Optimisation of a Launch Vehicle Stack Model Using ...

Optimisation of a Launch Vehicle Stack Model Using an Evolutionary Algorithm G.P.Briggs, Tapabrata Ray & J.F.Milthorpe School of Aeronautical, Civil and Mechanical Engineering University of New South Wales at Australian Defence Force Academy, Northcott Drive, Canberra, ACT 2600 Abstract. Conceptual design of a satellite launch vehicle is a multidisciplinary task which must take into account interactions of disciplines such as propulsion, aerodynamics, structures, guidance and orbital mechanics. We discuss the initial modelling of a clean sheet design for a satellite launch vehicle (SLV) capable of placing an Ariane-44L equivalent payload into geostationary transfer orbit. While the Ariane-44L vehicle is/was a three and a half stage vehicle the hypothetical design is for a straight three stage vehicle. The delta-V capability of the AR44L is first derived from published data then the proposed design is modelled using a spreadsheet. The Gross Lift-Off Weight of the vehicle is then minimised for the same delta-V as Ariane. Various differences between the two vehicles are discussed. The initial design of an SLV as presented is based on a simple stack model optimised automatically using an evolutionary algorithm. The efficiency of the proposed approach is discussed along with future developments in the areas of vehicle model and multiobjective formulations of the design optimisation problem.

INTRODUCTION Our interest in the School of Aerospace, Civil and Mechanical Engineering (ACME) is to develop a framework that supports multidisciplinary conceptual design of expendable satellite launch vehicles. For a start, a simple stack model of the launch vehicle was considered to understand the behavior of the optimization algorithms and problem formulations. The final aim being the ability to optimise designs using a detailed model of a launch vehicle including optimised trajectory, structure and engine design parameters. This simple model was sufficiently valid to be able to be give first order vehicle mass estimates to be input into the more detailed program. It was also able to be used to give students a feel for the initial sizing of a launch vehicle. As a first step towards the more detailed model, the Gross Lift-Off Weight (GLOW) of the vehicle was required to be minimised on the initial assumption that the lowest cost vehicle corresponded to the lightest weight vehicle. Evolutionary optimisation techniques were used to choose masses of the three stages

Figure 1: Diagram of Ariane 44L & HLV

that would minimise the GLOW. A number of other parameters, as described below, were chosen as being constant and were not optimised. It was initially desired to model a clean-sheet design for a three stage launch vehicle with payload and much of the design equivalent to the Ariane 44L (AR44L) vehicle. While AR44L is/was a three and a half stage vehicle due to four liquid fuel strap-on boosters being staged in parallel with the main first stage of the vehicle, the proposed design was to be a straight three stage vehicle (Figure 1, HLV = Hypothetical Launch Vehicle). The question to be answered was how much heavier this design would be than that of the AR44L. In addition, the model was required to be able to be used to

design other proposed launch vehicles moving away from the AR44L design parameters. METHOD In order to obtain realistic mass and delta-V values for the model, the Ariane 44L was used as a baseline against which to compare results from the model. Analysis of the performance capability of AR44L was carried out by applying the rocket equation to each stage and obtaining the delta-V capability of the entire vehicle (Table 1). The delta-V required from the hypothetical three stage vehicle (HLV) stack was the same as that of the Ariane vehicle.

Table 1: Mass and performance breakdown figures for Ariane 44L as used in the Delta -V analysis Component Payload Payload Adapter Vehicle Equipment Bay Third stage structure Third stage margin propellant Third stage Delta-V increment Third stage mass ratio Vehicle Mass at 3rd stage shutdown Third stage nominal propellant Vehicle Mass at 3rd stage start Third stage Ivac Second stage Delta-V after fairing jettison Second stage mass ratio after fairing jettison Second stage structure mass Vehicle mass at 2nd stage shutdown Second stage propellant after fairing jettison Vehicle mass after fairing jettison Fairing mass Second stage Delta-V before fairing jettison Second stage mass ratio before fairing jettison Vehicle mass at fairing jettison Second stage propellant before fairing jettison Vehicle mass at 2nd stage start-up Second stage Ivac First stage structure mass Delta-V from 2nd half of first stage Mass ratio 2nd half of 1st stage Mass at stage-1 burnout 1st stage propellant after strap-on jettison Mass after strap-on jettison Delta-V before strap on jettison Mass Ratio during strap-on burn Mass at Strap-on burn-out First stage propellant during strap-on burn Strap-on Propellant Mass at lift-off (GLOW) Zeroth (strapon) and first stages Ivac Total Delta-V

Component Vehicle Delta-V, Mass, kg Mass, kg m/sec 4768 200 400 1250 200 4327.41

Comments

Estimate only: 1.7% of propellant 2.687 Nominal flight

6818 11700 18318

446.5 sec 1709.70 1.811 3500 21818 Jettison at 50% of 2nd stage burn

17700 39518 900 1045.35

1.438 40418 17700 58118 293.5 sec 17900 1822.01 1.951 76018 72293 148311 2889.66 2.886 166311 155707 158000 480018 278 sec 11794.1

While quoted masses of AR44L components vary from reference to reference the masses used were the best available from easily obtainable sources (being mainly the Ariane User’s Manual, Issue2, Rev 0) along with specific impulse performance and structure factor data for each stage .from the same source The payload and accomodation masses used for both the AR44L and HLV vehicle models were identical and are shown in table 2. Table 2. Payload and equipment masses used for both AR-44L and HLV models

Item Fairing Payload Adapter Vehicle Equipment Bay

Mass, kg 900 4768 200 400

For the payload configuration given the analysis performed gives an AR44L GLOW of 480018 kg and a delta-V capability of 11794.1 m/sec. A hypothetical straight three stage vehicle (HLV) was then modelled on a PC using an MS Excel spreadsheet (STAGEX) to obtain the mass breakdown of the stack, its GLOW and its delta-V capability by using the rocket equation for each stage. A basic optimisation function was included in the spreadsheet model in order to maximise the delta-V capability by enabling mass to be transferred from one stage to another while keeping the GLOW constant. Parallel to the spreadsheet a Fortran program (ROKOPT) was written to duplicate the results of the spreadsheet. This program was to be the basis of a full launch vehicle model and it was used here to test the evolutionary optimisation methods to be used later on the full vehicle model. Spreadsheet Method: The initial methodology involved displaying the HLV stack mass breakdown on an MS-Excel spreadsheet (STAGEX) and carrying out an optimisation to maximise the delta-V delivered by the stack. The propellant specific impulses and the stage structure factors were assumed to be identical to the AR44L vehicle. Implicit in this scenario is the assumption that the propellant combinations used in the two vehicles are also identical. In reality this is not likely to occur because of the greatly differing sizes of the stages of the two vehicles.

The AR44L analysis above includes an allowance of 200kg for third stage contingency or margin propellant. This is an estimate based on hearsay only as the actual figure could not be determined. This amounts to ~1.7% of the third stage propellant and this is the percentage that is allowed in the HLV even though the third stages of the two vehicles differ greatly in size. In addition the Ariane payload fairing is jetisoned during the second stage burn at a height that depends on the aerothemal flux that can be tolerated by the payload. This can make a difference of up to 14kg in payload and a variation of 14 secs in the mission time of fairing jettison. In the HLV we consider the fairing to be always jettisoned just before or at second stage burnout. Initial stage masses, including propellant, chosen by any of a number of methods (including guessing) were entered into the spreadsheet and modified until the desired delta-V was obtained. The stack was then modified by entering mass exchange values into dedicated cells to investigate the effect on the delta-V of moving mass from one stage to another. Once the delta-V was maximised the whole vehicle was resized to once again give the target delta-V. The process was then repeated until minimum GLOW was achieved. This method illustrates the optimisation of a launch vehicle stack with a single objective function (GLOW) and a single constraint to be met (delta-V). The results of the manual spreadsheet optimisation gave the HLV a GLOW of 473800 kg for the target delta-V of 11794.1 m/sec, which is about 6 tonnes lighter than the AR44L. The resultant stage masses are shown in table 3 Table 3: Optimal stage masses for the HLV as determined by STAGEX

HLV Stage 3 2 1 GLOW Delta-V

Stage Mass, kg 29700 91600 346200 473800 11794.1 m/sec

The result of the STAGEX optimisation shows that as a consequence of effectively reducing the size of the first stage by

eliminating the four liquid strapon boosters the HLV vehicle requires that its two upper stages be considerably larger than those of AR44L while the first stage is about 81% the size of the combined AR44L first stage and strapons.

6.6C compiler. Mixed language projects can be supported in this environment so that although the main code was written in Fortran-77 with Fortran 90/95 extensions, C++ subroutines can also be automatically compiled and linked within the project.

Velocity Budget: The equations for launch vehicle sizing are well known and can be found in many texts, for example, White 1963, Appendix-D, develops the equations for single, multiple and infinite numbers of stages with like and unlike stage parameters. There is however no analytical method which allows the equations describing a multistage launch vehicle to be solved for the optimum mass ratios of each stage if the effects of drag and gravity are considered (Koelle, 1961). As the design of the vehicle is dependent on the delta-V to be achieved, the delta-V is partly dependent on the trajectory flown and the trajectory flown is dependent on the vehicle design there is no closed form solution to the problem. If the velocity requirement (or budget) is known however, a preliminary launch vehicle design can then be produced to provide the required delta-V.

The initial part of the project was to calculate the stack GLOW and delta-V using the data from the HLV vehicle as approximated by the STAGEX spreadsheet. Optimisation code was then added to choose stage masses to minimise the GLOW and constrain the HLV delta-V to the AR44L capability.

Approximations to the velocity budget to be flown can be made including allowances for thrust-atmospheric loss, drag loss, gravity loss, earth rotation gain and orbital velocity required. Full Launch Vehicle Model: The STAGEX spreadsheet described above was developed to assist with estimating the velocity budget. While the estimates are not exact they are sufficient to allow an estimate of the stage sizes to be made and used as inputs to a more exact optimising computer model which will eventually include integration of the trajectory to give guidance parameters and the flight profile. Engine nozzle expansion ratio, thrust atmospheric loss effects, drag, the effects of mixture ratio on the delivered specific impulse, the propellant density and hence the structure mass are also to be included. These further developments of the model will include many more variables to be optimised than the simple stack model. It will not then be possible to optimise the variables manually so an automatic method must be found. In order to test optimisation methods for the full computer model (ROKOPT) a project was set up to run on a Pentium 4 desktop computer under the Windows XP operating system. The development environment used was MS Visual Studio with an integrated Compaq Visual Fortran

Evolutionary Approach: Prior to using EA, we observed that Nedler and Mead Simplex converged to different optimal solutions which were dependent on the starting guess. This highlights the possiblity of the function being multimodal and hence a population based EA was choosen. Gradient search methods require a continuuity of function and slope and can work with problems involving continuous values. The decision to investigate a change of propellant types or a move from a three stage to a two or four stage vehicle is an example of where the variables are discrete and integer in nature respectively. As the stack minimisation problem is only the testbed for the full launch vehicle model that is proposed, it was decided to use evolutionary methods from the outset to avoid difficulties that could arise further down the software model development path from having to change methods depending on the type of the variable to be optimised. Evolutionary optimisation methods are one of the keys to providing efficient discovery of minimum lift-off weight and hence lowest cost without having to resort to multiple methods depending on the nature of the problem. In the application of the launch vehicle stack minimisation the variables are the stage masses with specific impulse and structure factor as constants. The objective function is minimum GLOW and the constraints to be observed are the upper and lower delta-V velocity values. In this study, initial populations with random values of stage masses lying within specified mass ranges were created and the corresponding vehicle GLOW and delta-V capability was determined. Typically, the delta-V values may not be within the allowed tolerance for the intial set of solutions but as the evolution proceeds, the set of solutions moves towards the feasible region satisfying the delta-V constraint.

The evolutionary optimisation method used in this study is described in detail in Ray and Sarker (2006). The evolutionary algorithm as developed by Ray and Sarker (2006) is a variant of NSGA-II (Deb, K., Pratap, A., Agarwal, S. and Meryarivan, T, 2000, 2002) with a modified method of population reduction which insists on maintaining the diversity of solutions both in the objective and the variable space. The method is computationally more expensive than NSGA-II but maintains the diversity of the variable space more effectively than NSGA-II. The Calculations : The problem was to test the optimisation algorithm to determine the number of members of the population and the number of generations required in the evolutionary computation method to obtain a minimum GLOW with the stack delta-V capability constrained to 11794.1 m/sec. The calculation of the stack mass and delta-V delivered was coded to include the payload configuration of table 1, the 1.7% 3rd stage margin propellant and the fairing jettison at 2nd stage burnout. While the delta-V target represents one physical constraint, the minimisation routine was given two constraints, viz., delta-V+e and delta-V-e, where e is the Table 4: Stage mass ranges for the HLV as input to ROKOPT Run-A

HLV Stage Stage Mass Range, kg 3 26800 – 32900 2 83400 – 102000 1 316000 - 386500 GLOW To be minimised Delta-V 11794.1 m/sec residual allowed in the delta-V at convergence. The routines were therefore working with three variables (viz., the three stage masses), one objective function to be minimised (GLOW) and two constraints to be observed.

While an acceptable residual was deemed to be 0.1 m/sec, in all calculations e was set to be 0.01 m/sec as it was found that the constraint was quickly reached during the optimisation with the minimum GLOW taking longer to be found. In the first test the initial stage size estimates were derived from STAGEX and the ranges used allowed for an approximately ±10% spread of input solutions to the problem (table 4). The random number generator used to generate the random solutions for this method was RAN0 following the method of Park and Miller as described in (Press et al, 1999). A run of 64 optimisations was carried out using all the combinations of the values of the evolution parameters shown in table 5. The number of input solutions generated was 30 as Table 5: Values of evolutionary parameters used to evaluate the optimal stage masses – Run-A

Parameter No of solutions Max number of generations Random Seed Probability of crossover Probability of mutation Distribution index crossover Distribution index mutation

Values 30 50 0.2, 0.3, 0.4, 0.5 0.90, 0.92 0.05, 0.07 10, 15 10, 15

it is normally estimated that 10N should be used where N is the number of variables (viz: the three stage masses). Initial test runs demonstrated that 50 full generations (parent and child) were more than sufficient for convergence. The 64 runs were completed in 37.4 seconds on the Pentium 4: a run time of 0.58 seconds per optimisation. The values of the stage masses found for the optimum GLOW of

Table 6: Optimal stage masses for the HLV as determined by ROKOPT Run-A Vehicle 1st stage 2 nd stage 3 rd stage GLOW, mass, kg mass, kg mass, kg kg Maximum GLOW 344844 97472 26845 475430 Minimum GLOW 343940 93874 29707 473789 Spread of values 904 3598 -2862 1641 Percentage spread 0.26% 3.83% 9.63% 0.35% Mean Solution 344746 92925 30482 480739 Median Solution 343228 93641 30864 474418 STAGEX Solution 346200 91600 29700 473768 STAGEX residuals -2260 2274 7 21

Table 7: Values of evolutionary parameters used to evaluate the optimal stage masses – Run-B

Parameter No of solutions Max no of generations Random Seed Probability of crossover Probability of mutation Distribution index crossover Distribution index mutation

Values 100 50 0.05, 0.1, … , 0.95 0.90, 0.92, 0.94 0.05, 0.07, 0.09 10, 15, 20 10, 15, 20

each run are listed in table 6. The minimum values of GLOW found for the vehicle vary by approx 1100kg. While the minimum value reported is 67kg above that obtained by the manual optimisation using STAGEX. The first stage masses found vary by 904kg (~0.3%), the second stage masses by 359kg (~3.8%) and for the third stage by 2862kg (~9.6%). While the minimum GLOW found is acceptable, being only 21g above the STAGEX optimum solution the range of stage masses is not. A first stage design could not for instance be based on a mass uncertain by 2.5 tonnes. In order to visualise the distribution of the values of the optimal solutions found, the GLOWs were plotted in ascending order (figure 2). The plot demonstrates a straight line segment in the distribution of members of the optimal GLOW solutions.

Figure 3: Optimal stage mass solutions against their optimal GLOW solutions – Run-A

there are several combinations of stage mass values that give the GLOW value. As the GLOW approaches the minimum value towards the left of the graph the range of stage mass values decrease but still does not reach a single optimal solution. It is thus not apparent whether the method has not yet found the minimum or whether there are a number of minimal solutions. Examination of the plots for the first and second stage solutions would in fact indicate that there may be two distinct branches to the plot at lower GLOW values.

Figure 2: Members of optimal GLOW population of Run-A demonstrating straight line segments in distribution of optimal values.

Figure 3 shows the optimal stage masses plotted against their optimal GLOW found. For values of GLOW greater than the minimum

In order to test the hypothesis that the evolutionary algorithm had not found the minimum GLOW a program run (Run-B) with a much larger number of optimisations was carried out. Table 7 shows the optimisation parameters used for this larger run this time using a population of 100 generated solutions and again 50 generations was sufficient for each optimisation run to converge.

Table 8: Optimal stage masses for the HLV as determined by ROKOPT Run-B

Vehicle Maximum GLOW Minimum GLOW Spread of values Percentage spread Mean Solution Median Solution STAGEX Solution STAGEX residual

1st stage mass, kg 353001 345814 7188 2.08% 345436 345508 346200 -386

This corresponded to a total of 1539 optimisation runs and took a total elapsed time of 3495.3 seconds on the Pentium-4 computer. Figure 4 shows that this time the distribution of optimal solutions is not linear but demonstrates some members at extreme high GLOW values. Table 8 shows the maximum and minimum optimal GLOW solutions found by Run-B. The minimum optimal vehicle is now 15kg lighter than the solution found by STAGEX so we conclude that Run-A had not found the minimum optimal solution.

2 nd stage mass, kg 83555 91907 8352 9.09% 92466 92613 91600 307

3 rd stage GLOW, mass, kg kg 32860 475684 29765 473753 3095 1931 10.40% 0.41% 29980 474149 29917 474073 29700 473768 65 -15

Confirmation that Run-B is reaching a minimum optimal solution can be obtained by examining figure 5 which is the equivalent plot for Run-B as figure 3 was for Run-A. Now we can see that at the lower GLOW values the spread of stage mass values is lower and eventually at the extreme left side of the plot

Figure 4: Members of the optimal GLOW population of Run-B demonstrating several points at high values.

The spreads of the optimal stage masses found is still large but this is due to the large number of high GLOW optimal solutions in the population of 1539 runs. The last line in the table presents the difference between the STAGEX spreadsheet solution and the minimum optimal solution found by ROKOPT in Run-B. The residuals are now much smaller than they were for Run-A.

Figure 5: Optimal stage mass solutions against their optimal GLOW solutions – Run B

the spread reduces to almost zero. The stage mass residuals with respect to the STAGEX solution (Table 8) have reduced to the several hundred kilogram level and the ROKOPT solution is better than that of STAGEX by 15 kg. It is also apparent from figure 5 that the solutions are bounded by an envelope. The exact nature of the envelope is unclear as the statistical and evolutionary nature of the process makes intuitive deductions difficult. All that can be said is that feasible solutions occur inside the envelope and outside the solutions are unfeasible and do not appear. Starting each optimisation with a larger number of members of the initial population beyond 30 did not help in our study. The optimisation using STAGEX became difficult around the minimum as the solution wandered around the basin of the local minimum It was not obvious to the operator which way to move the solution to reach the absolute minimum. The evolutionary method on the other hand will eventually choose a minimum and given enough time will automatically do better than STAGEX. DISCUSSION While the exact parameters of the AR44L launch vehicle are probably only known precisely to the mission analysis staff at Arianespace, by using the data from the user’s manual we have developed a stack model of the vehicle that gives a mass breakdown and enables a delta-V capability figure to be calculated. While this value is probably not the true value, it is close enough to allow a stack model of an equivalent straight three stage vehicle (HLV) to be developed. In order to minimise the lift-off weight of the HLV the choice of stage masses to provide the required delta-V capability must be chosen correctly. To this end an operator can use the STAGEX spreadsheet to produce a nearly optimum stage mass allocation. The ROKOPT program can be used to further optimise the mass allocation between stages and can do it automatically rather than relying on operator intervention. The development of software to model expendable launch vehicles requires multidisciplinary tasks amongst which trajectory,

guidance, propuls ion and structure, for example, are as important as the initial vehicle sizing. In fact all the disciplines are more or less strongly dependent on each other. In order to optimise the vehicle a multidisiplinary model must be constructed to optimise the variables of all the disciplines together. In the project planned at ACME the simplest trajectory and guidance model will require at least nine variables while propulsion will require three for each stage. Optimising so many variables is beyond the ability of a human operator so an automatic method must be found that is reliable not only that it works efficiently but that it can also distinguish between different local minima and find the lowest value. To reach the absolute minimum, the evolutionary algorithm may however take a very large number of trials and once the basin of the global minimum within the search region has been found it will be more efficient to switch to an alternative search method to reach the minimum of the basin. While a stack model is not a particularly complicated task it is being used as a test bed for evaluating optimisation methods before proceeding to more detailed launch vehicle desription. At ACME it is intended that the ROKOPT program will continue to be developed with a number of additional selection methods added to supplement the evolutionary algorithms to make the process more efficient. In addition to improving the optimisation methods trajectory integration software is to be added as the next step in the multi-disciplinary description of the launch vehicle. This will allow a more accurate evaluation of mission delta-V requirements for the HLV vehicle. Following trajectory, drag losses and atmospheric thrust loss models are to be added to the program which will then allow optimisation of nozzle expansion ratio on the first stage. This is to be followed by structure mass models. And so the work ever expands and the design becomes ever more detailed.

REFERENCES Arianespace, Ariane-4 User’s Manual. Issue 2, Rev-0 Deb, K., Pratap, A., Agarwal, S. and Meryarivan,T. (2000). A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multiobjective Optimisation: NSGAII, Proceedings of the Parallel Problem Solving from Nature VI Conference, Paris, France, pp849-858 Deb, K., Pratap, A., Agarwal, S. and Meryarivan,T. (2002). A Fast and Elitist Multi-objective Genetic Algorithm: NSGAII. IEEE Transaction on Evolutionary Computation, 6(2), pp181-197 Koelle, H.H., ed., Handbook of Astronautical Engineering, §22.23 Multistage Optimisation, McGraw-Hill, 1961 Nelder, J.A., Mead, R., A Simplex Method For Function Minimization, Computer Journal, 7, 308-313 [1], 1965 Press, W.H., et al, Numerical Recipes in Fortran77, second edition, “Chapter 7 Random Numbers”, Cambridge 1999 Ray, Tapabrata; Sarker, Ruhul; Multiobjective Evolutionary Approach to the Solution of Gas Lift Optimisation Problem, IEEE Proceedings of the World Congress on Evolutionary Computation, Vancouver, July 2006. White, J.F, ed. Flight Performance Handbook for Powered Flig ht Operations, John Wiley & Sons Inc., 1963

BIOGRAPHIES Gordon Briggs graduated BSc(Hons) from Melbourne University and MSc(Astron) from Swinburne University. He has had experience in explosives, ammunition and propellants at DSTO, Maribyrnong, Victoria and worked as mission analyst and project manager on comsat and upper stage design at space and communications division of British Aerospace, Stevenage, UK. Returning to Australia he was Manager of Space Systems at BAe(Aust) where he instituted the Cape York Spaceport concept and the solid propellant Australian Light Launch Vehicle (“Capricorn”) projects. He is currently carrying out further postgraduate work on an Australian medium launch vehicle. Tapabrata Ray obtained his Bachelors, Masters and Ph.D from the Indian Institute of Technology (IIT) Kharagpur, India. He is currently a Lecturer with the School of Aerospace, Civil and Mechanical Engineering (ACME) at the University of New South Wales at Australian Defence Force Academy, Canberra. His research interests are in all forms of multidisciplinary design optimization and bio-inspired models for constrained and multiobjective optimization. John Milthorpe BSc (Syd), BE Hons (Syd), Grad DipHEd (UNSW), MEngSc (Syd), MDefStud (UNSW), PhD (Syd), SMAIAA. He also holds a private pilot license. His professional and teaching experience includes time as engineer at the Commonwealth Aircraft Corp, Research Fellow University of Sydney Royal Australian Engineers (Army Reserve), President of Australian Chapter of American Helicopter Society, MIE Aust, Member of IEAust Mechanical College Board. He is now Senior Lecturer at ADFA where his research interests include computational fluid dynamics, transonic aerodyanmics and helicopter aerodynamics. His consulting interests include guided weapons, sustainable power sytems and mechanical systems.