,$e$) are the approximation model coefficients, dependent on the oxidizer–fuel pair used [21].

The sizing of individual stage components is performed using analytical and semi-empirical mathematical models available in the literature [1,15,16,17,18,19,22], a short overview being given here for each main component.

For the tanks (both oxidizer and fuel tanks use a 2000 series aluminum alloy as material [23]), the mathematical model computes the following data: required volume [15], operating pressure [22], tank thickness [17], tank shape [17], lateral surface area [17] and finally its mass [1]. Regarding the shape of the fuel/oxidizer tanks, two possible scenarios occur. For the sizing of a large tank (typical for launcher lower stages which are more elongated), the output geometry is that of a cylindrical tank with spherical caps at the ends. If the length of the tank is not large enough to allow the use of a cylindrical portion and spherical end caps, then the output geometry of the tank is spherical (typical for launcher upper stages which do not have high length/diameter ratios).

The main role of the feed system is to increase the propellant pressure from the existing value in the tanks to the required value in the combustion chamber. The usual constructive solution of the feed system is the one based on centrifugal turbopumps, being implemented on most small, medium and large launchers. Based on existing works [15,17,22], a semi-empirical model has been implemented, modeling the fuel and oxidizer turbopumps separately. This mathematical model is detailed in [17] and computes the following data: mass flow rate, turbopump power requirement and finally its mass.

The liquid propellant rocket engine is a highly complex assembly. Depending on the combustion chamber pressure, the mass of the main subassemblies (combustion chamber $M_{{combustion}_{chamber}}$ and nozzle $M_{nozzle}$) is increased by a correction factor ${\xi }_{engine}$ between 2.5 and 5 (according to [15,22]) to include the mass of any additional components (injector, ablative shield, etc.). The final mass of the engine assembly can be estimated using

$$M_{engine}={\displaystyle \frac{M_{{combustion}_{chamber}}+M_{nozzle}}{{\xi }_{engine}}}$$

(3)

with the correction factor ${\xi }_{engine}$ having the following form [15]:

$${\xi }_{engine}=\left\{\begin{array}{cc}0.2& ,\mathrm{i}\mathrm{f} P_c\le 20 bar\\ {\displaystyle \frac{0.2P_c+2}{30}}& ,\mathrm{i}\mathrm{f} 20 bar<P_c<50 bar\\ 0.4& ,\mathrm{i}\mathrm{f} P_c\ge 50 bar\end{array}\right.$$

(4)

For the sizing of the combustion chamber, the model presented in [15,22] is implemented, where the following data are computed: critical area of the nozzle and corresponding diameter, the length of the combustion chamber (which is based on the characteristic length $L^{*}$, for which a conservative value was used $L^{*}=1.51$ [17,24]), the combustion chamber cross-sectional area and its corresponding diameter, the thickness of the combustion chamber wall (a cost-effective solution of stainless steel is implemented as a preliminary material) and, finally, its mass.

For a liquid propellant rocket engine, different types of nozzles can be used, depending on the requirements of the launcher mission: conical nozzle, bell nozzle (partial, total or double), multi-position nozzle, aerospike, etc. Details regarding the constructive solutions, advantages and disadvantages of each type of nozzle are presented in [22,25,26,27,28,29]. For the case of microlaunchers, the primary selection criterion is most often its costs. From the list of nozzles mentioned above, the simplest technical architecture which has an associated low cost corresponds to the conical nozzle, being implemented in the current paper. The sizing process of the conical nozzle is presented in detail in [17] and computes the following data: the nozzle expansion ratio, the cross-sectional area, the length of the nozzle (a nozzle half-angle value of 15° is used [15,17]), the thickness of the nozzle (material similar to the combustion chamber) and, finally, its mass.

The mass of any additional components inside an expendable launcher stage (Thrust Vector Control System, internal pipes, exterior shell, actuators, etc.) is computed using

$$M_{ad}=M_{propellant}\left(-2.3·{10}^{-7}{M}_{propellant}+0.07\right)$$

(5)

An extra dry mass of 5% has been implemented inside the Preliminary design module of the MDO algorithm for an expendable stage as a safety margin, together with a 10% length safety margin. These are used only in the preliminary stages of design [16].

Within the lower structure of a partially reusable microlauncher, the integration of at least one reusable stage is necessary. Since the microlauncher concept investigated in this work is based on a two-stage architecture, the lower stage (first stage) is the one that is subject to the recovery process.

The complexity of a reusable stage is much higher than an expendable one, additional systems being necessary to successfully recover the stage. The breakdown scheme used in the Preliminary design module for a reusable microlauncher stage is shown in Figure 4a.

Besides all of the components associated with a standard, expendable stage mentioned earlier in Figure 3, the architecture of a reusable microlauncher stage must contain a reusable concept dry mass contribution; the following five additional systems (shown in Figure 4b) needs to be integrated, as each one of them has a specific task to accomplish:

• An aerodynamic control system (ACS) is needed to modify the attitude of the first stage at low altitudes (a grid fin-based system is implemented);
• An extra-atmospheric control system is needed to change the attitude of the first stage at high altitudes (a cold gas thruster-based Reaction Control System (RCS) is implemented);
• An enlarged interstage (which is used to include the new RCS and ACS);
• A landing system is needed to safely land the first stage (a foldable system is envisioned to reduce its aerodynamic impact during the ascent phase of the microlauncher);
• A heat shield placed on the bottom part of the first stage is needed such that the reentry phase does not thermally damage it.

The first major system required for lower-stage recovery is the aerodynamic control system (ACS), which is essential for guiding the lower stage to the landing site at low altitudes, where the Earth’s atmosphere is dense. Grid fins are preferred over conventional (planar) fins, as they are very efficient for the typical flight profiles of space launchers [30,31,32]. Thus, a solution based on four grid fins positioned in an X configuration has been implemented in the MDO algorithm, similar to the Falcon 9 launch vehicle [33].

Based on internal sizing studies realized in INCAS, simple formulas can be used to preliminary estimate the dimensions and total mass of the ACS. For the ACS height computation, the following relation is proposed:

$$H_{ACS}=k_{ACS}·D_{stage}$$

(6)

where $D_{stage}$ is the outer diameter of the stage (considered an optimization variable) and $k_{ACS}$ is a scale factor (the value 0.65 is used).

To estimate the mass of the aerodynamic control system $M_{ACS}$ [kg], the following relation is proposed:

$$M_{ACS}=a·{D_{stage}}^k$$

(7)

where a = 77.1 [kg/m], $D_{stage}$ is measured in [m] and the exponent $k$ is dependent on the size of the launch vehicle. For launchers with $D_{stage}<1 \mathrm{m}$, a value of $k=2.9$ has provided excellent results, while for bigger launchers a value of $k=2.3$ is more realistic. An intermediary value of $k=2.5$ is suggested to be used as a preliminary value, being subject to further investigations during advanced phases of microlauncher design.

The second major system required for lower-stage recovery is the extra-atmospheric reaction control system (RCS), which is primarily used to change the attitude of the lower stage after separation from the microlauncher assembly at very high altitudes. The most common solution for such a control method is that of a system based on cold gas thrusters; for the current paper a solution based on nitrogen is implemented, which is readily available on the market (and easily storable) [34]. The architecture of the implemented RCS is based on a constructive solution with four gas expulsion zones, of which two zones have three thrusters (for roll and pitch control) and the other two simpler zones have only one thruster each (for yaw motion).

Based on internal sizing studies realized in INCAS, simple formulas can be used to preliminary estimate the dimensions and total mass of the RCS. For the RCS mass computation, the following relation is proposed:

$$M_{RCS}=k_{RCS}·\left(M_{{gas}_{RCS}}+M_{{tank}_{RCS}}\right)$$

(8)

where $M_{{gas}_{RCS}}$ is the pressurized gas (nitrogen), $M_{{tank}_{RCS}}$ is the mass of the gas tank(s) and $k_{RCS}$ is a scale factor which includes the contribution of auxiliary RCS components such as valves, circuits, nozzles, sensors, actuators, connecting supports and other smaller parts (the value of $k_{RCS}=1.5$ is proposed for a reusable first stage).

The mass of the pressurized gas can be approximated using

$$M_{{gas}_{RCS}}={\displaystyle \frac{I_{RCS}}{I_{SP}·g_0}}$$

(9)

where $I_{RCS}$ is the impulse generated by the RCS, $I_{SP}$ is the specific impulse of the gas used and $g_0$ is the standard gravitational acceleration [35].

A sensitive parameter of the proposed model is that of estimating the required first-stage impulse needed to be generated by the RCS ($I_{RCS}$). This total impulse can be split into pre-programmed impulses $I_{{RCS}_{prog}}$ and impulses required for any unwanted rotation correction $I_{{RCS}_{per}}$ (various major and minor perturbations during the trajectory propagation). Thus, one can write

$$I_{RCS}=I_{{RCS}_{prog}}+I_{{RCS}_{per}}$$

(10)

The RCS impulses required for both wanted and unwanted stage rotations are modeled as follows:

$${I_{RCS}}_{prog}=J_{stage}·{\omega }_{prog}; {I_{RCS}}_{per}=J_{stage}·{\omega }_{per}$$

(11)

where $J_{stage}$ is the moment of inertia of the lower stage with respect to the relevant axis and ${\omega }_{prog}$ and ${\omega }_{per}$ [rad/s] are angular velocities.

The moment of inertia for the lower stage can be precisely computed only after all the internal components are placed in the correct positions and cannot be estimated with a high degree of accuracy at this point in the design. However, if we use an engineering assumption, namely that the mass of the stage is uniformly distributed over the shape of a cylinder, we can express the term $J_{stage}$ as being one of the following, depending on the relevant rotation axis:

$$\begin{gathered} {J_{stage}}_{roll}={\displaystyle \frac{1}{2}}·m_{stage}·{r_{stage}}^2; \\ {J_{stage}}_{pitch\_yaw}={\displaystyle \frac{1}{4}}·m_{stage}·{r_{stage}}^2+{\displaystyle \frac{1}{12}}·m_{stage}·{L_{stage}}^2 \end{gathered}$$

(12)

where $m_{stage}$ is the mass of the stage (which changes over time when fuel or pressurized gas is consumed; conservatively one can use the maximum mass of the recovery configuration, that from the moment of stage separation), $r_{stage}$ is the radius of the stage (external) and $L_{stage}$ is the length of the stage (with interstage).

The most important RCS maneuver needed to recover the first stage is the flip-over maneuver, which needs two RCS pitch impulses, one at the start and one at the end of the maneuver (details in Section 2.4.2). Since the completion time of the flip-over maneuver is not a strictly imposed criterion, one can imagine using a low-intensity rotation movement, in the order of 5–10°/s (or around 0.1 rad/s), so that the rotation is performed within a range of 10–30 s (at least for the case of rotation at the apogee of the trajectory, when we are in an extra-atmospheric area and the aerodynamic impact is reduced, so the stage is not prone to accelerated destabilization due to the rapid change in attitude). This assumption is considered valid, as a more realistic way to estimate the impulse requirement for the RCS sizing is not available in the early phases, such as that of the multidisciplinary pre-sizing/optimization of a reusable microlauncher.

To be conservative and not underestimate the mass of the RCS, we will impose stricter requirements related to the number of impulses generated by the RCS during the entire flight profile. The final first-stage impulse needed to be generated by the RCS ($I_{RCS}$) can be estimated using

$$I_{RCS}={J_{stage}}_{pitch\_yaw}·\left(n_1·{\omega }_{prog}+n_2·{\omega }_{{per}_{large}}\right)+{J_{stage}}_{roll}·\left(n_3·{\omega }_{{per}_{small}}\right)$$

(13)

where $n_1$ is the number of RCS impulses considered for pitch rotations (set as $n_1=4$ to be conservative), $n_2$ is the number of RCS impulses considered for yaw rotations (set as $n_2=4$), $n_3$ is the number of RCS impulses considered for roll rotations (set as $n_3=4$), ${\omega }_{prog}$ is the programmed angular rate for the pitch rotations (${\omega }_{prog}=0.1 [\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}]$), ${\omega }_{{per}_{large}}$ is the angular rate used to nullify any perturbation on the yaw axis (${\omega }_{{per}_{large}}=0.1 [\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}]$) and ${\omega }_{{per}_{small}}$ is the angular rate used to nullify any perturbation on the roll axis (${\omega }_{{per}_{small}}=0.05 [\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}]$).

Having now access to the amount of pressurized cold gas required (computed with Equations (9)–(13)) we can proceed to evaluate the mass of the tank(s) required for the RCS. The mass of the RCS tank is approximated using similar relations to that used for the oxidizer and fuel tanks, the main difference being the material used for the tank (Ti-6Al-4V [36]) and tank pressure (around 240 bar [37]).

Regarding the space requirement for the RCS within the interstage, it was observed that there is no need to create a separate area above or below the ACS, as the RCS components can be integrated alongside those of the aerodynamic control system, so that a dedicated RCS height no longer needs to be defined. This will result in no additional changes to the adapter height (in addition to its increase for the integration of the ACS).

The third major system needed for first-stage recovery is an enlarged interstage. The interstage is usually a cylindrical structure (if the joined stages are of the same diameter, otherwise frustoconical if the lower stage has a larger diameter) that connects two consecutive stages of a launch vehicle. This interstage provides protection for certain components during flight, including auxiliary parts such as cable connectors, pipes, and fasteners, but also critical components such as the upper-stage posterior part of the engine/nozzle. In the case of a reusable stage concept, in addition to the components listed above, we need the interstage to also house all the additional components of the ACS and RCS.

Since the architecture considered in this paper is that of a two-stage constant-diameter microlauncher, the shape of our interstage is cylindrical and thus a simple mathematical model can be imagined to estimate its mass. The basic formula for the adapter mass $M_{interstage}$ is given by

$$M_{interstage}={\rho }_{interstage}·V_{interstage}·f_{auxiliary}$$

(14)

where ${\rho }_{interstage}$ is the density of the material used, $V_{interstage}$ is the volume of the interstage and $f_{auxiliary}$ is a correction coefficient that is used to include the mass of any auxiliary components that appear (a value of $f_{auxiliary}=1.1$ is proposed).

The volume for a thin-walled cylinder shell is computed as the difference between the volume of the outer and inner cylinders to account for the wall thickness, as follows:

$$V_{interstage}=\pi H_{interstage}\left({r_{ext}}^2-{r_{int}}^2\right)$$

(15)

where $H_{interstage}$ is the height of the interstage, $r_{ext}$ is the outer radius of the interstage and $r_{int}$ is the inner radius of the interstage.

To compute the outer and inner radius of the interstage, an engineering hypothesis can be used, namely that of the existence of a constant adapter wall thickness $t_{interstage}$:

$$r_{int}=r_{ext}-t_{interstage}$$

(16)

For the Falcon 9 launcher, the average shell thickness is approximately 4–5 mm [33]. The outer radius of the adapter is equal to the outer radius of the stage, being indirectly one of the optimization variables of the developed MDO algorithm (here the outer diameter of the stage is optimized, detailed in Section 2.6.2).

Since, in reality, an adapter concept also presents some additional stiffening elements [38], it is expected that using only a constant average shell thickness will underestimate the total mass of the adapter. According to [38], for an optimized interstage structure of a small launcher, the shell mass represents only 40% of the total interstage mass, the rest being associated with ribs, laminates and skin reinforcements.

Therefore, a correction factor will be applied to the average thickness to include the contribution of additional stiffening elements for the interstage structure. Taking into account the above information, an average thickness ($t_{interstage}$) of 10 mm (4 mm × 2.5 correction factor) is used to estimate the mass of the adapter for a reusable space launcher. A carbon-based composite material will be used as baseline (${\rho }_{interstage}$ ≈ 1750 kg/m³ [39]).

Since the expendable launcher stage concept already contained an adapter-type component (integrated in the additional components section of the list depicted in Figure 3), it is important to note that the introduction of the reusability concept from the perspective of the interstage height is reduced to

$${H_{interstage}}_{reusable}={H_{interstage}}_{clasic}+{{∆H}_{interstage}}_{reusable}$$

(17)

For the expendable microlauncher concept, the interstage height ${H_{interstage}}_{clasic}$ is equal to the length of the upper-stage nozzle. The remainder interstage height which is needed to house the ACS and RCS is computed with

$${{∆H}_{interstage}}_{reusable}=H_{ACS}·f_{safety\_margin}$$

(18)

where $f_{safety\_margin}$ represents a safety margin implemented to avoid collision between the ACS components and the upper-stage nozzle (a value of $f_{safety\_margin}$ = 1.1 is proposed) and $H_{ACS}$ is the height of the ACS (computed with Equation (6)).

The fourth major system needed to successfully recover the first stage of a reusable microlauncher is a landing gear system and based on internal studies in INCAS (where the materials used were aerospace grade aluminum (7000 series) and a carbon fiber-reinforced polymer (CFRP)); the following estimate is proposed for the height of the landing system:

$$H_{landing\_system}=k_L·D_{stage}$$

(19)

where $D_{stage}$ is the outer diameter of the stage (it is considered an optimization variable) and $k_L$ is a scale factor (the value 2.5 is used).

To estimate the mass of a foldable landing system $M_{landing\_system}$ for a reusable lower-stage concept, the following formula (power type) is proposed:

$$M_{landing\_system}=a·{D_{stage}}^k$$

(20)

where $a=124.45$ [kg/m], $D_{stage}$ is measured in [m] and the exponent $k$ is dependent on the size of the launch vehicle. For launchers with $D_{stage}<1 \mathrm{m}$, a value of $k=2.9$ has provided excellent results, while for bigger launchers, a value of $k=2.1$ is more realistic. An intermediary value of $k=2.5$ is suggested to be used as a preliminary value, being subject to further investigations during advanced phases of microlauncher design.

The fifth and last major system needed for first-stage recovery is the heat shield, which is essential in protecting the lower stage from the extreme temperatures generated during atmospheric reentry. Computing the weight of the heat shield involves determining several critical factors, such as the type of material used, the shield surface area and the thickness required for adequate thermal protection.

The basic formula used to estimate the mass of the heat shield $m_{heat\_shield}$ is

$$m_{heat\_shield}={\rho }_{heat\_shield}·V_{heat\_shield}$$

(21)

where ${\rho }_{heat\_shield}$ is the density of the material used for the heat shield and V_{heat_shield} is the volume of the entire heat shield required.

The volume of the heat shield $V_{heat\_shield}$ depends on the thickness of the material and the surface on which it is applied. If the thickness of the heat shield is considered uniform, the formula simplifies drastically, taking the following form:

$$V_{heat\_shield}=A_{heat\_shield}·t_{heat\_shield}$$

(22)

where $t_{heat\_shield}$ is now the average thickness of the heat shield and $A_{heat\_shield}$ is the total area of the vehicle to be covered (exposed to a critical thermal load).

The total surface area covered with a heat shield $A_{heat\_shield}$ is computed as

$$A_{heat\_shield}=\pi r^2·f_{nozzle}+2\pi r{h}_{heat\_shield}·f_{landing\_legs}$$

(23)

where $r$ is the radius of the lower stage, $h_{heat\_shield}$ is the height of the heat shield, $f_{nozzle}$ is a correction factor for the area of the stage base occupied by the nozzle (considered 0.5) and $f_{landing\_legs}$ is a correction factor necessary for the leg coating (considered 1.1).

The definition of $h_{heat\_shield}$ was based on CFD results obtained from the reentry analysis of a similar reusable lower-stage concept, the simulations being carried out for three external flow regimes (subsonic, supersonic and hypersonic up to Mach 10) [40]. It was found that a heat shield height of approximately 30–35% of the lower-stage length is necessary, since the thermal load arising during reentry is concentrated in this area.

The thickness of the thermal shield $t_{heat\_shield}$ can vary depending on the reentry altitude and associated Mach number, with the paper [41] suggesting a value of 1 mm for an insulating material based on carbon fiber and 15 mm for an ablative cork-type material (for example PICA-X). For this paper, the use of a carbon fiber-based material (${\rho }_{heat\_shield}$ ≈ 1750 kg/m³) [39] with a conservative thickness of 1.5 mm is proposed.

2.2. Propulsion Module

The second main MDO module that is called during an optimization loop (details in Figure 1) is the Propulsion module in which the liquid propellant rocket engines’ performances are estimated. It is important to mention the fact that regardless of the type of stage used (expendable or reusable), the methodology for estimating the propulsive performance of the engines remains the same.

During the propulsive assessment, the main output is related to the thrust curves of the engines. Because there is no major difference between the methodology used for estimating the engine performance during the main mission (microlauncher ascent flight) and recovery mission (first-stage ascent and descent flight), the mathematical models used in the developed MDO are identical and briefly presented below (more details can be obtained from papers [1,21]).

The thrust ($T$) generated by the rocket engine is computed using the following simple, analytical model:

$$T=q·g_0·I_{sp}$$

(24)

where $q$ is the propellant mass flow rate (derived from an optimization variable), $g_0$ is the standard gravitational acceleration [35] and $I_{sp}$ is the specific impulse.

It is possible to use different engine regimes by modifying the propellant mass flow rate because the thrust generated by the engine is directly proportional to the quantity of propellant that is ignited in the combustion chamber. Nevertheless, in this paper, the engine throttle setting was set to 100%, meaning that the mass flow rate is constant.

The specific impulse is the most important parameter associated with a rocket engine combustion efficiency. The accurate modeling of the specific impulse is a difficult process, but also necessary in obtaining accurate propulsive characteristics of the launch vehicle. This specific impulse is strictly dependent on the oxidizer and fuel pair, their mixture ratio, and also on the atmospheric operating conditions. According to [15,22,28], the specific impulse can be approximated using

$$I_{sp}={\eta }_n·{\displaystyle \frac{C^{*}}{g_0}}\left(\gamma \sqrt{\left({\displaystyle \frac{2}{\gamma -1}}\right)·{\left({\displaystyle \frac{2}{\gamma +1}}\right)}^{\left({\displaystyle \frac{\gamma +1}{\gamma -1}}\right)}·\left(1-{\left({\displaystyle \frac{P_e}{P_c}}\right)}^{\left({\displaystyle \frac{\gamma -1}{\gamma }}\right)}\right)}+{\displaystyle \frac{\epsilon }{P_c}}\left(P_e-P_a\right)\right)$$

(25)

where ${\eta }_n$ is the nozzle efficiency (considered 98% [21]), $C^{*}$ is the propellant characteristic velocity, $\gamma$ is the isentropic coefficient of the exhaust gas, $P_c$ is the pressure in the combustion chamber, $P_e$ is the exhaust pressure, $P_a$ is the atmospheric pressure and $\epsilon$ is the nozzle expansion ratio (obtained in the Preliminary design module).

The last term in Equation (25) represents the altitude impact on the specific impulse. At low altitudes, the atmospheric pressure is high and the thrust generated by the engine is significantly lower compared to that generated after leaving the atmosphere.

The propellant characteristic velocity can be approximated using [1,42]

$$C^{*}={\eta }_c·{\displaystyle \frac{\sqrt{\gamma R{T}_f}}{\gamma {\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2\gamma -2}}}}$$

(26)

where ${\eta }_c$ is the combustion efficiency (98% for the LOX/methane pair [21]), $R$ is the exhaust gas constant and $T_f$ is the flame temperature.

For the exhaust gas constant, the following relation is used:

$$R={\displaystyle \frac{R_u}{M_w}}$$

(27)

where $R_u$ is the universal gas constant [43] and $M_w$ is the gas molecular weight.

Based on the model presented above, the following four propulsive parameters are needed to obtain the propulsive characteristics of a liquid propellant rocket engine:

• The optimal oxidizer/fuel mixture ratio (R_m);
• The flame temperature (T_f);
• The gas molecular weight ($M_w$);
• The isentropic coefficient of the exhaust gases ($\gamma$).

The term $R_m$ does not explicitly appear in the mathematical model, but is needed because the other three parameters ($T_f, M_w$ and $\gamma$) are computed based on it.

The four propulsive parameters of interest can be approximated in multiple ways. The simplest way is by calling classical combustion charts [44]. These combustion curves were generated following a thermochemical equilibrium analysis with the help of the STANJAN program [45,46] but other similar ones can be used [47,48,49,50]. The thermochemical reaction constants required for the analysis are taken from the JANAF databases [51]. With this approach, the only necessary input data needed to assess the propulsive parameters of interest are the combustion chamber pressure ($P_c$) and exhaust pressure ($P_e$), these being among the optimization variables used within the MDO algorithm (Section 2.6.2).

While the typical approach seems to be directly calling the combustion charts and interpolating the data, this is not practical in terms of computational time required in the context of the MDO algorithm (in the order of multiple million function calls). Thus, the need arises for a simpler model that does not require multidimensional interpolation of the data.

In [1,21], nonlinear approximation functions are developed for multiple oxidizer/fuel pairs that provide accurate results for the pressure range of interest, making possible the implementation of four combustion surfaces, one for each propulsive parameter. The approximation functions (used in the current MDO) have the following definition:

$$f(x,y)=a+b\cdot x^c+d\cdot y^e$$

(28)

where $f=(R_m, T_f, M_m,\gamma )$, $x$ is associated with the combustion chamber pressure ($P_c$), $y$ is associated with either the exhaust pressure ($P_e$) or the optimal mixture ratio ($R_m$) and $(a,b,c,d,e)$ are the model coefficients.

For the determination of approximation model coefficients $(a,b,c,d,e)$ several nonlinear regressions were performed using the Trust-Region [52,53,54,55,56,57] and Levenberg–Marquardt [58,59,60,61] algorithms, which minimize the sum of residual squares. The influence of extreme values (usually occurring at the boundaries) is minimized using the robust LAR and Bisquare methods [62,63]. The developed approximation model is valid for combustion pressure values in the range of 10–250 atm and exhaust pressure values in the range 0.1–1 atm, being in line with most of the current rocket engines in use.

Finally, the values of the coefficients $(a,b,c,d,e)$ necessary for Equation (28) are presented in

Table 1. Approximation model coefficients, LOX/methane pair [21].

Function	Variables		Coefficient
$f$	$x$	$y$	$a$	$b$	$c$	$d$	$e$
$R_m$ [-]	$P_c$ [atm]	$P_e$ [atm]	−0.88149	3.17521	0.03058	0.03777	−0.3413
$T_f$ [K]	$P_c$ [atm]	$R_m$ [-]	−283,663.2214	200,294.10397	0.00045	84,639.97654	0.01976
$M_w$ [-]	$P_c$ [atm]	$R_m$ [-]	−332.49854	372.70801	0.00045	−36.22978	−0.5195
$\gamma$ [-]	$P_c$ [atm]	$R_m$ [-]	2.1738	−2.66329	0.00105	1.78059	−0.0395

[1,21] for the LOX/methane pair. An average error of 1.3% was observed between the results obtained with the current propulsive model (sea-level and vacuum specific impulse/thrust) and experimental results [64,65,66] for the LOX/methane pair.

2.3. Aerodynamics Module

The third main module that is assessed during the MDO loop (see Figure 1) is the Aerodynamics module. From an aerodynamic point of view, for the correct definition and optimization of a reusable microlauncher concept, the generation of aero databases for each of the unique configurations that appears during both main mission (payload insertion into orbit) and recovery mission (of first stage after separation from the microlauncher) is needed. These aerodynamic databases are necessary to accurately integrate the equations of motion during reference trajectory generation (details in Section 2.4).

For the main mission of the microlauncher (with a two-stage architecture) the unique configurations which appear during trajectory propagation are presented in

Table 2. Microlauncher configurations (main mission).

Configuration	Description	Lower Structure Components	Upper Structure Components
M1	Take-off configuration	Stage 1 + Stage 2	Payload + Adapter + VEB + Fairing
M2	After first-stage separation	Stage 2	Payload + Adapter + VEB + Fairing
M3	After fairing separation	Stage 2	Payload + Adapter + VEB
M4	After satellite separation	Stage 2	Adapter + VEB

.

It is worth mentioning that the last configuration from Table 2 (configuration M4) appears after the payload (satellite) is inserted into the predefined orbit and is associated with the de-orbit configuration of the second stage. This configuration is not investigated during the current MDO loop, a future analysis being envisioned in a latter phase [20].

In the case of an expendable microlauncher, the generation of the aerodynamic databases of interest for the first three configurations presented in Table 2 is enough to fully simulate the launcher trajectory (as achieved in [13,16]). Because we are now investigating a reusable microlauncher concept, additional lower-stage recovery capabilities are required, which translate into more configurations that must be aerodynamically assessed. A quick visual comparison between an expendable microlauncher concept [13] and a partially reusable microlauncher concept can be seen in Figure 5 as depicted in [40,67,68,69].

The new unique configurations which appear during recovery mission trajectory propagation are presented in

Table 3. First-stage configurations (recovery mission).

Configuration	Description	Aerodynamic Control System	Landing System
R1	Baseline first-stage configuration	Undeployed	Folded
R2	After flip-over maneuver	Deployed	Folded
R3	Aerodynamic guidance	Active	Folded
R4	Landing configuration	Active	Deployed

.

2.3.1. Microlauncher Configurations

The microlauncher configurations which appear during the main mission are presented in Table 2 and are known as M1, M2 and M3 (M4 is associated with the de-orbit configuration of the second stage and will be the subject of investigation in a further paper). The M3 configuration is obtained after fairing separation, which usually occurs during the coast period, after first-stage separation (details in Section 2.4). Because the altitude of the microlauncher at that time is very high (for example the fairing jettisons at around 70 km for the expendable microlauncher presented in [13]), the aerodynamic impact of the configuration is quite low and some simplifications can be made. To lower the number of aerodynamic databases that are generated in the Aerodynamics module for each of the unique values of the optimization variable vector, the M3 and M2 configurations are considered to be identical in terms of the aerodynamic database associated.

In addition, it is worth mentioning that, due to the use of a 3DOF dynamic model (Section 2.4), the number of aerodynamic coefficients of interest is low, being related only to the aerodynamic forces which appear during flight. The aero databases generated inside the Aerodynamics module must contain the following [1,70,71]: axial force coefficient ($C_A$), normal force coefficient ($C_N$), drag coefficient ($C_D$) and lift coefficient ($C_L$).

Analytical and semi-empirical methods are preferred because they present high flexibility, but are also associated with a reduced computation time. From the four coefficients presented earlier, only two are strictly needed to generate the required aero database, this being the drag coefficient and lift coefficient (both dependent on the angle of attack and Mach number). The other two can be obtained using the following relations [72]:

$$C_A={\displaystyle \frac{C_D\mathit{cos}\alpha -\frac{1}{2}{C}_N\mathit{sin}2\alpha }{1-{\mathit{sin}}^2\alpha }}; C_L=C_N\mathit{cos}\alpha -C_A\mathit{sin}\alpha$$

(29)

where $\alpha$ is the angle of attack of the microlauncher.

Because the proposed relations are based on linearized models, the superposition principle can pe applied and thus the final coefficients can be broken down into smaller contributions (which will be later added), each corresponding to a specific main component of the studied configuration. This ensures that the mathematical models used in earlier papers regarding the aerodynamic assessment of expendable launchers [13,16] does not change significantly. In these papers, the mathematical models are detailed only for the “clean configuration”, which can still be of use in our case; the only difference is that we will need to add the contribution of the landing system and that of the ACS.

The difference between the exact microlauncher configuration (CAD model) and the aerodynamic “clean configuration” (based on the Aerodynamics module of the MDO) can be seen in Figure 6 for the M1 configuration. Even though the output of the Aerodynamics module of the MDO (Matlab) seems to not include any exterior components, they are included in the final aerodynamic databases which are used in the Trajectory module.

There is no standard external geometry for launch vehicles, as they can come in different shapes and sizes. Therefore, it is practical to break down the launcher into geometrically simple components. Thus, a microlauncher can be seen as an assembly consisting of the following components [71]:

• Fairing (which can have multiple geometries based on different nose cone generating profiles—conical, Haack series, ogive, ellipsoid, etc.);
• Cylindrical stage or interstage (if both the lower and upper connecting stages have the same diameter);
• Positive transition (if the lower stage has a greater diameter than the upper stage);
• Negative transition (if the lower stage has a smaller diameter than the upper stage).

In the case of a constant-diameter launcher, the input geometry has a significantly reduced complexity (one fairing, two stages and an interstage with the same base diameter), which ensures that the number of geometrically simple components used is also reduced and, in the end, the computational time needed for the aero database is very low.

The first major aero coefficient that is assessed is the drag coefficient $C_D$ (which is dependent on the angle of attack $\alpha$ and Mach number $M$), being estimated with

$$C_D\left(\alpha ,M\right)=C_{d_0}\left(M\right)+C_{d_i}\left(\alpha \right)$$

(30)

where $C_{d_0}$ is the zero angle of attack drag coefficient and $C_{d_i}$ is the alpha drag coefficient.

The microlauncher zero angle of attack drag coefficient is broken down into multiple smaller contributions, each one corresponding to every individual simple component:

$$C_{d_0}=\sum _i^N\left({\displaystyle \frac{A_i}{A_{ref}}}\cdot C_{{d_0}_i}\right)$$

(31)

where N is the number of individual components, $A_i$ is the local reference area of component $i$, $A_{ref}$ is the global reference area and $C_{{d_0}_i}$ is the individual zero angle of attack drag coefficient of component $i$.

Multiple ways of expressing the zero angle of attack drag coefficient (of each component) exist in the literature [73,74,75], the following one being preferred in the MDO:

$$C_{{d_0}_i}=C_{{d_0}_{pressure}}+C_{{d_0}_{friction}}+C_{{d_0}_{base}}$$

(32)

where $C_{{d_0}_{pressure}}$ is the body pressure drag coefficient, $C_{{d_0}_{friction}}$ is the skin friction drag coefficient and $C_{{d_0}_{base}}$ is the base drag coefficient.

The body pressure drag coefficient $C_{{d_0}_{pressure}}$ is modeled using analytical [76,77] and semi-empirical [78] relations for components that have different fore and aft diameters (fairing, positive and negative transitions). Additionally, experimental data provided by [78,79] are also used for standard fairings (for which the tip has not been blunted). For more details on the exact correspondence between the Mach number and the body pressure drag, together with the influence of tip bluntness (considerable reduction in the thermal load with minimal increase in drag) one can consult papers [70,71].

The skin friction drag coefficient $C_{{d_0}_{friction}}$ is computed with the aid of several models used in the literature [80,81,82,83,84], based on a dependency with the skin friction coefficient $C_f$. This term is further modeled as a function of Reynolds number, surface roughness and launcher length. The detailed models are presented in papers [70,71], while additional relevant information can be gathered from [80,81,82,83,84].

The base drag coefficient $C_{{d_0}_{base}}$ is the most difficult term to analytically estimate, as it is strongly influenced by the flow conditions before and after the separation zone. The paper [78] presents a theoretical distribution of this coefficient as a function of the Mach number for a three-dimensional axially symmetric body. In paper [79], experimental results of interest are presented only starting from the supersonic regime. Inside the Aerodynamics module of the MDO, a hybrid model is implemented, using the data from [78,79], together with a connection model that is proposed in [71].

The alpha drag coefficient $C_{d_i}$ needed in Equation (30) is estimated using [72,85]

$$C_{d_i}=2\delta {\alpha }^2+{\displaystyle \frac{3.6\eta \left(1.36l_l-0.55l_v\right)}{\pi d_b}}{\alpha }^3$$

(33)

where $\delta$ and $\eta$ are empirically obtained parameters, $l_l$ is the total length of the launcher, $l_v$ is the length of the launcher fairing, $d_b$ is the launcher diameter (maximum) and $\alpha$ is the angle of attack (measured in radians).

The next major aero coefficient that is assessed is the normal force coefficient $C_N$ (which is dependent on the angle of attack $\alpha$ and Mach number $M$), using

$$C_N={\sum }_i^N\left(C_{N_i}\right); C_{N_i}=C_{{N_i}_{\alpha }}\alpha ; C_{{N_i}_{\alpha }}\left(\alpha ,M\right)=C_{{{N_{incomp}}_i}_{\alpha }}\left(\alpha \right)\cdot F_{comp}\left(\alpha ,M\right).$$

(34)

where $N$ is the number of individual components, $C_{N_i}$ is the normal force coefficient of component $i$, $C_{{N_i}_{\alpha }}$ is the normal force coefficient derivative of component $i$, $\alpha$ is the angle of attack, $C_{{{N_{incomp}}_i}_{\alpha }}$ is the incompressible normal force coefficient derivative and $F_{comp}$ is a compressibility factor.

The term $C_{{{N_{incomp}}_i}_{\alpha }}$ is estimated with the Barrowman model [80], together with the Galejs extension [86], while the compressibility factor $F_{comp}$ is computed using [87,88,89,90,91].

The validity limits of the mathematical model used to assess the aerodynamic characteristics of the microlauncher “clean configuration” are in line with the expected flight regimes of space launchers (Mach number < 10 and angle of attack < 10°) [1,70]. The aerodynamic contribution of the landing gear system and ACS are presented separately in Section 2.3.3 and Section 2.3.4 and are added on top of the current generated aero database.

2.3.2. Reusable First-Stage Configurations

The first-stage configurations that can appear during the recovery mission are presented in Table 3 (being known as R1, R2, R3 and R4) and depicted in Figure 7.

It is worth mentioning the fact that the R2 and R3 configurations are practically the same, the main difference being the fact that for R3 the ACS is active and each grid fin can be deflected independently to guide the lower stage to the landing location, while for the R2 configuration, the ACS is deployed but the grid fins are not deflected.

As presented earlier in Section 2.3.1, the best way to approach the aerodynamic database generation is to split the studied configuration into a “clean configuration” and later quantify the aero contributions of any external components (ACS and landing system in the case of the reusable first stage). This breakdown can be seen in Figure 8 for the case of R2 configuration.

The mathematical models previously presented for the case of an expendable launcher concept “clean configuration” (Section 2.3.1) must be now extended to include new geometrically simple components. A breakdown of the lower stage into its main components is presented in Figure 9 and consists of the following: interstage (open-ended cylindrical geometry), cylindrical stage and nozzle (truncated cone geometry).

The component list now contains two new entries associated with the component which will be in direct contact with the airflow (interstage or nozzle). During the recovery mission of the first stage, the angle of attack will no longer be always close to 0° (as for the microlauncher case), this only being the case for the R1 configuration up until ballistic trajectory apogee (details in Section 2.4.2). For R2–R4 configurations, the angle of attack $\alpha$ (AoA) will be close to 180° because of the flip-over maneuver realized at apogee.

To ensure that the mathematical models developed earlier for the microlauncher “clean configuration” case are still usable (Section 2.3.1), one must introduce a way to asses both AoA = 0° and AoA = 180° cases in a similar matter. This is realized with the introduction of the term “relative angle of attack” ${\alpha }^{\prime }$ (rAoA) [67] defined as

$${\alpha }^{\prime }=\left\{\begin{array}{ll}\alpha & \mathrm{if} \alpha < 90°\\ 180°-\alpha & \mathrm{if} \alpha > 90\end{array}\right.$$

(35)

Thus, both the ascent (R1) and descent first-stage configurations (R2–R4) can be assessed in a similar matter using the same mathematical model. For further clarification, the flow direction is presented in Figure 10, with (a) showing an angle of attack $\alpha$ close to 0°, while in (b), the angle of attack $\alpha$ is close to 180°. For both cases, the relative angle of attack ${\alpha }^{\prime }$ is close to 0°.

Similar to the model presented in Section 2.3.1, only the drag coefficient $C_D$ and normal force coefficient $C_N$ must be assessed for the clean configuration, the other two being obtained using Equation (29). Most of the mathematical models used for the microlauncher assessment remain the same, with the difference being that the angle of attack is replaced by the relative angle of attack.

For example, the drag coefficient $C_D$ is now estimated with

$$C_D\left({\alpha }^{\prime },M\right)=C_{d_0}\left(M\right)+C_{d_i}\left({\alpha }^{\prime }\right)$$

(36)

where ${\alpha }^{\prime }$ is the relative angle of attack (Equation (35)), $M$ is the Mach number, $C_{d_0}$ is the zero relative angle of attack drag coefficient and $C_{d_i}$ is the alpha drag coefficient.

Equations (31) and (32) remain valid, the only difference between the models occurring for the estimation of the term $C_{{d_0}_{pressure}}$. This is due to the fact that the fore part of the configuration is not aerodynamically profiled anymore (the first component to interact with the airflow is now either an open-ended interstage or a nozzle). As the tip component (fairing) no longer exists, a particularization of the mathematical models used to asses $C_{d_0}$ must be made for the first-stage configurations.

For the R1 configuration (presented in Figure 10a), an analogy with the flow over a blunt cylinder [77] is used as a starting point for the model implemented inside the Aerodynamics module of the MDO. For the subsonic regime, a simple formula can be used to assess the drag coefficient (at null angle of attack) based on the data from [77]:

$$C_{d_0}=0.05+0.03+0.74\left(1+0.25M^2\right)$$

(37)

where the first term (0.05) is associated with the friction drag, the second term (0.03) is associated with the base drag, the last term is associated with the body pressure drag (the main component) and $M$ is the Mach number.

The blunt cylinder in a transonic flow exhibits a completely detached shock wave, the body pressure drag coefficient on the front face of the cylinder being closely related to the stagnation pressure. The following relation can be used to estimate the body pressure drag coefficient for the blunt cylinder case [77]:

$$C_{{d_0}_{pressure}}=0.85\cdot {\displaystyle \frac{q_s}{q}}; {\displaystyle \frac{q_s}{q}}=\left\{\begin{array}{c}1+{\displaystyle \frac{M^2}{4}}+{\displaystyle \frac{M^4}{40}},if\hspace{0.33em}M\le 1\\ 1.84-{\displaystyle \frac{0.76}{M^2}}+{\displaystyle \frac{0.166}{M^4}}+{\displaystyle \frac{0.035}{M^6}},if\hspace{0.33em}M>1\end{array}\right.$$

(38)

Just for the transonic regime, a coefficient of $k=0.9$ is recommended to be used, as it has been shown to provide a better overall accuracy [77]. It is also estimated here that the friction drag coefficient becomes negligible, while the base drag coefficient is approximately 0.2, remaining in this region for flows with Mach number up to 2.

For a provisional mathematical model in the supersonic regime, the base drag component can be assumed to have a value of 0.6 times the maximum $C_{{d_0}_{base}}$ (details in paper [71]), according to [77]. At high Mach numbers, the drag coefficient at zero angle of attack has a ceiling at a value of 1.65–1.7, being consistent with results from the literature. The above model is ideal for the situation where the analyzed configuration does not have an aerodynamically profiled tip and the fineness ratio of the cylindrical component is large, which is the case of the lower stages of a microlauncher.

The next step is the extension of the aero model to include the contribution of the interstage and nozzle. The contribution of the nozzle component can be neglected because of its placement in the wake of the blunt body (where the base drag does not significantly change). In order to quantify the aerodynamic impact generated by the interstage adapter, the notion of an open-ended cylinder “tip” (stage adapter component) was introduced in [13] for which a new contribution $C_d$ appears that can be estimated using

$$C_{{d_0}_{adapter}}=0.2e^{-0.243M}$$

(39)

For the R2-R4 configurations (presented in Figure 10b), the order in which the first-stage components interact with the airflow are now reversed, the first one being the nozzle (with the exhaust side first). We must now address this issue and present a reliable mathematical model to assess the aero characteristics. Here, an analogy can be made with the flow over a blunt cylinder with frontal probe. Again, paper [77] offers a compilation of experimental data for three configurations of blunt bodies with frontal probes, of which two are sharp probes and one is a blunt circular cylinder probe.

If we adopt the hypothesis that the geometry of the truncated cone nozzle can be approximated by that of a cylinder having an equivalent diameter [67], one can propose the following model for descent first-stage configurations (R2–R4):

$$C_{d_{{pressure}_{nozzle}}}={\displaystyle \frac{A_{l_{nozzle}}}{A_{ref}}}·C_{d_{{pressure}_{blunt.cylinder}}}$$

(40)

where $A_{l_{nozzle}}$ is the local reference area of the nozzle-type component, $A_{ref}$ is the reference area of the first-stage configuration and $C_{d_{{pressure}_{blunt.cylinder}}}$ is computed using (37).

For the computation of the term $A_{l_{nozzle}}$, one can use

$$A_{l_{nozzle}}=\pi {\displaystyle \frac{{D_{{nozzle}_{equivalent}}}^2}{4}}$$

(41)

where $D_{{nozzle}_{equivalent}}$ is the equivalent diameter of the nozzle component when approximated as a blunt cylinder probe and has the following form [67]:

$$D_{{nozzle}_{equivalent}}=\left\{\begin{array}{c}{D}_{{nozzle}_{exhaust}}-D_{{nozzle}_{throat}},if M<1\\ D_{{nozzle}_{exhaust}}+D_{{nozzle}_{throat}},if M\ge 1\end{array}\right.$$

(42)

with $D_{{nozzle}_{exhaust}}$ being the nozzle diameter at the exhaust zone and $D_{{nozzle}_{throat}}$ being the nozzle diameter in the critical zone.

For the second component of R2 configuration (semi-blunt cylinder/main stage), it can be considered that its pressure drag coefficient is zero (because we have another component preceding it—nozzle) [67]. For the friction and base drag coefficients, the same models mentioned in Section 2.3.1 remain valid and, again, the last component of the first-stage descent configurations (R2–R4) can be neglected because of its placement in the wake (as was the case of the nozzle placement in R1 configuration).

The alpha drag coefficient $C_{d_i}$ found in Equation (36) remains unchanged for the first-stage configuration where the interstage is forward-facing (R1), while for the cases where the nozzle is forward-facing (R2–R4) the following formula is proposed [40,67]:

$$C_{d_i}=2\delta {{\alpha }^{\prime }}^2+{\displaystyle \frac{3.6\eta \left(1.36l_l-0.55l_v\right)}{\pi d_b}}{{\alpha }^{\prime }}^3+∆C_{d_{i_{nozzle}}}$$

(43)

where $\delta$ and $\eta$ are obtained from [72,85], $l_l$ is the total length of the configuration, $l_v$ is the length of the first component (either interstage or nozzle), $d_b$ is the configuration diameter (maximum) and ${\alpha }^{\prime }$ is the relative angle of attack (measured in radians).

Comparing Equations (33) and (43), the impact of using a nozzle as the first component can be quantified by the addition of the term $∆C_{d_{i_{nozzle}}}$ which can be computed with [13]

$$∆ C_{d_{i_{nozzle}}}={\displaystyle \frac{A_{{max}_{nozzle}}}{A_{ref}}}\left(0.1{\alpha }^{\prime }-0.01M{\alpha }^{\prime }\right)$$

(44)

where $A_{{max}_{nozzle}}$ is the maximum nozzle area (exhaust section) and ${\alpha }^{\prime }$ is measured in °.

The model presented in Section 2.3.1 for estimating the normal force coefficient of the microlauncher remains valid also for the first-stage configurations, with the difference that the angle of attack $\alpha$ is replaced by the relative angle of attack ${\alpha }^{\prime }$ (similar to the drag coefficient model presented earlier for R1–R4 configurations). Thus, we can write

$$C_{N_i}=C_{{N_i}_{{\alpha }^{\prime }}}{\alpha }^{\prime }; C_{{N_i}_{{\alpha }^{\prime }}}\left({\alpha }^{\prime },M\right)=C_{{{N_{incomp}}_i}_{{\alpha }^{\prime }}}\left({\alpha }^{\prime }\right)\cdot F_{comp}\left({\alpha }^{\prime },M\right)+{∆C}_{N_{blunt}} \left({\alpha }^{\prime },M\right).$$

(45)

The additional term ${∆C}_{N_{blunt}}$ only appears for blunted components. For the ascent configuration (R1), the term ${∆C}_{N_{blunt}}$ is computed using [67]

$${∆C}_{N_{blunt}}=C_{N_{adapter}}={\displaystyle \frac{A_{l_{adapter}}}{A_{ref}}}\left(0.026{\alpha }^{\prime }-0.002M{\alpha }^{\prime }-0.002{{\alpha }^{\prime }}^2\right)$$

(46)

where $A_{l_{adapter}}$ is the local reference area of the adapter (maximum frontal area).

For the descent configurations (R2–R4), the term ${∆C}_{N_{blunt}}$ is computed using [67]

$${∆C}_{N_{blunt}}=C_{N_{nozzle}}={\displaystyle \frac{A_{{max}_{nozzle}}}{A_{ref}}}\left(0.11{\alpha }^{\prime }-0.008M{\alpha }^{\prime }-0.005{{\alpha }^{\prime }}^2\right)$$

(47)

where ${\alpha }^{\prime }$ measured in °.

2.3.3. Aerodynamic Control System Contribution

For the aerodynamic control system (ACS), a simple architecture was envisioned, the ACS consisting of four square-pattern cells, unswept grid fins with another four quasi-cylindrical corresponding outer supports placed in an X configuration, similar to the SpaceX Falcon 9 launch vehicle [33]. The sizing process is presented in Section 2.1 while more details related to the close-up geometry are given in [68].

During the main mission of the microlauncher (orbital insertion of payload), the ACS is part of only the M1 configuration (details in Table 2 and Figure 5b) and is in undeployed position (shown in Figure 11a), no aerodynamic forces being needed to guide the launcher. During the main microlauncher mission, the control and guidance of the launcher is performed exclusively by means of the engine TVC system.

After the lower stage separates from the launcher, the possible configurations of the recoverable first stage are presented in Table 3. For the R1 configuration, the ACS is set to undeployed (Figure 11a); for the R2 configuration the ACS is set to deployed (Figure 11b, where the individual grid fins will have a null deflection angle, and thus only drag will be generated and no lateral forces), while for the R3 and R4 configurations, the ACS is set to active, meaning that the grid fins can be deflected as needed to guide the lower stage to the landing/recovery location (Figure 11b–d).

The flight regimes expected for configurations where the ACS is needed to be in a deployed state are associated with flows with Mach number ≤ 5, while the control authority needed for the ACS can be obtained with grid fin deflections up to 20° [30,31,92]. To match the size of the aerodynamic database generated for the “clean configuration”, the upper Mach number limit was extended to the value 10 for the ACS as well.

For the case of undeployed ACS (Figure 11a), the aerodynamic impact of the outer components of ACS will have a minor impact on the characteristics of the “clean configuration” (the ACS geometry is streamlined); thus it can be considered that the aero database generated earlier for M1 and R1 configurations (with the mathematical models detailed in Section 2.3.1 and Section 2.3.2) does not require the addition of any ACS contribution. These two configurations will, however, need the addition of a landing system aero contribution (presented in Section 2.3.4).

For descent first-stage configurations (when the angle of attack is close to 180°—R2–R4 configurations), the aerodynamic influence of the ACS (which is now deployed) can no longer be neglected. A detailed CFD analysis was performed in [68]; only the numerical databases which are provided there as output were integrated into the Aerodynamics module of the developed MDO (lift and drag coefficients vs. Mach number at different grid deflection angles). The global reference used to scale the coefficients is the maximum frontal area of the configuration (the same as for the “clean configuration”).

The final ACS aero database is generated (inside the Trajectory module for lower computational time) based on four individual grid fin contributions, as each grid fin can be deflected separately, depending on the desired output (symmetrical deflection for yaw/pitch axis control and asymmetrical deflection for roll axis control). Also, the magnitude of the deflection angle can vary from one grid fin to another.

2.3.4. Landing Gear System Contribution

The next outer system that must be aerodynamically assessed in order to obtain the final microlauncher/first-stage configuration aero database is the landing system, for which a simple architecture was envisioned, consisting of four landing legs (together with additional auxiliary components), placed in an X configuration, similar to the SpaceX Falcon 9 launch vehicle [33]. It is worth mentioning the fact that the landing system is placed at a 45° rotation with respect the ACS to not disturb the airflow towards the grid fins when the first stage trajectory is descensional (angle of attack close to 180$°$).

The landing system can be in either of two distinct positions (folded and unfolded), together with a transitionary state (during the unfolding sequence). To better understand the possible states of the landing system, one can see Figure 12 [69].

During the main mission of the microlauncher (orbital insertion of payload), the landing system is a part of only the M1 configuration (details in Table 2 and Figure 5b) and is in a folded position. A streamlined exterior surface was used for the leg outer shell such that the drag increase is reduced for the ascent flight phases (with angle of attack close to 0°).

After the lower stage separates from the launcher, the possible configurations of the recoverable first stage are presented in Table 3. For R1–R3 configurations, the landing system is in a folded position, while for the R4 configuration, the landing system unfolds towards its final resting position needed for stage recovery.

To lower the number of landing systems that must be aero-assessed (for which an aerodynamic database must be generated in the Aerodynamics module of the MDO), it can be considered that the R4 configuration is not of current interest as the unfolded landing system position (seen in Figure 7c and Figure 12) only occurs in the final landing phase, a phase that has a brief duration (a few seconds) compared to the mission itself. Thus, the landing system is aerodynamically assessed only in folded position in the MDO.

A detailed CFD analysis is performed in [69], where a baseline reusable lower-stage configuration is numerically analyzed and the contribution of the landing system is extracted for both ascent flight (AoA = 0°, which appears in M1/R1 configurations) and descent flight (AoA = 180°, which appears in R2/R3 configurations). Only the numerical databases which are provided there as output were integrated into the Aerodynamics module of the developed MDO, similar to the procedure used for the ACS in Section 2.3.3.

It was observed in [69] that the main difference between a “clean configuration” and the one with a folded landing system architecture is regarding the total drag coefficient of the configuration $C_D$. The aerodynamic impact of the landing system is low for the ascent configurations (AoA = 0°), being in the range of 0.5–5.9%. This is due to the use of a streamlined outer shell in the Preliminary design module of the MDO. For the descent configurations, the aerodynamic impact is higher, being in the range of 7–8.5%, having an overall positive effect on the first-stage recovery performance, as a higher drag will decrease the velocity of the stage during the evolution towards the landing location.

2.4. Trajectory Module

In the last main module of the MDO algorithm (block scheme presented in Figure 1) entitled Trajectory module, the microlauncher evolution is simulated (by integrating the equations of motion) during its main mission (ascent towards the target orbit), along with the generation of the reference recovery trajectory for the lower stage towards the landing location (which is different from the launch site in the case of a downrange recovery).

As the MDO tool used is based on an iterative process, during the launcher optimization, the durations of the key evolution phases will constantly change (along with other parameters closely related to the control schemes). To write the full set of equations of motion specific to the case of a microlauncher (a variable mass body), the theorems of impulse and kinetic momentum must be applied. This corresponds to using a complex dynamic model with six degrees of freedom (6DOF), a model which is detailed in [1,93] and for the case of a guided launch vehicle consists of 21 differential equations.

However, the engineering approach is to introduce assumptions to simplify the system of equations. In preliminary design activities, when the technical information of the microlauncher is not entirely defined, but also in the case of trajectory optimization applications (which require a large number of successive evaluations), it is more appropriate to use a simplified dynamic model with three degrees of freedom (3DOF). Here, the dynamic model describes only the translational motion of the launcher, as the launcher is now considered to be a point of variable mass. Thus, only the dynamic and kinematic translational equations describing the velocity and position are needed (six equations).

Many different 3DOF implementations are used in the literature; in this paper, a null bank angle $\mu$ model is implemented as detailed in [94]. In this model, the GNC system ensures that the launcher central body axis is aligned with the thrust vector. Between the velocity vector and the launcher body frame, two aerodynamic angles exist ($\alpha , {\beta }^{*}$) that can be non-zero, while the third aerodynamic angle (bank angle) is always zero $\mu =0$.

The six equations of motion are written in the quasi-velocity frame [94]. The origin of this frame is in the launch vehicle center of mass (participating in the diurnal rotation). The x axis is along the velocity vector, with the y axis in the vertical plane (up direction), and the z axis completes the right trihedron. The other main coordinate system needed for this 3DOF dynamic model is the body frame, in which the forces acting on the launcher are usually written. Further details on all of the coordinate systems of interest for a launch vehicle trajectory assessment (Earth frame, local frame, start frame, Geographical Mobile Frame, Geocentric Spherical Frame, quasi-velocity frame, and body frame) are given in papers [1,95]. The final system of differential equations integrated inside the Trajectory module of the MDO is

$$\begin{array}{c}\dot{V}={\displaystyle \frac{N_x}{m}}-g_r\mathit{sin}\gamma -g_{\omega }\left(\mathit{cos}\phi \mathit{cos}\chi \mathit{cos}\gamma +\mathit{sin}\phi \mathit{sin}\gamma \right)\\ \dot{\gamma }={\displaystyle \frac{N_y}{mV}}-{\displaystyle \frac{g_r}{V}}\mathit{cos}\gamma -{\displaystyle \frac{g_{\omega }}{V}}\left(-\mathit{cos}\phi \mathit{cos}\chi \mathit{sin}\gamma +\mathit{sin}\phi \mathit{cos}\gamma \right)+{\displaystyle \frac{V}{r}}\mathit{cos}\gamma -2{\Omega }_p\mathit{cos}\phi \mathit{sin}\chi \\ \dot{\chi }=-{\displaystyle \frac{N_z}{mV\mathit{cos}\gamma }}+{\displaystyle \frac{g_{\omega }\mathit{cos}\phi \mathit{sin}\chi }{V\mathit{cos}\gamma }}+{\displaystyle \frac{V}{r}}\mathit{tan}\phi \mathit{sin}\chi \mathit{cos}\gamma +2{\Omega }_p(\mathit{cos}\phi \mathit{cos}\chi \mathit{tan}\gamma -\mathit{sin}\phi )\\ \dot{\phi }={\displaystyle \frac{V}{r}}\mathit{cos}\chi \mathit{cos}\gamma ; \dot{\lambda }=-{\displaystyle \frac{V\mathit{sin}\chi \mathit{cos}\gamma }{r\mathit{cos}\phi }}; \dot{r}=V\mathit{sin}\gamma .\end{array}$$

(48)

where $V$ is the vehicle velocity (relative to the atmosphere), $\gamma$ is the flight path angle (FPA), $\chi$ is the path track angle, $\phi$ is the geocentric latitude, $\lambda$ is the geocentric longitude (relative), $r$ is the distance between Earth and the vehicle center of mass, $\left[N_x,N_y,N_z\right]$ are the applied force components, $m$ is the instantaneous mass of the vehicle, $\left[g_r,g_{\omega }\right]$ are the gravitational acceleration components and ${\Omega }_p$ is the angular velocity of the Earth.

The gravitational model used is the J2 model [95], where the components of gravitational acceleration (radial $g_r$ and polar $g_{\omega }$) are computed as functions of the current radius $r$ and the geocentric latitude $\phi$. During the integration of the system of Equation (48), the current radius or distance from the vehicle (launcher or first stage depending on the evolution phase) $r$ is computed using [95]

$$r=R_p+H; R_p\cong a(1-\alpha {\mathit{sin}}^2\phi )$$

(49)

where $R_p$ is Earth’s radius, $H$ is the altitude of the vehicle, $a$ is Earth’s semimajor axis and $\alpha$ is Earth’s flattening coefficient (being computed using the WGS-84 database [96]).

Depending on the altitude $H$, the atmospheric data necessary to estimate the aerodynamic and thrust forces are extracted from the US76 standard atmosphere [97,98,99].

2.4.1. Main Mission

For the case of a reusable microlauncher, two distinct missions are performed, the main mission (which is that of inserting the payload into the predefined orbit) and the lower-stage recovery mission. The equations of motions described with the aid of the system (48) are identical for both missions, but differences appear when defining some of the terms. It is of great importance to highlight the way in which the external forces acting on the vehicle (microlauncher or first stage) are computed, this being closely related to the type of evolution phase which the vehicle is in. Interestingly, the main mission is the same for both an expendable and a reusable microlauncher; thus, the way in which the external forces are computed in this paper is similar to that of a classical launcher.

First, the aerodynamic and propulsive forces, which are necessary to obtain the projections of the applied force $\left[N_x,N_y,N_z\right],$ are detailed for the case of the microlauncher main mission. The two forces are initially evaluated in the body frame (detailed in Figure 13); then, using the appropriate transformation matrix (details in [94,95]), their projections in the quasi-velocity frame are computed. As the microlauncher configuration changes over time (details in Table 2), so does the body frame (Figure 13a vs. b).

Since one of the assumptions of the 3DOF model used is that the thrust vector is along the launcher central axis, the definition of the propulsive force is very simple. In the body frame, the propulsive force $T_{{body}_{frame}}$ has the components $X^T$, $Y^T$,$Z^T$ defined by

$$T_{{body}_{frame}}=\left[\begin{array}{c}{X}^T\\ Y^T\\ Z^T\end{array}\right]=\left[\begin{array}{c}T\\ 0\\ 0\end{array}\right]$$

(50)

where $T$ is the engine thrust (computed inside the Propulsion module).

Similar, according to Figure 13, the aerodynamic force $F_{{body}_{frame}}$ has three components $X^F$, $Y^F$,$Z^F$ that can be defined by

$$F_{{body}_{frame}}=\left[\begin{array}{c}{X}^F\\ Y^F\\ Z^F\end{array}\right]=\left[\begin{array}{c}-A\\ N\\ -Y\end{array}\right]=\left[\begin{array}{c}-q\cdot S_{ref}\cdot C_A\\ q\cdot S_{ref}\cdot C_N\\ -q\cdot S_{ref}\cdot C_Y\end{array}\right]$$

(51)

where $q$ is the dynamic pressure, $S_{ref}$ is the reference area and $C_A$, $C_N$ and $C_Y$ are the aerodynamic coefficients (computed inside the Aerodynamics module).

Additional attention must be given to the computation procedure for the aero forces because the aero contribution of the ACS and landing system must also be added on top of the “clean configuration” aero database. After computing the two external forces acting on the launcher (propulsion and aerodynamic) in the body frame, their projection in the quasi-velocity frame is realized with the aid of the rotation matrix $B_{\mu {\beta }^{*}\alpha }$, with the particularity $\mu =0$ [1,94,95].

The orientation of the thrust vector relative to the velocity vector is described by the aerodynamic angles $\alpha$ and ${\beta }^{*}$, which can be viewed as control parameters of the system with which the trajectory flight path angle $\gamma$ and the track angle $\chi$ can be controlled through feedback loop schemes, such as

$$\alpha =-k_1\left(\gamma -{\gamma }_d\right); {\beta }^{*}=-k_1\left(\chi -{\chi }_d\right)$$

(52)

where the reference target angles are ${\gamma }_d$ and ${\chi }_d$ (both optimization variables, details in Section 2.6.2) and $k_1$ is a setting parameter.

For the orbital insertion maneuver, the system control parameters $\alpha$ and ${\beta }^{*}$ are computed differently (with an increase in orbital performance) using [1,16]

$$\alpha =-k_2\left(\gamma -{\delta }_1\right); {\beta }^{*}=-k_3\left(i-i_d\right)$$

(53)

where $i_d$ represents the target orbit inclination, $k_2$ and $k_3$ are tuning/setting parameters and ${\delta }_1$ is the command of the orbit-related reference system obtained by imposing the optimal maneuver condition (decrease in orbital eccentricity in minimum time). The parameter ${\delta }_1$ is computed as follows [1,16]:

$${\delta }_1=atan\left({\displaystyle \frac{\left(1+e\cos \theta \right)\sin \theta }{e\left(1+{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta \right)+2\cos \theta }}\right)$$

(54)

where $e$ is current orbit eccentricity and $\theta$ is the true anomaly. The correlations between the position and velocity vectors and the six classical orbital parameters are given in [1].

The way in which the trajectory is generated and optimized inside the Trajectory module of the MDO algorithm is closely related to the altitude of the target orbit and the mass of the satellite, the main mission profile being slightly different depending on the chosen insertion method. For the case of microlaunchers (expendable or reusable), the use of a direct ascent to orbit (DATO) trajectory for the main mission is preferred due do the low altitudes associated (mainly Low Earth Orbit), together with the reduced complexity of the upper-stage engine which does not require consecutive restarts [16]. Thus, the main mission profile implemented in this paper is that of a DATO trajectory.

Multiple flight phases occur during the main mission of the launcher, each one having a particular impact on the system of Equation (48). The order in which they appear for a two-stage microlauncher concept [94] is presented in Figure 14 and mentioned next:

• Vertical flight is the period right after launch, when the flight path angle is $\gamma =90°$. The aerodynamic angles $\alpha \mathrm{a}\mathrm{n}\mathrm{d} {\beta }^{*}$ are both null.
• Active guidance (primary): The launcher is actively tilted towards the imposed flight path angle ${\gamma }_d$. For a short period of time, the flight path angle is maintained constant such that the velocity vector aligns with the thrust vector. A path track angle ${\chi }_d$ is also imposed if the launch direction cannot allow the payload to be inserted into an orbit of certain inclination.
• Gravity turn (primary): The normal load factor is zero as no applied lateral forces act on the launcher. The flight path angle $\gamma$ decreases naturally due to the gravitational acceleration. The aerodynamic angles $\alpha \mathrm{a}\mathrm{n}\mathrm{d} {\beta }^{*}$ are both null.
• First-stage separation (start of coast period): After the first-stage main mission propellant is used, the lower stage is separated to decrease the launcher mass. For an expendable launcher, the separated stage will follow a ballistic trajectory (as in Figure 14). For a reusable first stage, this moment corresponds to the start of the recovery mission.
• Fairing jettison: The fairing is separated from the microlauncher at an imposed condition (associated with either altitude, dynamic pressure or thermal heat flux [94]), preferably in the coast period to limit any dynamic instabilities upon the launcher.
• Second-stage ignition (end of coast period): At the end of the coast phase, the second-stage engine is ignited and starts to generate thrust.
• Gravity turn (secondary): The gravity turn is continued for the new microlauncher configuration (without stage 1 and fairing).
• Active guidance (orbital insertion): The flight path angle approaches the value $\gamma =0°$ (for a circular target orbit). The aerodynamic angles $\alpha \mathrm{a}\mathrm{n}\mathrm{d} {\beta }^{*}$ are computed based on Equation (53) and can be non-zero.
• Payload separation: After the second-stage engine consumes all of the main mission propellant, the payload is separated and inserted into orbit.

2.4.2. Recovery Mission

As presented in Figure 14, the main mission of the microlauncher does not include any computations related to the trajectory of the first stage after separation. In the case of an expendable launch vehicle, the lower stage usually follows a ballistic trajectory towards the ocean and is destroyed during atmospheric reentry and at impact. It is now of interest to present the recovery mission envisioned for the first stage in the case of a partially reusable microlauncher.

As methods of possible recovery, one can observe the rise of autonomous landing techniques versus more traditional methods, such as with the aid of parachutes in either an aerial capture recovery (RocketLab) or water splashdown recovery (PLD Space). In the current paper an autonomous landing is investigated as a baseline approach.

Two distinct mission profiles can be used for the recovery mission of the microlauncher lower stage, one in which the autonomous landing is performed at a secondary location (usually a barge in the ocean) and another one in which the first stage returns to the launch site (or in the close vicinity of it). Both mission profiles have been successfully performed by a launch vehicle (albeit not of small dimensions), in the case of the Falcon 9 launcher developed and operated by SpaceX [33].

Figure 15a shows a representation of the reference trajectory (not to scale [100]) used for the successful completion of the OneWeb F16 mission (with a return to launch site recovery), while Figure 15b shows the same representation but now for the Crew-6 mission (with a downrange secondary location landing and recovery).

It can be easily seen that the complexity of the return to launch site recovery mission is much higher than that associated with the downrange recovery of the first stage. Both recovery missions are of course of interest, but as a starting point, the somewhat easier one will be implemented inside the MDO tool (secondary location recovery).

Regardless of the recovery mission profile, a stage flip-over maneuver is required, where the aerodynamic angle of attack $\alpha$ increases rapidly from 0° to 180°. This flip-over maneuver after the lower stage has been separated from the microlauncher assembly is performed by an extra-atmospheric control system based on cold gas thrusters (RCS).

In the case of a stage downrange recovery (usually at a barge), after the flip-over maneuver, the lower stage has a (temporary) ballistic evolution towards the landing zone, which is briefly interrupted for the reentry burn, followed by an actively controlled evolution, through the activation of the aerodynamic control system (grid fin-based ACS), engine ignition for landing burn and deployment of landing gear.

Similar to the breakdown of the main mission trajectory into smaller flight phases, it is now of interest to explicitly mention critical events and evolution phases that occur during the recovery mission of the first stage. These are presented in

Table 4. Recovery mission flight phases.

Flight Phase	Description	Engine	RCS	ACS	Landing System
1	First-stage separation	Off	Off	Undeployed	Folded
2	Gravity turn/Ballistic evolution 1 (AoA = 0°)	Off	Off	Undeployed	Folded
3	Flip-over maneuver	Off	On	Undeployed	Folded
4	ACS deployment	Off	On for minor corrections	Deployed	Folded
5	Gravity turn/Ballistic evolution 2 (AoA = 180°)	Off		Deployed	Folded
6	Reentry burn	On		Deployed	Folded
7	Aerodynamic guidance	Off		Active	Folded
8	Landing maneuver	On		Active	Deployed
9	Touchdown and engine shutdown	Off		Deployed	Deployed

.

The start of the recovery mission corresponds to the first-stage separation which is performed after the first-stage propellant reserved for the main mission is burned. This moment is considered T0 and will be used as a reference point in the propagation of the recovery trajectory (where the first stage has a certain altitude, velocity, orientation, etc.).

After stage separation, the first ballistic evolution occurs, where the equations of motion are similar to the ones used in the gravity turn phases of the main mission. The flight path angle $\gamma$ decreases naturally due to the gravitational acceleration, and the aerodynamic angles $\alpha \mathrm{a}\mathrm{n}\mathrm{d} {\beta }^{*}$ are both null.

The third flight phase corresponds to the flip-over maneuver of the first stage, a maneuver which is performed using the RCS at the apogee of the ballistic trajectory. After this moment, the RCS is considered to be in use only for small corrections. The angle of attack increases from 0° to 180°, while the sideslip angle ${\beta }^{*}$ is null.

The next important phase is the deployment of the aerodynamic control system (ACS), performed right after the flip-over maneuver is finalized. From this moment, the ACS is capable of modifying the attitude of the lower stage, but realistically is not activated (non-zero deflection angles) at high altitude because of the low performance.

The fifth flight phase corresponds to the second ballistic evolution, where the first stage continues its gravity turn. The flight path angle $\gamma$ decreases naturally due to the gravitational acceleration. A value of $\gamma =-90°$ is desired at stage touchdown.

The next phase of the recovery mission is the reentry burn, in which the rocket engine is restarted at a target altitude $H_{reentry}$ (optimization variable, details in Section 2.6.2) to slow down the lower stage. This maneuver is performed in a more rarefied atmosphere to limit the thermal load and protect the structural integrity of the reusable stage. The rocket engine is shut down at the completion of the reentry maneuver, when the amount of propellant available for this maneuver is consumed.

The seventh recovery mission phase is the aerodynamic guidance flight and corresponds to the period immediately following the reentry maneuver. The evolution can be seen as a semi-ballistic one, the rocket engine being turned off, the only method of changing the launcher’s attitude being by using the ACS.

The next milestone is the landing maneuver which is performed when the stage altitude drops below a preset threshold $H_{landing}$. At the start of the maneuver, the rocket engine is restarted. The maneuver is considered complete after the available propellant ${M_{landing}}_{max}$ is consumed. During the last portion of the maneuver (a few seconds), the landing system is deployed. RCS and ACS allow small corrections for the stage attitude.

The final recovery mission phase (number 9) corresponds to stage touchdown and engine shutdown. When the altitude of the lower stage hits zero, the stage is considered to have landed. It is checked whether the vertical velocity of the stage at the moment of touchdown is below a critical threshold ${v_{landing}}_{max}$. If a soft touchdown is realized (${{v_{landing}<v}_{landing}}_{max}$) then the recovery mission is considered successful; otherwise, the landing maneuver has caused stage structural damage (how the violation of set constraints impacts the overall objection function is detailed in Section 2.6.3).

Since the dynamic model used in the Trajectory module of the MDO is based on a 3DOF formulation and the studied configuration is approximated as a material point of variable mass, the system of differential Equation (48) can also be used for the case of a stage recovery mission. However, the major difference between studying a microlauncher configuration and a lower-stage configuration is given by the type (and magnitude) of applied force components $\left[N_x,N_y,N_z\right]$. For an easier definition of the applied forces, it is best to graphically represent the new body frame (previously presented in Figure 13 for the microlauncher case) now for the lower-stage case. This is achieved in Figure 16.

Since one of the assumptions of the 3DOF dynamic model used is that the thrust vector is along the vehicle central axis (either launcher or first stage), the propulsive force $T_{{body}_{frame}}$ is computed exactly the same as for the microlauncher case defined in Equation (50).

In order to compute the aerodynamic force in the body frame $F_{{body}_{frame}}$ for the lower-stage case, we use a similar procedure to the one described for the microlauncher case (Section 2.4.1). Equation (51) is still valid, but only for the flight phases where the ACS is not deployed (first-stage “clean configuration” + landing system contribution).

Additionally, one must express the aerodynamic forces generated by the grid-fin-based aerodynamic control system (ACS), noted with ${ACS}_{{body}_{frame}}$. The grid fins are numbered as shown in Figure 17a, while angle convention is shown in Figure 17b.

In the body frame, the aerodynamic force given by the ACS (${ACS}_{{body}_{frame}}$) has the components $X^{ACS}$, $Y^{ACS}$, $Z^{ACS}$ defined by

$${ACS}_{{body}_{frame}}={\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin A}+{\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin B}+{\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin C}+{\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin D}$$

(55)

By using the sign convention from Figure 17, grid fin contributions are written as

$${\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin A/B}=\left[\begin{array}{c}q\cdot S_{ref}\cdot C_x\\ 0\\ q\cdot S_{ref}\cdot C_z\end{array}\right]; {\left.\left[\begin{array}{c}{X}^{ACS}\\ Y^{ACS}\\ Z^{ACS}\end{array}\right]\right|}_{Grid fin C/D}=\left[\begin{array}{c}q\cdot S_{ref}\cdot C_x\\ q\cdot S_{ref}\cdot C_y\\ 0\end{array}\right]$$

(56)

where $q$ is the dynamic pressure, $S_{ref}$ is the reference area and $[C_x,C_y,C_z]$ are the aerodynamic coefficients in the body frame (computed based on the data from [68]).

The way in which the control is achieved during the recovery mission is directly related to the flight phases presented in Table 4. When the engine is on, full-throttle setting is considered and no additional control scheme dependent on the aerodynamic angles is used during reentry or landing burns (like Equations (52) and (53) for the microlauncher).

For this paper, it was considered that the RCS manages to rotate the lower stage to predefined values instantaneously, the rotational dynamics of the lower stage not being simulated (because of the 3DOF environment); thus, the attitude variation is treated as a sudden change in the angle of attack (for example, the angle of attack increases from 0° to 180° during the flip-over maneuver instantaneously).

The ACS is deployed (but not activated) after the flip-over maneuver, the first stage angle of attack being approximately 180° for the rest of the descent trajectory. At the moment, no control laws are implemented for the active aero phases of the recovery mission (as it was observed later that a successful landing and recovery of the first stage from a secondary location can be achieved without active aero control, the grid fin being considered to be at a constant 0° deflection angle). This is mainly because the landing location (or downrange) was not imposed before the mission definition, but rather defined after trajectory optimization and used as reference for any real-world application.

2.5. Cost Estimation Module

For the current MDO version, the use of a mathematical model derived from TransCost [101,102,103,104,105] is proposed, which is based on system-level cost estimation relations (CER) and can be used in the conceptual design phase of a space launch vehicle (either expendable or reusable). The cost estimation is broken down into three smaller interconnected submodels: development costs, production costs and operation costs.

This model breakdown structuring offers the advantage of estimating costs separately or combined depending on the mission application area. Development and production costs are estimated based on CER, with the aid of relations of the following type:

$$C=a·M^x$$

(57)

where $a$ is a system constant, $x$ is a specific sensitivity factor correlating the cost $C$ (expressed in Man-Year [MYr]) and $M$ is the mass (expressed in kg).

The transformation between MYr and local currency can be made afterwards to obtain more palpable data when comparing different launchers. The total cost per launch ($C_{pL}$) is obtained by amortizing the development costs ($C_{DEV}$) over a fixed number of estimated flights ($N_{amortization}$) and summing them with the production ($C_{PROD}$) and operations costs (direct $C_{DOC}$, indirect $C_{IOC}$ and refurbishment costs $C_{RSC}$). Thus, one can use

$$C_{pL}={\displaystyle \frac{C_{DEV}}{N_{amortization}}}+C_{PROD}+C_{DOC}+C_{IOC}+C_{RSC}$$

(58)

2.5.1. Development Costs

Development costs are the non-recurring costs which include all activities from the early design phases (such as the optimization process realized in this paper) to the more advanced phases. The total development costs ($C_{DEV}$) are given by [101]

$$C_{DEV}=f_0\left(\sum _1^N{H}_E+\sum _1^N{H}_{V }\right)f_6{f}_7{f}_8$$

(59)

where $f_0$ is a system integration factor, $N$ is the number of microlauncher stages, $H_E$ is the engine-related development costs (for each stage), $H_V$ is the vehicle system development costs (for each stage) and $(f_6, f_7, f_8)$ are programmatic cost impact factors.

The system integration factor $f_0$ can be expressed as [101]

$$f_0={1.04}^N$$

(60)

The first programmatic cost factor $f_6$ corresponds to the increase in cost due to deviations from optimum time schedule. A unit value of this factor suggests the project will be completed on time, but in practice the deadline is not always met, which is why the factor takes values between 1 and 1.5 [101]. The value $f_6=1.1$ has been used in the MDO.

The second programmatic cost factor $f_7$ quantifies the inefficiencies that arise when there are multiple parallel contractors. Its value is defined as follows [101]:

$$f_7=n^{0.2}$$

(61)

where $n$ is the number of contractors. In the MDO context, the value $n=1$ is used.

The final programmatic cost factor $f_8$ corresponds to a personnel correction factor and highlights the differences in productivity between different regions or countries. A list of values for this coefficient is given in [101], a value of $f_8=0.86$ (Europe) being used.

For a liquid propellant rocket engine (with turbopumps as feed system), the following CER is implemented inside the Cost estimation module of the MDO for $H_E$:

$$H_E=277{M_{engine}}^{0.48}{f}_1{f}_2{f}_3$$

(62)

where $M_{engine}$ is the engine mass [kg] and $\left(f_1, f_2, f_3\right)$ are technical correlation factors.

The first correlation factor is $f_1$ which corresponds to a technical development standard factor that quantifies whether the current vehicle brings a completely innovative approach, is based on a state-of-the-art design or brings modifications to an already existing system. A list of values for this coefficient is given in [101], the value $f_1=1.1$ being chosen for our case (associated with a standard project).

The next correlation factor used in Equation (62) is $f_2$ and corresponds to a technical quality factor (specific to each main subsystem of the launcher). It has been previously shown that the greatest impact on the development cost of the rocket engine is due to the number of engine tests, rather than the type of propellant [106,107]. Based on existing data for multiple engines, the following relation was determined in [101]:

$$f_2=0.026{\left(ln{N}_Q\right)}^2$$

(63)

where $N_Q$ is the number of tests (preliminary, qualification, operational and other tests needed) performed for the rocket engine ($N_Q=500$ is implemented as proposed in [101]).

The final correlation factor is $f_3$ which quantifies the relevant experience of the team within the project, as a team with no or limited experience will significantly increase the cost associated with engine development. A list of values for this coefficient is given in [101], a value of $f_3=1.1$ (partially new project activities) being chosen for this paper.

The vehicle system development costs are broken down into propulsion module costs ($H_V$) and stage costs (either $H_{VP}$ for an expendable stage or $H_{VB}$ for a reusable stage), which are added up to obtain the final value of $H_V$.

For the definition of a propulsion module costs, the following relation can be used:

$$H_{VP}=14.2{M_{engine}}^{0.577}{f}_1{f}_3$$

(64)

For an expendable stage, the cost associated with the entire stage structure (minus the propulsion module which was estimated separately) can be estimated using

$$H_{VE}=100{{(M}_{dry}-M_{engine})}^{0.555}{f}_1{f}_2{f}_3$$

(65)

where $M_{dry}$ is the stage dry mass (without propellant) and the quality factor $f_2$ depends on the stage net mass fraction [101] (it can be considered that $f_2=1$ in the preliminary phase of design, but a further investigation should be considered in detailed phases).

In the case of a reusable stage, one can use

$$H_{VB}=803.5{(M_{dry}+M_{recovery}-M_{engine})}^{0.385}{f}_1{f}_2{f}_3$$

(66)

where $M_{recovery}$ is the propellant mass reserved for the recovery mission.

2.5.2. Production Costs

Production costs include the main costs of materials, processing, manufacturing, assembly and compliance testing, together with other costs associated with quality assurance and technical support. Production costs are substantially lower than development costs, for the case of an expendable launcher being approximately 1.5% of the total development costs [101]. The production costs (per unit) can be estimated using [101]

$$C_{PROD}=f_0\left(\sum _1^N{F}_E+\sum _1^N{F}_{V }\right)f_8$$

(67)

where $f_0$ is a system integration factor ($f_0={1.02}^N$), $F_E$ is the engine production cost (for each stage), $F_V$ is the vehicle system production cost (for each stage) and $f_8$ is the productivity correction factor ($f_8=0.86$, similar to the development costs).

Depending on the liquid propellant pair used, one of the following equations can be used to assess $F_E$ (for every stage) [101]:

$$F_E=1.9n{M_{engine}}^{0.535}{f}_4; F_E=5.16n{M_{engine}}^{0.45}{f}_4.$$

(68)

where $n$ is the number of identical engines per stage ($n>1$ for medium and large launchers, usually $n=1$ for microlaunchers), $M_{engine}$ is the engine mass [kg] and $f_4$ is the cost reduction factor derived from the learning factor $p$ (the $f_4=1$ was used) [101].

The vehicle system production cost $F_V$ is computed with the aid of

$${F_V=F_{VP}+F}_{VE/VB}$$

(69)

The propulsion module production cost $F_{VP}$ can be estimated using

$$F_{VP}=4.65n{M_{engine}}^{0.49}{f}_4$$

(70)

The entire stage structure (both for expendable and reusable stages) production cost is dependent on the liquid propellant pair used and can be estimated using one of the following equations [101]:

$$F_{VE/VB}=0.83n{M^{*}}^{0.65}{f}_4; F_{VE/VB}=1.3n{M^{*}}^{0.65}{f}_4.$$

(71)

with $M^{*}=M_{dry}-M_{engine}$ (similar to Equation (65)).

2.5.3. Operation Costs

The following categories of operation costs are independently assessed in the Cost estimation module of the MDO (based on the TransCost model [101]): Direct Operation Costs ($C_{DOC}$), Indirect Operation Costs ($C_{IOC}$) and Refurbishment Costs ($C_{RSC}$).

The first main category of costs analyzed is the Direct Operation Costs ($C_{DOC})$, which are further divided into five main contributions: Ground Operations ($C_{GOC}$), Materials and Propellants ($C_{FOC}$), Launch, Flight and Mission Operations ($C_{LFMO}$), Transport and Recovery ($C_{TREC}$), and Fees and Insurance ($C_{FIC}$). Thus, one can write

$$C_{DOC}=C_{GOC}+C_{FOC}+C_{LFMO}+C_{TREC}+C_{FIC}$$

(72)

For the Ground Operation cost, the following relation is used [101]:

$$C_{GOC}=8{M_{GLOW}}^{0.67}{L}^{-0.9}{N}^{0.7}{f}_V{f}_C{f}_4{f}_8$$

(73)

where $M_{GLOW}$ is the lift-off mass of the launcher (in metric tons), $L$ is the launch rate (per year), $N$ is the number of stages, $f_V$ is a coefficient that quantifies the impact of the launcher type (between 0.7 and 1), the coefficient $f_C$ quantifies the impact of the integration method (between 0.5 and 1) and $(f_4,f_8)$ have been previously defined.

The propellant cost has a small contribution to the total launch costs (around 0.5–1.5%) and depends on the type of oxidizer/fuel pair and annual demand. An average cost of 0.2 EUR/kg for LOX and 8.8 EUR/kg for methane has been implemented inside the MDO [108].

For the Launch, Flight and Mission Operation cost, the following relation is used:

$$C_{LFMO}=20\sum Q·L^{-0.65}{f}_4{f}_8$$

(74)

where $Q$ is a coefficient dependent on the complexity of the launcher architecture [101].

Transport and Recovery cost is approximated using

$$C_{TREC}={\displaystyle \frac{1.5}{L}}\left(7L^{0.7}+{M_{recovery}}^{0.83}\right)f_8$$

(75)

where $M_{recovery}$ is the recovery mass of the stage [kg].

Fees and insurance costs are very hard to estimate in the preliminary stages of design and are considered to be negligible compared to the total costs per launch; thus $C_{FIC}=0$. The Indirect Operation costs can be estimated with the aid of [109]

$$C_{IOC}=40S+22.5L^{-0.379}$$

(76)

where $S$ represents the ratio between the administrative effort of subcontractors compared to the total effort. In the current project, $S=0$ can be used given the intention to generate concepts for locally operated launchers (Europe).

For the Refurbishment cost estimation, the following model is proposed based on the statistical data presented in [101]:

$$C_{RSC}=0.053{C_{PROD}}_{reusable stage}$$

(77)

where ${C_{PROD}}_{reusable stage}$ is the production cost of the reusable stage (from Equation (67)).

2.6. Additional Modules

Alongside the five main modules presented so far, within the architecture of the MDO algorithm developed (Figure 1), an additional four modules are needed to successfully generate a reusable microlauncher concept. During this section, the main observations related to the contents of the additional modules will be presented, with special attention being given to the Optimization Variables Section (Section 2.6.2) and Objective Function (Section 2.6.3).

2.6.1. Requirements and Input Data

The various input data required to run the MDO algorithm are defined within this module. This module is called only once at the beginning of a microlauncher optimization process. Here, the target orbit requirements, any trajectory constraints to be applied (main mission or recovery mission), and the design requirements are defined. The solution search space is defined by specifying clear upper and lower bounds for the optimization vector variables.

The launch site (and landing location if predetermined, such as the case of a return to launch site recovery) is also defined through geographical coordinates. Within this module, the separation condition of the satellite’s protective aerodynamic fairing is defined. The materials used for different components of the launcher, together with the mechanical and thermal characteristics, are also specified here.

2.6.2. Optimization Variables

The global optimization process of the microlauncher concept can be seen as the optimization of key parameters (optimization variables) that define its main characteristics, along with some key parameters through which the main mission and recovery mission reference trajectories are generated. Thus, the microlauncher optimization is achieved by obtaining an adequate optimization variable vector (close to optimal, but which cannot be proven mathematically). The launcher is then completely defined by means of global input data (overview given in Section 2.6.1) and optimization variables.

The optimization variables are necessary for the mathematical models developed in the five main modules (details in Section 2.1, Section 2.2, Section 2.3, Section 2.4 and Section 2.5). Where the implementation was convenient, a dimensionless formulation of these parameters was considered in order to reduce the search space of the optimal solution as much as possible (for example, the burn time of the rocket engine is not optimized, but rather the thrust/weight ratio at start-up).

The optimization vector structure can be split into four main subsets, with each subset being responsible for a distinct part of the optimization process:

• First-stage definition, based on the optimization variables presented in

  Table 5. Optimization variables, subset 1 (first-stage definition).

  Index in Optimization Variable Vector	Name	Unit	Description
  1	${M_p}_1$	[t]	First-stage propellant mass
  2	${D_e}_1$	[m]	First-stage exterior/outer diameter
  3	${P_c}_1$	[bar]	Combustion chamber pressure of the first-stage engine
  4	${P_e}_1$	[bar]	Exhaust pressure of the first-stage engine
  5	${\left({\displaystyle \frac{T}{W}}\right)}_1$	[-]	Thrust to weight ratio at first-stage engine start

  ;
• Second-stage definition, based on the optimization variables presented in

  Table 6. Optimization variables, subset 2 (second-stage definition).

  Index in Optimization Variable Vector	Name	Unit	Description
  8	${M_p}_2$	[t]	Second-stage propellant mass
  9	${D_e}_2$	[m]	Second-stage exterior/outer diameter
  10	${P_c}_2$	[bar]	Combustion chamber pressure of the second-stage engine
  11	${P_e}_2$	[bar]	Exhaust pressure of the second-stage engine
  12	${\left({\displaystyle \frac{T}{W}}\right)}_2$	[-]	Thrust to weight ratio at second-stage engine start

  ;
• Main mission definition, based on the optimization variables presented in

  Table 7. Optimization variables, subset 3 (main mission definition).

  Index in Optimization Variable Vector	Name	Unit	Description
  6	${{\gamma }_d}_1$	[°]	Desired flight path angle for the primary active guidance phase
  7	${{\chi }_d}_1$	[°]	Desired track angle for the primary active guidance phase
  13	$t_{vertical}$	[s]	Vertical ascent flight duration (after lift-off)
  14	$t_{coast}$	[s]	Coast phase duration (time between first-stage separation and second-stage engine ignition)
  15	$\Delta t_1$	[-]	Ratio between the active guidance phase and total first stage guidance phases (active guidance + primary gravity turn)
  16	$\Delta t_2$	[-]	Ratio between active guidance phase and total second stage guidance phases (orbital insertion + secondary gravity turn)

  ;
• Recovery mission definition, based on the optimization variables presented in

  Table 8. Optimization variables, subset 4 (recovery mission definition).

  Index in Optimization Variable Vector	Name	Unit	Description
  17	$M_{p_{recovery}}$	[t]	Propellant mass reserved for recovery mission (from first-stage tanks)
  18	$H_{reentry}$	[km]	The altitude at which the reentry maneuver begins
  19	$\Delta M_{p_{landing}}$	[-]	Ratio between the propellant mass reserved for the landing maneuver and recovery mission ($M_{p_{recovery}}$)
  20	$H_{landing}$	[km]	The altitude at which the landing maneuver begins

  .

As can be seen from Table 5, Table 6, Table 7 and Table 8, the total number of optimization variables needed for the generation of a reusable microlauncher concept is 20. Of these, 10 are used to define the main characteristics of the microlauncher (dimensions, masses, engine performances), 6 are used to define the main mission of the launcher (orbit insertion of payload into predefined target orbit), and the last 4 are needed to define the recovery mission (from a secondary location).

The recovery mission can be fully defined with the aid of just four optimization variables (presented in Table 8). Stage separation occurs when the propellant reserved for the first-stage ascent flight (of the maim mission) is fully consumed (${M_p}_1-M_{p_{recovery}}$) and is considered the reference starting point of the recovery mission definition.

When the lower stage drops below the threshold altitude $H_{reentry}$, the reentry burn starts. The burn ends when the propellant mass reserved for this maneuver is consumed (computed based on the optimization variable $\Delta M_{p_{landing}}$).

When the lower stage drops below the $H_{landing}$ threshold, the landing burn starts and finishes when either the altitude has reached 0 or the reserved propellant mass of the recovery mission has been consumed.

2.6.3. Objective Function

The objective function establishes the criteria by which the solution is selected and advanced (by means of the numerical values inside the optimization variable vector presented earlier in Section 2.6.2). Possible formulations of the objective functions suitable for satellite microlaunchers are those of minimizing the lift-off mass, minimizing the cost of the vehicle, maximizing the mass performance index (the ratio between the payload and the launcher mass) or maximizing the payload mass (if the vehicle is already defined).

Since the current worldwide trend is that of miniaturization [1,16], an objective function $f_{objective}$ is defined and used within the MDO algorithm in which the dominant criterion is the minimum lift-off mass of the launcher, based on the following formulation:

$$f_{objective}=\left(M_{start}+I_{mission}+I_{recovery}\right)\cdot I_{constraints}$$

(78)

where $M_{start}$ is the lift-off mass of the launcher (obtained with the aid of the Preliminary design module described in Section 2.1), $I_{mission}$ is the main mission performance index, $I_{recovery}$ is a recovery mission performance index and $I_{constraints}$ is a constraint index (computed based on both main mission and recovery mission data).

The formulation of the objective function is similar to the one used in previous papers [1,13,16] adapted to the current application by introducing a recovery mission performance index $I_{recovery}$.

The main mission performance index is used to quantify whether the mission has been completed successfully by assessing the difference between the orbit in which the payload has been inserted versus the target one (earlier defined in the Requirements and input data module). The formulation used for the definition of $I_{mission}$ is the following:

$$I_{mission}=\sqrt{w_a{\left(a-a_t\right)}^2+w_V{\left(V-V_t\right)}^2+w_{\gamma }{\left(\gamma -{\gamma }_t\right)}^2+w_i{\left(i-i_t\right)}^2}$$

(79)

where $\left(w_a,w_V,w_{\gamma },w_i\right)$ are parameter weights, $a$ is the semimajor axis, $V$ is the inertial velocity, $\gamma$ is the flight path angle, $i$ is the orbit inclination and $t$ indices correspond to target values (based on the predefined target orbit).

More details on the correlations between the position and velocity vectors (which are obtained from the system of Equations (48)) and the six classical orbital parameters can be found in [1,16]. For the parameter weights, the following values have been used:

$$w_a=w_V=1; w_{\gamma }=w_i=10$$

(80)

In the case of a perfect main mission of the launch vehicle, the payload is inserted into the target orbit and thus the value of the $I_{mission}$ performance index is null. During the optimization process, the values of $I_{mission}$ will decrease as the selection and advancement algorithm (details in Section 2.6.4) updates the optimization variable vector, and it is expected that it converges to a numerical value close to 0 (usually $I_{mission}<{10}^{-2}$).

The recovery mission performance index is used to quantify whether the recovery mission has been accomplished successfully by assessing the error/deviation from the required landing location $E_{{landing}_{location}}$ (if imposed) and the surplus of unused propellant mass $M_{p_{unused}}$. The following formulation is used to asses $I_{recovery}$:

$$I_{recovery}=M_{p_{unused}}+E_{landing}$$

(81)

The first term used in Equation (81) $M_{p_{unused}}$ corresponds to the unused propellant mass which has been already transported from the launch location and not used during the recovery mission of the first stage (effectively a mass that has no purpose and will negatively impact the performance of the launcher by means of increasing its lift-off mass). The second term $E_{landing}$ corresponds to the position error between the target landing location and the one obtained.

For the current paper, we can consider $E_{landing}=0$ because the landing location is not strictly imposed but rather defined after trajectory optimization and used as a reference. In the case of a return to launch site recovery mission, or in the case the landing location is imposed to be at a maximum downrange from the launch site, then $E_{landing}$ could be computed as a function of the haversine distance [110]:

$$d_{1-2}=2r·\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left(\sqrt{{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left({\displaystyle \frac{∆\phi }{2}}\right)+\cos {\phi }_1·\cos {\phi }_2·{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left({\displaystyle \frac{∆\lambda }{2}}\right)}\right)$$

(82)

where $d_{1-2}$ is the minimum distance between two points on the Earth’s surface taking into account its spherical curvature, $∆\phi ={\phi }_2-{\phi }_1$ is the difference between the latitudes of the two points, $∆\lambda ={\lambda }_2-{\lambda }_1$ is the difference between the longitudes of the two points and $r$ is the radius of the Earth.

Similar to the main mission performance index $I_{mission}$, the recovery mission performance index $I_{recovery}$ tends to the value 0 for a successful recovery mission (though without checking whether any imposed constraints have been violated).

The constraint index is used to quantify the validity of the obtained trajectories compared to the imposed requirements. The following formulation is used to asses $I_{constraints}$:

$$I_{constraints}=\prod _{i=1}^{N_{constraints}}{I}_{{constraint}_i}$$

(83)

where $N_{constraints}$ is the total number of constraints implemented and $I_{{constraint}_i}$ is the performance index corresponding to constraint $i$.

If constraint $i$ is respected, then the term $I_{{constraint}_i}$ takes unitary value and does not alter the objective function detailed in Equation (78). If the imposed constraint $i$ is not respected (the constraint is violated for a particular set of optimization variables), then the term $I_{{constraint}_i}$ is computed using

$$I_{{constraint}_i}=1+\left|{\displaystyle \frac{v-v_{imposed}}{v_{imposed}}}\right|·f_{penalty}$$

(84)

where $v_{imposed}$ is the imposed value of constraint $i$, $v$ is the obtained value of the constraint parameter and $f_{penalty}$ is a penalty factor ($f_{penalty}=100$ for the main mission constraints and $f_{penalty}=1$ for the recovery mission constraints).

2.6.4. Selection and Advancement Algorithm

This algorithm is very important as it is responsible for generating the numerical values within the optimization variable vector, which are needed for the multidisciplinary analysis (within the five main modules of the MDO algorithm). In the specialized literature there are numerous selection and advancement methods that are used for optimization problems [15], but not all of them can be applied to a space launch vehicle problem. Due to the complexity of the mathematical models used in the main modules of the MDO algorithm, the number of optimization variables is quite high (for example 20 variables for the case of optimizing a partially reusable two-stage microlauncher). Thus, solution and advancement methods are needed that can work efficiently with a high number of variables, the most suitable being the heuristic category [1,15,111].

There are numerous heuristic or evolutionary search algorithms, the most robust and most widely used in the aerospace field being the genetic one, presented in detail in [112,113,114,115]. This algorithm is based on Darwin’s evolutionary theory and is inspired by the biological processes that occur in nature (selection, reproduction/crossover and mutation). This type of algorithm is often used to solve problems in which obtaining the optimal solution using gradient algorithms is inefficient due to the large number of variables [15] and will also be implemented in the current study.

3. Mission Requirements and Studied Microlauncher Architecture

When addressing possible microlauncher solutions, the main requirement is the definition of the payload that will be housed inside the upper structure. As the name implies, the major category of satellites that will be inserted into orbit with the aid of a microlauncher corresponds to microsatellites, which usually hover in the 10 kg–100 kg interval [116]. Most satellites in this category have as their primary objective Earth observation missions; thus the orbits of interest for such small satellites are Low Earth Orbits (LEO) of high inclination (the best one being the polar orbit as it provides full Earth coverage).

For these orbits, the target altitudes usually range between 200 km and 800 km, with a concentration in the 400 km zone. Detailed marketing analyses show that there is a continuously growing market for compact satellites in the 100 kg mass range, which ensures that the development of a dedicated launcher for this category of satellites is of current interest (more details are given in [1]).

Considering all of the above, the main mission of the reusable microlauncher optimized in this paper (shown in

Table 9. Main mission target orbit.

Requirement	Type
Orbit type	Circular
Orbit altitude	400 km
Orbit inclination	Polar (90°)
Payload mass	100 kg

) is that of inserting a 100 kg payload into a circular LEO. The target altitude was set at 400 km, which is approximately the orbital altitude of the International Space Station (ISS) [1]. Details related to the target parameters (used in the objective function assessment) are given in

Table 10. Main mission target parameters (payload orbit insertion).

Parameter	Target Value
Semimajor axis	$a_{target}=6778.137$ km
Eccentricity	$e_{target}=0$
Orbit inclination	$i_{target}=90°$
Velocity (inertial)	$V_{target}=7.6686$ km/s
Flight path angle	${\gamma }_{target}=0°$

.

Table 11. Reusable microlauncher architecture.

Specification	Type
Number of stages	2
Recoverable stages	1 (first stage)
Stage diameters	Constant
Propellant	LOX + Methane
Component length safety margin	10%
Dry mass safety margin	Stage 1: 0%; Stage 2: 5%
Propellant mass safety margin	Stage 1: 5% of recovery propellant; Stage 2: 0%

presents the most important input data related to the reusable microlauncher architecture implemented. The architecture studied is similar to that of the Falcon 9 launcher [33], with the mention that the fuel pair used corresponds to the new generation of SpaceX engines (Raptor). The safety margins (length and mass) are used to further increase the confidence in the proposed constructive solution of the launcher.

The launch location considered in this study was Andøya Space Centre, Norway, which provides adequate launch infrastructure, has a clear launch corridor and is a typical European launch site for LEO. The most important initial conditions used to initialize the Trajectory module are presented in

Table 12. Reusable microlauncher initial conditions.

Initial Condition	Specification
Launch location	Andøya Space Centre, Norway
Longitude	${\lambda }_{initial}=16.0198°$
Latitude	${\phi }_{initial}=69.2944°$
Altitude	$H_{initial}=0 \mathrm{m}$
Flight path angle	${\gamma }_{initial}=90°$
Track angle	${\chi }_{initial}=1.2285°$
Launcher velocity	$V_{initial}=1$ m/s

. A small value (1 m/s) for the start velocity was used to avoid the singularity point of the system of Equation (48).

Since the inclination of the target orbit is much higher than the initial latitude (that of the launch location), one can use the following relations [117] to obtain the initial track angle corresponding to the optimal launch direction:

$$\begin{gathered} {\beta }_{inertial}=arcsin\left({\displaystyle \frac{cos\left(i_{target}\right)}{cos\left({\phi }_{initial}\right)}}\right); \\ {\beta }_{launch}=arctan\left({\displaystyle \frac{V_{target} sin\left({\beta }_{inertial}\right)-V_{eqrot} cos\left({\phi }_{initial}\right)}{V_{target }cos\left({\beta }_{inertial}\right)}}\right); {\chi }_{initial}=-{\beta }_{launch}. \end{gathered}$$

(85)

where $V_{eqrot}$ is the Earth’s rotation velocity at the equator (approximately 465.1 m/s [1]).

For the fairing separation condition implemented inside the MDO, a formulation based on a dynamic pressure threshold is used (similar to [118]), where the fairing is jettisoned when the dynamic pressure drops below 0.5 kPa. Additionally, to avoid stability problems, a constraint is introduced related to the moment of fairing separation; namely, separation occurs during the coast period between stages (when the microlauncher assembly does not have the rocket engine turned on).

For the active control phases of the launcher defined by Equations (52) and (53), the following tuning parameters are used [1]:

$$\begin{array}{c}{k}_1=7;\\ k_2=7;\\ k_3=\left\{\begin{array}{c} -700, \mathrm{i}\mathrm{f} cos\left(\chi \right)>0\\ 700, \mathrm{i}\mathrm{f} cos\left(\chi \right)<0\end{array}\right..\end{array}$$

(86)

To ensure that the TVC can deflect the engine nozzle in a 6DOF environment in such a way that the thrust vector is modeled correctly in a 3DOF environment, similar limitations are implemented regarding the control variables used in the MDO (aerodynamic angles $\alpha$ and ${\beta }^{*}$). The aerodynamic angles have been limited to the following values during trajectory propagation (for main mission) [1]:

$$\begin{gathered} -7°\le \alpha \le 7°; \\ -7°\le {\beta }^{*}\le 7°. \end{gathered}$$

(87)

The lower-stage recovery mission can be viewed as a separate optimization problem, with well-defined initial and final conditions. The initial conditions of the lower-stage recovery trajectory optimization problem are those occurring at the moment of stage separation (from the microlauncher assembly), while the final conditions correspond to stage touchdown. Of course, the optimization of the lower-stage recovery trajectory is not performed independently of the optimization of the microlauncher main mission ascent trajectory, but rather realized intrinsically by obtaining an optimal optimization variable vector (Section 2.6.2) that minimizes the implemented objective function (Section 2.6.3).

The most important constraints implemented for the reusable microlauncher optimization problem are presented in

Table 13. Implemented constraints.

Parameter	Constraint	Description
$n_{axial}$	≤$11$	Axial load factor below 11
$n_{normal}$	≤$0.75$	Normal load factor below 0.75
$\left|{\displaystyle \frac{D_{e_1}-D_{e_2}}{D_{e_2}}}\right|$	≤$0.0001$	Very small diameter difference between consecutive stages (≈ stages of the same diameter)
${\displaystyle \frac{L_{e_1}}{D_{e_1}}}$	≤$10$	Maximum first stage fineness ratio of 10 (without interstage)
${\displaystyle \frac{L_{e_2}}{D_{e_2}}}$	≤$5$	Maximum second stage fineness ratio of 5
$\epsilon$	$5\le \epsilon \le 90$	Nozzle expansion ratio between 5 and 90
${\gamma }_{d1}$	$\subset {\gamma }_{{active}_{\mathrm{guidance}1}}$	Achievement of the imposed trajectory flight path angle and track angle during primary active guidance phase
${\chi }_{d1}$	$\subset {\chi }_{{active}_{\mathrm{guidance}1}}$
$\left|{\displaystyle \frac{{\gamma }_{d1}-{\gamma }_{fina{l}_{guidance1}}}{\Delta t_{\approx constant}}}\right|$	≤$0.25$ °/s	Deviation from the required flight path and track angles less than 0.25 deg/s (for a short period of time, these angles are kept quasi-constant such that the velocity vector aligns with the thrust vector)
$\left|{\displaystyle \frac{{\chi }_{d1}-{\chi }_{fina{l}_{{active}_{\mathrm{guidance}1}}}}{\Delta t_{\approx constant}}}\right|$	≤$0.25$°/s
$V_{landing}$	≤$3$ m/s	Recoverable first-stage landing velocity below 3 m/s
$\Delta {\gamma }_{landing}$	≤$2$°	Maximum deviation from vertical position at the moment of landing of 2°

. The last two constraints correspond to the recovery mission (being necessary for a soft landing), while the rest are actively considered for the main mission.

The search space considered for the optimization problem is shown in

Table 14. Solution search space.

Nr.	Lower Bound	Optimization Variable	Upper Bound	Nr.	Lower Bound	Optimization Variable	Upper Bound
1	5	${M_p}_1$	25	11	0.01	${P_e}_2$	0.15
2	1	${D_e}_1$	2	12	0.4	${\left({\displaystyle \frac{T}{W}}\right)}_2$	1.5
3	60	${P_c}_1$	80	13	2	$t_{vertical}$	100
4	0.3	${P_e}_1$	1	14	2	$t_{coast}$	100
5	1	${\left({\displaystyle \frac{T}{W}}\right)}_1$	5	15	0.01	$\Delta t_1$	0.5
6	50	${{\gamma }_d}_1$	85	16	0.01	$\Delta t_2$	0.5
7	${\chi }_{initial}-5$	${{\chi }_d}_1$	${\chi }_{initial}+5$	17	0.1	$M_{p_{recovery}}$	1.0
8	0.5	${M_p}_2$	5	18	50	$H_{reentry}$	100
9	1	${D_e}_2$	2	19	0.01	$\Delta M_{p_{landing}}$	1
10	60	${P_c}_2$	80	20	0.01	$H_{landing}$	5

. It can be easily seen that the search limits of the solution are very extensive, a large initial space being desired in order not to exclude the optimal solution. This search space will be narrowed during the evolution of the MDO algorithm, and upon convergence, a single optimal solution (a single vector of optimization variables) will be obtained.

4. Results

4.1. MDO Convergence

For the convergence of the reusable microlauncher concept, 3996 generations of the genetic algorithm were required. An initial population of 200 individuals was used (10 times the number of optimization variables), while the number of elites was set to 5%. The optimization was performed on a workstation with an Intel i9-14900K processor (24 cores) and 128 GB RAM, using MathWorks Matlab version R2024a (with parallel computing active). Convergence was achieved after approximately 2 h. The solution was considered converged when the objective function had not improved after 100 consecutive generations (after 19,000 new microlauncher concepts and associated trajectory evaluations).

The convergence of the objective function is presented in Figure 18a,b. At the end of the multidisciplinary optimization process, the minimum value $f_{objective}=14.8309$ was obtained. A rapid decrease in the value of the objective function occurs in the first several hundred generations, followed by a slow process of solution refinement towards the optimal value (the last approximately two-thirds of the total generations).

It is now of interest to show how every term of Equation (78) varies to better understand the MDO convergence process. This process is also depicted in Figure 18. The MDO first assesses low-mass microlauncher concepts ($M_{start}$ has low values—Figure 18c) to keep the objective function as low as possible. Of course, these solutions are not feasible as the amount of propellant is insufficient to insert the payload into desired orbit ($I_{mission}$ has high values—Figure 18d). The MDO iteratively increases the lift-off mass of the microlauncher (together with minor adjustment to the target guidance parameters and flight phase duration) to steadily decrease the main mission performance index, which states how well the reference trajectory can be used to insert the payload into orbit. At the moment of convergence, $I_{mission}=0.0057$ was obtained, corresponding to a very good insertion, close to the ideal one, while the lift-off mass was $M_{start}=14.8251$ [t].

The recovery mission performance index is shown in Figure 18e. For the current case (recovery from a secondary location which is not predefined), this term is equal to the mass of unused propellant (measured in tons, similar to the term $M_{start}$). This propellant mass is seen as a penalty on the overall performance of the microlauncher concept, not being actively needed for the recovery of the lower stage, being the equivalent of an artificial increase in the structural mass of the first stage. It can be seen that the value of $I_{recovery}$ decreases rapidly towards 0 (final value being $I_{recovery}={8·10}^{-13}$), meaning that no extra propellant mass is in the first-stage tanks at touchdown (beside the 5% margin implemented, as per Table 11).

The last term included in the objective function definition is the constraint index, its convergence being shown in Figure 18f. One can observe that the MDO has a tendency to use configurations that respect all imposed constraints from the first hundred generations, restricting the 20-dimensional search space to one of interest for the problem studied. Upon solution convergence $I_{constraints}=0$.

The numerical values obtained after MDO convergence are presented in

Table 15. Reusable microlauncher concept, final optimization variables.

Nr.	Optimization Variable	Value	Unit	Nr.	Optimization Variable	Value	Unit
1	${M_p}_1$	10.8690	[t]	11	${P_e}_2$	0.0586	[bar]
2	${D_e}_1$	1.3853	[m]	12	${\left({\displaystyle \frac{T}{W}}\right)}_2$	0.7045	[-]
3	${P_c}_1$	79.9843	[bar]	13	$t_{vertical}$	13.7100	[s]
4	${P_e}_1$	0.5018	[bar]	14	$t_{coast}$	2.3009	[s]
5	${\left({\displaystyle \frac{T}{W}}\right)}_1$	2.1168	[-]	15	$\Delta t_1$	0.3955	[-]
6	${{\gamma }_d}_1$	51.9744	[°]	16	$\Delta t_2$	0.3308	[-]
7	${{\chi }_d}_1$	2.4987	[°]	17	$M_{p_{recovery}}$	0.1913	[t]
8	${M_p}_2$	1.5391	[t]	18	$H_{reentry}$	51.8140	[km]
9	${D_e}_2$	1.3852	[m]	19	$\Delta M_{p_{landing}}$	0.9978	[-]
10	${P_c}_2$	79.9310	[bar]	20	$H_{landing}$	0.1848	[km]

.

4.2. Proposed Microlauncher Concept

The entire constructive solution of the reusable microlauncher can now be generated based on the data in Table 15, with the aid of the mathematical models implemented inside the Preliminary design module. General specifications of the obtained microlauncher concept can be seen in

Table 16. Reusable microlauncher, specifications.

Specification	Value
Lift-off mass [t]	14.83
Payload mass [kg]	100
Payload performance index [%]	0.67
Microlauncher length [m]	20.50
Exterior diameter [m]	1.39
Upper structure mass [kg]	197.01
Upper structure length [m]	2.03
Propellant type	$O_2+{CH}_4$
Stage 2 mass [t]	1.75
Stage 2 length [m]	3.62
Stage 1 mass (+interstage) [t]	12.88
Stage 1 length (+interstage) [m]	15.38

. The preliminary Matlab “clean configuration” and more detailed CAD representations of the concept are shown in Figure 19.

The mass breakdown of the upper structure and both stages are given in

Table 17. Upper structure mass breakdown.

Component	Mass [kg]
Payload	100
Fairing	41.92
Fairing adapter	5.07
VEB	50.02
Total	197.01

,

Table 18. First-stage mass breakdown.

Breakdown Scheme	Mass [kg]
Propellant	10,869.00
Oxidizer	8005.40
Fuel	2863.60
Dry mass (expendable concept components)	1307.77
Oxidizer tank	132.17
Fuel tank	128.00
Oxidizer turbopump	62.01
Fuel turbopump	60.71
Engine	164.05
Additional components	760.83
Additional dry mass (due to stage recovery)	704.64
Aerodynamic control system	174.11
Reaction control system	59.62
Enlarged interstage	126.89
Landing system	281.09
Heat shield	62.90
Total Stage 1 Mass	12,881.42

and

Table 19. Second-stage mass breakdown.

Breakdown Scheme	Mass [kg]
Propellant	1539.07
Oxidizer	1137.27
Fuel	401.81
Dry mass	207.66
Oxidizer tank	34.05
Fuel tank	32.99
Oxidizer turbopump	8.46
Fuel turbopump	8.22
Engine	6.31
Additional components	107.33
Safety margin	9.89
Total Stage 2 Mass	1746.74

. For the second stage, a structural index (ratio between the dry mass and the total stage mass) of approximately 12% can be observed, which is conservative compared to the launchers currently in operation around the world (below 10%). For the first stage, due to the need for dedicated assemblies for stage recovery, the structural index is approximately 16%, the additional components summing up to approximately 700 kg (more than 50% of the expendable concept dry mass).

Some of the most important propulsive data (extracted for the main mission only) are shown in

Table 20. First-stage main propulsive data.

Specification	Value
Propellant	$O_2+{CH}_4$
Mixture ratio	2.80
Combustion chamber pressure [bar]	79.80
Exhaust pressure [bar]	0.50
Nozzle expansion ratio	16.64
Burn time [s]	97.32
Mean thrust [kN]	335.41
Mean propellant mass flow rate [kg/s]	109.71
Mean specific impulse	311.74

for the first stage and in

Table 21. Second-stage main propulsive data.

Specification	Value
Propellant	$O_2+{CH}_4$
Mixture ratio	2.83
Combustion chamber pressure [bar]	79.93
Exhaust pressure [bar]	0.06
Nozzle expansion ratio	89.29
Burn time [s]	392.64
Mean thrust [kN]	13.43
Mean propellant mass flow rate [kg/s]	3.92
Mean specific impulse	349.34

for the second stage. The influence of the low altitude on rocket engine performance can be clearly seen when comparing the mean specific impulse (311.74 vs. 349.34 s).

4.3. Mission Analyses and Reference Trajectories

At MDO convergence, based on the reference trajectories proposed, the mission analyses are realized for both the main mission and recovery mission. Details regarding the key events of the main mission of the reusable microlauncher proposed in this paper are given in

Table 22. Reusable microlauncher, main mission analysis.

Key Event	Start Reference Time [s]	Local Altitude [km]	Mach Number [-]	Microlauncher Mass [kg]
Lift-off	0	0	0	14,825.17
Active guidance phase	13.71	1.11	0.49	13,321.01
Max q	43.34	10.96	2.06	10,070.64
Gravity turn (primary)	46.78	12.81	2.32	9670.74
First-stage separation	97.32	65.18	9.62	4147.50
Fairing jettison	98.67	67.55	9.73	1943.75
Second-stage ignition	99.63	69.21	9.81	1901.83
Orbital insertion maneuver	362.37	379.23	5.64	871.29
Payload separation	492.27	421.16	8.91	362.75

. The duration of the mission is just over 8 min, this being a clear advantage of the DATO mission profile implemented (details in Figure 14).

Max Q (maximum dynamic pressure) occurs at almost 11 km and just above Mach 2. The fairing is jettisoned in the coast phase, ensuring that no stability issues arise from the separation maneuver. The deviation from the imposed target parameters (detailed in Table 10) is now presented in

Table 23. Payload insertion errors, main mission reference trajectory.

Parameter	Payload Insertion Error
Semimajor axis ${(a}_{target})$	−1.11 · 10−3 [km]
Eccentricity ${(e}_{target})$	5.58 · 10−7 [-]
Orbit inclination $\left(i_{target}\right)$	5.70 · 10−4 [°]
Velocity (inertial) ($V_{target})$	5.99 · 10−5 [m/s]
Flight path angle ${(\gamma }_{target}$)	3.17 · 10−5 [°]

, an accurate orbit insertion being observed.

The recovery mission is also optimized during the MDO by means of minimizing the amount of propellant needed. The recovery mission analysis is presented in

Table 24. Reusable microlauncher, recovery mission analysis.

Description	End Reference Time [s]	Stage Mass [kg]	Propellant Mass [kg]	Local Altitude [km]	Velocity [m/s]	Flight Path Angle [°]	Downrange [km]
First-stage separation	97.32	2203.74	191.33	65.18	2943.14	36.71	66.38
Ballistic evolution 1	304.88	2203.74	191.33	246.86	2288.66	0.00	533.98
Flip-over maneuver
ACS deployment
Ballistic evolution 2	520.85	2203.74	191.33	51.81	2974.22	−37.86	1020.37
Reentry burn	520.85	2203.33	190.92	51.81	2973.64	−37.86	1020.38
Aero guidance	582.59	2203.33	190.92	0.18	232.06	−73.43	1078.25
Landing maneuver	584.25	2021.98	9.57	0.00	0.14	−80.07	1078.30
Touchdown	584.49	2021.98	9.57	0.00	2.56	−89.52	1078.30

.

It can be easily observed from Table 15 that, in order to minimize the total propellant mass reserved for the recovery mission, most of it is dedicated to the landing burn (optimization variable number 19 $\Delta M_{p_{landing}}$ is almost 1). The rest of the propellant is used during the reentry burn, which has a brief duration. At this moment, there are no additional constraints imposed for the thermal loading (beside the ones used for the sizing of the heat shield—Section 2.1), dynamic pressure or load factors during the recovery trajectory; hence there is no reason to indicate the necessity of a longer-duration reentry burn (at least not quantifiable in the preliminary design phase). This reentry burn skip phenomenon was also seen in the SpaceX Starship Flight 11 test [119] and is viable for configurations that have low enough velocities at the moment of atmosphere reentry.

At touchdown, the first stage has a velocity of 2.56 m/s and a deviation from vertical position of 0.48°, which is in accordance with the imposed constraints. The 9.57 kg propellant mass still onboard the lower stage at the time of landing represents the desired 5% safety margin. The optimal landing location has been found to be at a distance of 1078.3 km from the launch location and can be used as the reference point for a recovery barge.

More insight can be obtained if we decide to plot out the reference trajectories for both main and recovery missions, together with the evolution of the payload. Figure 20 presents the variations in the parameters used in the MDO objective function definition (details in Section 2.6.3). A good insertion of the satellite can be observed, supported by the constant behavior of these parameters after satellite detachment.

One can observe from the data in Figure 20b that the first stage velocity decreases from 2.9 km/s (at stage separation) to almost 2.3 km/s (at ballistic trajectory apogee), increases again to 2.9 km/s (before reentry burn) then decreases as the secondary max Q moment approaches (recovery mission max Q occurs at t = 541 s after lift-off, at an altitude of 15.5 km). From the flight path angle variation with time shown in Figure 20c, the zone in which this parameter is maintained constant can be clearly seen during the first active guidance phase ($\gamma \cong {{\gamma }_d}_1\cong 52°$, as per Table 15). The orbit eccentricity variation with time depicted in Figure 20e validates the circularity of the orbit in which the payload has been inserted. Similar, the orbit inclination can be observed to be polar ($i=90°$).

Next, an additional set of parameters’ variation with time is depicted in Figure 21 to better understand the two distinct missions of the reusable microlauncher. First, the mass evolution over time is shown in Figure 21a. One can easily observe the jumps in mass associated with the separation of the first stage, the fairing jettison and the payload separation. It can also be observed that the mass of the lower stage is approximately constant until the landing maneuver, when the rocket engine is restarted and maintained at a 100% throttle setting until all of the propellant reserved for this maneuver has been consumed.

The next interesting representation is shown in Figure 21b, corresponding to the variation in the (local) altitude with the elapsed time. It can be seen that the satellite altitude is maintained at the desired one after its insertion (400 km measured from the equator, which corresponds to a local altitude of around 421.16 km). At the same time, during the lower-stage recovery mission, its trajectory has an apogee towards the 250 km mark.

The aerodynamic angle of attack variation with time is shown in Figure 21c. For the microlauncher case, the active guidance phases can be seen as this angle changes from null to non-zero values (being limited according to the input data used, considered ±7°). For the lower stage, the flip-over maneuver using the RCS can be observed, as the angle of attack varies from 0° to 180° in a short time interval (at t = 305 s after launch).

The thrust generated by liquid propellant rocket engines is shown in Figure 21d. The negative influence of the high atmospheric pressure near the ground can be observed, with propulsive performance at sea level being penalized by approximately 12% compared to high altitudes (348 kN vs. 308 kN). Since the propellant mass flow rate is constant throughout the stage operation, the thrust generated in the vacuum is also constant. The two moments when the lower-stage engine is restarted during the recovery mission (the reentry maneuver and the landing maneuver) can also be observed in Figure 21d. The durations of these maneuvers are very small compared to the entire recovery mission (in the order of seconds); thus the generated thrust profile is presented more like small spikes (during these maneuvers, the velocity of the first stage is reduced).

Even more information regarding the optimized trajectories, as well as the payload orbit, can be obtained by using a graphical representation in a 2D environment (Mercator projection) and in a 3D environment (globe representation). Figure 22 shows these two graphical representations from lift-off up until 5 min after payload separation. As the latitude of the launch location is quite high (${\phi }_{initial}=69.2944°$) and the track angle is towards the north, it can be seen that during its first minutes after orbit insertion, the satellite passes over the North Pole and continues its journey on the other side of the Earth.

If we extend the orbit propagation time (for which the 3DOF dynamic model is further integrated), then global coverage of using a polar orbit can be observed based on the results shown in Figure 23. Thus, the satellite coverage is presented in Figure 23a from payload insertion up to 24 h after and in Figure 23b for the first 7 days after orbit insertion. Having access to full coverage of the globe justifies the popularity of low Earth polar orbit among the small satellite category, satellites which are mainly focused on Earth observation missions.

4.4. Estimated Microlauncher Costs

As previously mentioned in Section 2, the multidisciplinary optimization algorithm consists of five main modules, of which only four are actually used during the optimization process of the reusable microlauncher concept. To reduce the computational effort, the Cost Estimation module is called only after the convergence of the MDO algorithm, the effective optimization loop thus containing the main Preliminary design, Propulsion, Aerodynamics and Trajectory modules. This is due to the lack of any output data of the Cost Estimation module that needs to be called later within the MDO algorithm.

Within this module, a list of costs for the development, production and operation of the small launch vehicle is generated, together with the estimation of a total cost per launch and a total price per launch valid for the considered operational lifetime.

Details on the mathematical models used within the module are provided in Section 2.5. Most of the input data necessary for this module have been mentioned there, now being defined the last terms of interest:

• The annual launch rate ($L$) needed for Equation (73): for this term, a once-per-month launch schedule is proposed for the reusable microlauncher; thus $L=12;$
• The total number of launches used to amortize the development costs ($N_{amortization}$) needed for Equation (58): For this term a 20-year operational lifetime is envisioned. Considering that 12 launches per year are considered, then $N_{amortization}=240$;
• The number of reuses of the first stage ($n_{stage.reuse}$) needed for the production costs estimation. Whenever the first stage is reused, the total production costs is drastically reduced as another lower stage is not built from scratch, but rather the old one is refurbished. For this term a five-time reuse is seen as viable; thus $n_{stage.reuse}=5$;
• The value associated with MYr (Man-Year) is needed to convert all costs related to the reusable microlauncher to an actual monetary evaluation. For this term, the average salary in Romania has been used; thus ${MYear}_{Romania}=EUR 24000$.

The final numerical data related to the cost estimations for the partially reusable microlauncher concept developed in this paper are shown in

Table 25. Reusable microlauncher, estimated costs.

Cost	$\mathbf{Value} [\mathbf{M} \mathit{E}\mathit{U}\mathit{R}$]
Development	630.15
Total number of launches	240
R&D amortization per launch	2.63
Production	1.79 (average)
Stage 1 (5 reuses)	4.36
Stage 2	0.92
Direct Operations	0.32
Ground operations	0.07
Materials and propellants	0.03
Launch, Flight and Mission Operations	0.11
Transport and Recovery	0.11
Indirect Operations	0.21
Refurbishment	0.23
Total cost per launch	5.17
Price per launch (30% profit margin)	6.73

.

Analyzing the cost results presented in Table 25, it can be observed that the greatest cost associated with the current reusable microlauncher concept arises from its development (diving even further, it was observed that the rocket engine development had the greatest contribution). By using a total number of 240 launches during the microlauncher operational lifetime (1 per month over 20 years), then the amortization of these development costs becomes acceptable, being approximately EUR 2.63 million per launch. Any additional funding from various sources (national or international projects) can lead to a substantial reduction in these development costs (which currently represent 51% of the total cost per launch).

A much lower production cost is noted for the reusable microlauncher (EUR 1.79 M on average compared to EUR 5.28 M for a new launcher), because the first stage is reused several times, thus only requiring its reconditioning for four of the five missions that the lower stage will perform (according to the assumption $n_{stage.reuse}=5$). However, even in this case, production costs represent a significant contribution to the total costs, being approximately 35% of the costs associated with a launch. The direct and indirect operations costs amount to approximately EUR 530 k per launch, thus being approximately 10% of the total costs associated with the microlauncher concept proposed in this paper.

As final data, an average total cost per launch of approximately EUR 5.17 M is obtained, and considering that the mission studied is that of inserting a satellite of 100 kg payload into a low Earth orbit, the cost per kg of payload inserted is approximately EUR 51.7 k, which seems to be on par with expendable European launch vehicle concepts. On top of these costs, the profit margin considered was added (30%), and thus, the final price per launch paid by the customer is estimated at EUR 6.73 M.

5. Conclusions

This paper has demonstrated the possibility of generating a partially reusable microlauncher concept by means of multidisciplinary optimization. An MDO algorithm has been proposed, yielding promising results for a microlauncher with a two-stage LOX/methane architecture, in which the first stage is recovered at a secondary location (being at 1078 km distance from the launch location). A presentation of the main mathematical models used to assess the five main disciplinary analyses needed for the generation of a microlauncher concept has been offered, each main module (Preliminary design, Propulsion, Aerodynamics, Trajectory, Cost estimation) being reserved a separate section.

The reusable microlauncher has two distinct missions to accomplish (compared to just one for an expendable launch vehicle), the main mission being the insertion of a payload/satellite into a predefined orbit, while the secondary mission is the recovery of the lower stage. The optimization of the microlauncher concept is performed via mass minimization (at lift-off), constraint verification, and successful completion of both main and recovery missions. This work proposes a preliminary design for a European reusable microlauncher, providing a foundation for subsequent advanced development phases if adequate interest and funding materialize.