A Multi-Step Sequential System Optimization Design Method for Upper Stages

ABSTRACT This paper proposes a new approach in multi-step sequential system optimization (MSSO) to implement a conceptual design for satellite upper stage with a maneuver in the conditions close to reality. In this method of design, there are two main cycles; trajectory optimization cycle and optimal design cycle, each one is correlated to each other in another cycle called configuration. In the trajectory optimization, the optimization objective is to place the upper stage in the destination orbit, using the minimum amount of fuel consumption. In this cycle, a new approach has been introduced for a three-dimensional trajectory using two genetic algorithms inside each other. In addition, selecting the suitable engine is carried out in this cycle. Convergence of design and exclusion of design are carried out in the configuration cycle. Convergence and optimization of subsystems design are carried out in the optimal design cycle. The innovations of this paper are implementation of the design according to multi-step sequential system design in which optimization is performed step by step, and orbital optimization is introduced according to a new approach. Choosing a desirable criterion for optimization process and proper coefficient for convergence in design, are among considerable characteristics of this paper. Validation has been performed using one of the upper stages in the world.


INTRODUCTION
The upper stages are responsible for transporting the satellites to the final orbit.Their performance trajectory is out of the atmosphere.Today, obtaining the design methods for upper stages, which have an optimization process and are less dependent on statistical studies, is an attractive subject.Accordingly, this study proceeds the researches performed based on designing the upper stage, aiming to optimize the transfer trajectory and applying its effect on the system design.Orbital transfer is an important issue, which has been investigated for years.It should be noted that most of the efforts were done for optimization and achieving the minimum amount of fuel consumption in space missions; because reducing 1 kg payload saves a lot of money.For this reason, investigating this subject is very important.It should be noted that impulsive maneuvers are not totally momentary and if there is not a proper understanding of the satellite position at the time of the impulse, a large amount of fuel will be spent for correction of the orbital position, causing less accuracy.
In this paper, it has been tried to design an optimized trajectory from the engine start moment to the moment of placement in the transfer orbit, using one of the optimization methods.So far, the optimized trajectory design has been implemented in different ways.The optimization is performed in a non-linear method by Vincent and Grantham (1997) and Brinda and Dasgupta (2005).Control optimization is carried out by Lu et al. (2003) and Brinda and Punnoose (2007) and the trajectory design dynamic optimization has been carried out by Betts (1998), Betts and Huffman (1993) and Ross et al. (2003).In this paper, according to Maani et al. (2013;2014), the orbital trajectory optimization has been carried out with three major differences; that are the solution method, reality of orbital mission and integration with the system design.In this paper, trajectory optimization has been performed inside an optimization cycle, which is a part of a comprehensive design approach called multi-step sequential system optimization design.

PROBLEM DESCRIPTION
The purpose of this paper is the implementation of the optimized design, including system design of all subsystems in order to pass the requirements for orbital mission in an optimal way.In many studies, the main objective of the optimization is reducing fuel consumption (Maani et al. 2013;2014).Here a question arises: Is the optimization acceptable and implementable in practice?
If the mass of the fuel changes, the dry and total mass of the upper stage also changes.Changing the mass of the fuel, at first it changes the mass and volume of the fuel tanks and then it influences many relevant parameters.By changing the total mass, the trajectory optimization loop should be run again.The results in this paper could be considered valuable for considering the relationship and interaction between optimization cycle and design cycle.Innovations presented in this article include, first, presenting a new approach in design; the concepts of design and interactive optimization are applied in a strategic computing environment to design an upper stage for transferring the satellite from the parking orbit to the destination orbit in a new approach.Second, trajectory optimization; in the paper, the trajectory optimization is performed by two genetic algorithms for continued orbital maneuvers in a three-dimensional environment.Third, providing a system design in a collaborative design environment, which is implementable as primary design; collaborative design is one of the successful methods in system design in a software environment, where the intervention of the user during the run time is taken into account (Boy and Gruber 1990).The main characteristic in this method is the presence of the expert human in the design environment and in the integration of all subsystems.In this method, just like multi-disciplinary design optimization methods (MDO), no mathematical based design is presented.In MSSO design, reviewing and changing the design parameters according to constraints could be performed in every point of the algorithm.In this method, optimization is done step by step and in the development procedure of the design process.Fourth, the methodology of MSSO design process, which is based on Fig. 1, as shown below.

Structure & Stifener
Feeding system

Configuration cycle
Source: Elaborated by the authors using data from Boy and Gruber (1990).

MSSO DESIGN PROCESS ALGORITHM
This algorithm consists of three basic cycles.Trajectory optimization cycle, configuration cycle, and optimal design cycle.Determining initial values and basic configuration of the upper stage is carried out using statistical Eq. 1-6.These equations are the result of data collection of upper stages from information websites (Brügge 2021;Mirshams et al. 2016;Wade 1996).The information required for determining initial configuration of design was obtained from Arianespace (2006) and Angelopoulos et al. (2014).
The following process has been done to achieve these equations: • Identify the parameters used in the design; • Drawing different graphs to obtain the relationship between two parameters; • Deriving equations from the behavior of graphs; • Selecting equations with at least 95% accuracy. (1) (2) where M 0 is the total mass, M f is the final mass, M pay is the payload mass, T is the thrust, n is the thrust to weight ratio, L is the length, D is the diameter and LD 2 is volume.
After defining the mission and creating the primary configuration of the upper stage, the trajectory optimization cycle would be carried out.In this cycle, trajectory optimization and selection of a proper engine for the upper stage are defined.The characteristics of this cycle include the following: • Choosing the engine by using the results of trajectory optimization and a list of available engines; • The advantage of the engine selection and trajectory optimization being common in optimization cycle is minimizing the errors of ballistic parameters in the simulation with actual ballistic properties of the usable engines; • Fuel mass and the selected engine are the most important outputs of this cycle.
System relations of trajectory optimization cycle is shown in Fig. 2. Mission and total mass are trajectory optimization cycle inputs.Fuel mass, thrust-time profile, and the characteristics of the selected engine are trajectory optimization cycle outputs.

Configuration optimization
Convergence

Selection motor
Source: Elaborated by the authors using data from Pan et al. (2012).where r is distance between Earth center and upper stage center, and θ and φ are the longitude and latitude of the upper stage.
V is the magnitude of the velocity vector, γ is the flight path angle, and ω is the velocity heading angle.ϑ and δ represent direction of thrust.ε is the out of plane thrust angle between thrust and the plane of position, and velocity vectors and δ is the in-plane thrust angle between velocity and the projection of thrust vectors in the vertical plane formed by position and velocity vectors of the upper stage.All lengths in equations are normalized by a reference length which is R 0 (radius of Earth), and all the times in equations are normalized by √(R 0 /g 0 ) where g 0 is gravitational acceleration.Therefore, velocity is normalized by the definitions above.M is normalized by initial mass and T is thrust to weight ratio and c is a constant which is c=I sp /√(R 0 /g 0 ).
( Pan et al. 2012).In a timed transfer, which is happening the system propulsion is turned on at a certain point of the first orbit, and is turned off after being placed in the second orbit.In this paper, orbital maneuver is considered in a real situation, meaning that both two orbits have no intersection and have rotation relative to each other at least in one direction.Therefore, the main objective is to solve orbital equations, flying from the first orbit, and placement in the second orbit with minimum errors.
Among the answers, which have minimum errors, the transfer with the lowest fuel consumption would be the final choice of trajectory optimization cycle.In other words, the aim of optimizing the trajectory is to reach the second orbit with the least amount of fuel.That's why, in this method, two optimization genetic algorithms have been used together.The optimization objective is optimizing fuel consumption and this purpose requires the controlled work to be minimized (Eq.14) (Kirk 2004;Motlagh et al. 2013): (14) Required orbital equations are obtained from Chobotov (1996) and Curtis (2005).Flight optimization algorithm is shown in Optimization 2 is performed inside optimization 1 and within each time step.The objective of optimization 2 is the minimum amount of (r -r 2 ) at any moment and the objective of optimization 1 is the minimum amount of fuel consumption in transfer.
After each use of optimization 2, the condition of the algorithm (r -r 2 ) < ϵ is checked.If the condition applies, the amount of fuel mass is calculated, and if the condition does not apply, the new range of ε and δ is obtained according to the Eq. 9 and 10 and optimization 2 is carried out again.The changes of δ and ε profile and the placement of upper stage in the second orbit according to Eqs. 15 and 16, is the result of intermittent performance of optimization 2.  The identification process of r and r 2 using the entries in Table 1 is explained as follows (Fig. 4.): Table 1.Specifications of optimizations in Fig. 4.

Subjected Population number Variable range Variable name Process
Minimum M p 400 Source: Elaborated by the authors.
• Calculating r, instantaneous position vector of upper stage (starting from parking orbit); • Calculating the angle difference of two orbits in three directions (Δi, ΔΩ, Δω); • Calculating the Anomaly difference between two orbits (Δθ); • r 1 : Rotation of position r in three values of two orbits angle differences (Δi, ΔΩ, Δω); • r 2 : Identification of corresponding point of r 1 in target orbit with preserving the anomaly difference (Δθ).Algorithms shown in Fig. 4 should be performed for each orbital maneuver.The final fuel mass is the sum of fuel mass required for each orbital maneuver according to Eq. 17.
(17)  The selection process and system performance analysis of the propulsion system in different space environments creates slightly different characteristics for a space propulsion (Kenny and Friedman 1965).The selection of the engine is completed according to the optimization cycle and other propulsion activities in the design cycle.Selecting a performance analysis of propulsion in this paper includes the following: • Selection of the appropriate engine (among the available engines) using thrust-time profile obtained from the simulation; • Receiving the nominal values of engine performance parameters such as thrust, pressure distribution of combustion chamber, fuel to oxidizer ratio; • Calculation of engine ballistic performance during flight and combustion chamber pressure distribution; • Calculation of fuel and oxidizer flow rate; • Engine mass-dimension and performance specifications.
Propulsion system design equations are available in Sutton and Biblarz (2001), Darabi et al. (2015) and Humble et al. (1995).Figure 5 shows how to achieve propulsion engine specifications.

T i t i n
Engine list

Select engine
Ballistic analyze Source: Elaborated by the authors.

Configuration cycle
This cycle connects the two other cycles.In this cycle, the optimal fuel mass is received from the trajectory optimization cycle and the optimized dry mass is received from the optimization design cycle.Exit from the system design algorithm is carried out in this cycle, and the reciprocating behavior between the cycles will continue until the convergence of mass values in this cycle.Mass configuration is composed of three main mass parameters: fuel mass, subsystems mass and structure mass.Fuel mass has a direct relationship with the optimization cycle, the mass of subsystems and the mass structures has a direct relationship with the design cycle.Mass of the upper stage in an integrating process between the two cycles of trajectory optimization, and the optimal design is obtained from the Eq.20 and the mass of subsystems is obtained from the total mass of subsystems according Eq. 21: where i is the Number of the orbital maneuvers, j and k are the repeatable integers.The mass of each subsystem is converged in the design cycle process.Fuel mass, engine mass from the trajectory optimization cycle and mass of subsystems from the optimization design cycle, are convergence cycle inputs.Fuel mass, Thrust-time profile, and Specifications of the selected engine are Convergence cycle outputs.

Optimal design cycle
Upper stage subsystem design is performed in this cycle.This cycle has the following system specifications: • Design cycle as shown in Fig. 1   Figure 6 shows the convergence process in the optimal design cycle.In this cycle, the convergence factor is γ and the optimization criterion is minimum α, which is shown in Eq. 22-26.Where μ p is the fuel mass ratio and μ F is the dry mass ratio.The reason for choosing α is to create optimized mass distribution for the upper stage.System relations of optimal design cycle is shown in Fig. 7:  Optimal design cycle uses the trajectory optimization cycle data to determine the optimization parameters.This cycle automatically acts according to the trajectory optimization cycle as long as configuration parameters do not change.If the mass configuration in the configuration cycle is not acceptable, the trajectory optimization cycle will operate according to new information based on the updated configuration.

SUBSYSTEMS DESIGN
Optimal design cycle includes the design of all subsystems that affect the final configuration.Proper design is a collection of balanced characteristics of subsystems (ILS 2004).After propulsion analysis, which was described above in this section, the design of other subsystems will be reviewed.

The design process of tanks
The shape of a tank is a function of weight, rate of tank outflow, the tank volume and its layout limitations.Tanks with more sphericity have the best empty weight ratio to loaded weight (Hutchinson Junior and Olds 2004).Design process of tanks can be summarized as follows: • Selection of the geometry and arrangement of tanks (layout limitations); • Calculation of the required volume of tanks; • Dimensional design of tanks (focus on sphericity with movement limitations); • Calculation of pressure distribution of ullage during a flight (control volume between the tanks and the combustion chamber); • Designing the tanks structure and layout of other components; • Tanks mass-dimensional distribution.
Figure 8 shows the process of achieving the tanks design model.The calculation design process in propellant tank modeling and conceptual estimation process of other components are carried out using feasibility studies in tank system analysis.

Feeding system design process
Simplicity of pressure regulation in the feeding system determines its high reliability (Huzel et al. 1992), thus on and off switching and flow control is easily possible.Helium as a gas or liquid goes under pressure in helium tanks and with the use of heater and pressure control valves, the expected flow rate output is obtained.Feeding system design process is as follows: • Calculation of the required volume for helium tanks; • Choosing the number of tanks (layout limitation); • Arrangement and layout of the tanks (Tanks are spherical); • Calculation of required pressure during the flight (control volume between the fuel and oxidant tanks with the helium tanks); • Tanks mass-dimensional distribution.
In Fig. 9, the method to achieve the feeding system design structure is presented.

The design process of structure
To determine the mass of the structure, the loads applied on each section of the structure should be determined during the different stages, from preparation to the end of the flight.After identifying critical situations, loading in these situations is used in structural analysis.According to the above, the structural process and stiffener design can be summarized as follows: • Identifying the critical situations; • Calculation of transverse and longitudinal loads in critical situations; • Calculation of equivalent stresses on the structures; • Calculation of body structures thickness; • Calculating the number, distance and rigidity of stiffener; • Selecting the stiffener profile after buckling test; • Structure mass-dimensional distribution.
In Fig. 10, the method to achieve the structure design model is presented.

Other divisions
A good mass-dimensional distribution estimation of other components of upper stages can be achieved using statistical and feasibility studies.Table 2 shows how to access other components such as actuators and case guides.Equations 27 and 28 show how to calculate the total length and diameter of the upper stage. ( (28)

MSSO PERFORMANCE ANALYSIS
In this paper, a mission similar to Cent.D-5 SEC upper stage has been selected.Table 3 shows the input parameters of this mission.Initial design configuration is shown in Tab. 4 using statistical equations.In this case, the code obtained from the algorithm of Fig. 1 was converged for 11 cycle consecutive times.In the following, the results of the last design loop are described for all three design cycles.

Trajectory optimization cycle
For this mission, according to the mission of Cent.D-5 SEC upper stage (Martin 1999), two orbital maneuvers (two phases) are considered.In both orbital maneuvers, orbits do not intersect with each other, and a change in the inclination is considered for the second maneuver.In the first phase, transfer from an altitude of 134 km with a specific speed is performed to 167 * 1418 km orbit.The optimal choice among different sets of solutions is achieved by the best placement of the upper stage in second orbit.The optimal trajectory of the first phase maneuver is shown in Fig. 11.In the first orbital maneuver, the engine is turned on in true anomaly of -22.97º in the parking orbit direction and turned off in the true anomaly of -8.22º of second orbit.The average value of the thrust force in this transfer is 92 kN, thrust vector angle changes is between 0.41º to 2.5º and upper stage final mass is 7004 kg.In the second phase , the transfer is carried out from 167 * 1418 km orbit to GTO orbit with an inclination change of 27º.The three-dimensional indication of transfer trajectory is shown in Fig. 12.In the second phase of the transfer maneuver, the engine is turned on in the true anomaly of 172.8º of first orbit and turned off in true anomaly of 188.31º of the destination orbit.The average value of the thrust force in this transfer is 87.7 kN, ε is in the range of -45º to -12º and δ is between -4º and 2º.The final mass is 3000 kg.Table 5 shows the results of comparison between the second phase of Atlas V (401) upper stage with the results of simulation presented in trajectory optimization cycle without using multi-step system design (OMO) and Hohmann transfer.Source: Elaborated by the authors.
Table 5 shows orbital maneuver optimization with a new and simple method.Besides, this results proves that in such special launches, Hohmann transfer is not a good estimation for the design process.

Optimal design cycle
According to Fig. 13, the minimum value of α is equal to line 1, but in the design process, line 2 is calculated for α, which is a fuel to oxidizer ratio of 4.8.The calculated value of α represents applying constraints and limitations in MSSO design.Convergence process for γ in the optimal design cycle, for optimal value of α and in the last loop of the MSSO design is shown in Fig. 14

Configuration cycle
Convergence and exit from the design process of algorithm in Fig. 1     Although mass coefficient (β) varies in the process of configuration cycle convergence, the gradual process of upper stage mass optimization is observable in Fig. 15.According to Figs. 15 and 16, it can be seen that if just orbital maneuver optimization (OMO) is used, the optimized mass will be obtained in point 1but the MSSO design will generate the optimum value of mass in point 12.
In this section, a comparison is provided between the obtained results of MSSO design, and the fuel usage consumption optimization process by Cent.D-5 SEC upper stage is shown in Table 6.• Design method presented in paper SDO (system design optimization) without convergence in configuration loop; • Multi-step collaborative system optimization presented in this paper (MSSO).According to the results obtained from the graph in Fig. 17, the amount of optimization in reducing the mass of the upper stage and the amount of optimization in the ability to carry the mass of the cargo is between 2 and 5 percent.

CONCLUSION
In this paper, a multi-stage optimal sequential design model is introduced based on the development of the MDO method for an upper stage.In this research, with a systemic approach, the relationships of the design parameters of an upper stage were extracted and considered.The model of this developed method includes inputs, optimization cycle, configuration cycle, optimal design cycle and criteria creation for design convergence and exit from the design process.The design is performed for an upper stage with basic system parameters according to the results of statistical design and for the assumed conditions of orbital transfer to the GEO orbit.The most important achievement of this study is to improve the MDO method and create an optimal design model for the upper stage in which all activities and parameters involved in the design are placed next to each other with the optimization step by step in convergence of all effective parameters in processes.Collective convergence of parameters involved in the design and optimization process, as opposed to point convergence (resulting from trajectory optimization, etc.) and local (subsystems separately) has been one of the most important results of this study.In this paper, specifically in the Cent D-5 SEC upper stage example, the design parameters are simultaneously converged and integrated.

Fig. 3 .
Fig. 3.In this study, flight optimization algorithm includes using the genetic algorithm two times.The characteristics of the two methods are listed below: by the authors.
by the authors.

Figure 5 .
Figure5.Selection and performance analysis of propulsion system.OF: oxide to fuel ratio; TVC: thrust vector control.
Convergence of structure efficiency (β) (Motlagh and Novinzadeh 2012) is the exclusion criterion from the algorithm of Fig. 1 by using Eqs.18 and 19.
includes six parts; • All subsystems of upper stage are designed in this cycle; • The output of this cycle is upper stage subsystem specifications; • The process of design optimization and convergence in this cycle is shown in Fig. 6. γ n-1 < ϵ α Source: Elaborated by the authors.

Figure 6 .
Figure 6.Optimization and convergence in optimization design cycle.OF: oxide to fuel ratio.

Figure 7 .
Figure 7. System relations of optimal design cycle.
by the authors.

Figure 11 .
Figure 11.Flight path in the first phase of transition.

Figure 12 .
Figure 12.Trajectory in the second phase of transition.

Figure 13 .
Figure 13.Changes of μ f and μ p due to OF. OF: oxide to fuel ratio.

Figure 14 .
Figure 14.Convergence of γ in optimal design cycle of MSSO design.
are performed in this cycle.The convergence criterion in this cycle is β.The changes of β is shown in Fig.15.

Figure 15 .
Figure 15.Convergence of β in configuration cycle of MSSO design.MSSO.The gradual process of upper stage total mass optimization in MSSO design is shown in Fig.16.

Figure 17
Figure17shows four situations of upper stages launching to transfer orbit, including:• Actual flight of Atlas V (401) upper stage;• Orbital maneuver optimization presented in this paper (OMO); by the authors.

Figure 17 .
Figure 17.Comparison performance to elliptical transfer orbit.

Table 2 .
Mass-dimension classification of other upper stage divisions.
Source: Elaborated by the authors.

Table 3 .
Upper stage design inputs.
Source: Elaborated by the authors.