CHAPTER 7 TRAJECTORY OPTIMIZATION OF A TYPICAL AIR-BREATHING LAUNCH VEHICLE

Transcription

1 CHAPTER 7 TRAJECTORY OPTIMIZATION OF A TYPICAL AIR-BREATHING LAUNCH VEHICLE 7.1. INTRODUCTION A Single Stage To Orbit (SSTO) vehicle that uses both air-breathing and rocket propulsion to accelerate to orbital velocity is referred to here as an Air Breathing Launch Vehicle (ABLV). ABLVs are being considered as promising candidates for the future low cost space transportation systems. These vehicles provide more effective way to launch satellites and other vehicles to Low Earth Orbits (LEO) than rockets. The near minimum fuel trajectory for an ABLV is, however, substantially different from that of a rocket powered vehicle. Whereas a rocket powered vehicle leaves the dense atmosphere quickly to minimize the drag losses, an ABLV dwells much longer in the dense atmosphere where the airbreathing propulsion is more efficient. Over this speed regime, air-breathing Scramjet engine must be the primary propulsive system, and in order to obtain high propulsive efficiency, the Scramjet engine must be operated at a high dynamic pressure. However since aerodynamic heating and drag also increase with dynamic pressure, the benefits of high propulsive efficiency must be balanced against aerodynamic drag, temperature and structural constrains. This chapter presents the problem of trajectory optimization for a typical ABLV. The particular objective of fuel optimal ascent is examined for constant dynamic pressure ascent segment of an air-breathing SSTO mission. Functional 111

2 dependence of air-breathing engine thrust on vehicle flight path (h-v) and angle of attack has major impact on the nature of fuel optimal ascent trajectories. 7.2 ABLV MISSION DESCRIPTION ABLV is a SSTO (Schoettle, U. M.,1985) fully reusable winged vehicle that takes off like an aircraft from any conventional airport. A typical flight path of a Single Stage To Orbit ABLV is shown in fig The trajectory may be divided into seven segments. Initially the vehicle takes off from rest and is accelerated as it rolls on the ground to take off speed. This is the first segment of the trajectory. Once the vehicle attains the take off speed, the elevator/horizontal tail is operated to rotate the vehicle about the center of gravity for getting the required angle of attack. When the flight path angle becomes equal to the required climb angle, the second segment of the trajectory is completed. The third segment of the trajectory ends when the dynamic pressure reaches the specified value. During the fourth segment of the trajectory, angle of climb is allowed to vary keeping the forward acceleration constant so as to maintain the same dynamic pressure through out. This segment ends at a prescribed Mach number 7. Above this Mach number, the engine operates in dual mode, scramrocket mode. The vehicle executes a pull up to the required hypersonic climb angle. This constitutes the fifth segment of the trajectory. Above Mach number 10 the engine operates as rocket producing the required thrust. During the last two segments of the trajectory the engine operates as rocket. Present study addresses the trajectory optimization and guidance strategy for the constant dynamic pressure 112

3 segment (Phase 4) of air breathing launch vehicle flight path. LEO Orbit Leveling off to Orbit 7 End of Atm mpl1 ere ->/'. Hypersonic Constant angle climb 5 6. Hypersonic pull-up <. mund Roll 2./ / Constant Dynamic -.( Pressure # Subsonic Constant / angle climb Subsonic pull-up Fig Typical Flight Path of ABL V 7.3 OPTIMAL TRAJECTORY SYNTHESIS FOR ABLV The objective of optimal trajectory synthesis is to do optimization for minimum fuel. The trajectory optimization can be carried out off-line or on-line depending upon the mission requirements and availability of powerful on-board computers. This chapter addresses both off-line and on-line trajectory optimization for a typical ABLV. Functional dependence of air-breathing engine thrust on vehicle flight path (h-v) and angle of attack has a major impact on the nature of fuel optimal ascent 113

4 trajectories. The amount of uncertainties in aerodynamic parameters is significantly greater for ABLVs compared to expendable rocket launch vehicles. Due to above reasons guidance and control technology dependent on pre-launch, predetermined trajectory as used in conventional launch vehicles is inadequate for ABLVs. In this research, optimal trajectory is generated using two methods. In the first study, reference optimal trajectory is generated off-line using Non Linear Programming (NLP) method. The NLP problem is solved using Sequential Quadratic Programming (SQP) method using the trajectory optimization tool available in MATLAB. The second study presents the on-line generation of optimal trajectories using energy state approximation approach (Arthur. E. Bryson Jr. and Mukund. N. Desai, 1969). 7.4 TRAJECTORY OPTIMIZATION TECHNIQUES IN BRIEF Generation of fuel optimal climb profile for constant dynamic pressure segment of ABLV ascent phase is addressed here. The trajectory generation problem can be posed as an optimal control problem. The vehicle will be obeying the equations of dynamics. The dynamics can be modified using the control vector. Time dependent optimization problems, where the state equation is constituted by a dynamical system, form an important subclass of optimal control problems. There are two general methods available for solving a deterministic optimal control problem, direct and indirect. Indirect methods proceed by formulating the optimality conditions according to the Pontryagin maximum principle and then 114

5 numerically solving the resulting Two Point Boundary Value Problem (TPBVP). Indirect methods are considerably more difficult to formulate and are very sensitive to the initial guess, but generate fast solutions with great accuracy. Direct methods reduce the optimal control problem to a single Non Linear Programming (NLP) problem. Direct methods discretize the original problem in time and solve the resulting parameter optimization problem and thus generate an approximate solution of the original problem. The strengths of the direct methods are that the formulation is significantly easier and the methods are relatively insensitive to the initial guess. A particular drawback of direct method is large execution times. However, as computing power of present day computers is quite high, direct methods offer a better choice for off line generation of fuel optimal trajectories. 7.5 A GENERAL OPTIMAL CONTROL PROBLEM Optimal Control is defined as the one that minimizes or maximizes the performance measure. The formulation of an optimal control problem requires 1. A mathematical model of the process to be controlled 2. A statement of the constraints. 3. Specification of a performance criterion. The optimization process can be represented in general as shown in fig

6 2.Constraint to he met Optimization Process Optimal Control History Optimal Trajectory r , I.Mathematical Model 3.Performance ' Fig Optimization Process 7.6 OFF-LINE GENERATION OF OPTIMAL TRAJECTORY Trajectory optimization problems are dynamic optimization problems in which, the optimization depends not only on a set of variables, but also on their trajectories throughout some sort of a space. Trajectory Optimization Problems are also constrained optimization problems. In constrained optimization, the general 116

7 aim is to transform the problem into an easier sub-problem that can then be solved and used as the basis of an iterative process. In the present study, reference optimal trajectory is generated off-line using NLP method. The NLP problem is solved using Sequential Quadratic Programming (SQP) method. In SQP method, the cost function is approximated by a quadratic function and the constraints are linearly approximated. At each major iteration, a Quadratic Problem (QP) sub-problem is formulated based on a quadratic approximation of the Lagrangian function. The solution of QP sub-problem is used to form a search direction for a line search procedure. SQP tool available in MATLAB is used to generate the optimal trajectory. Optimal Control Problem formulation Constraints are imposed on the following parameters for the air-breathing ascent phase of a typical ABLY mission. i) Path constraints Constraints are imposed on dynamic pressure(q), angle of attack (a.) and qa.. These constraints ensure the controllability and structural integrity of the vehicle. Constraint of a. ensures that the Scram jet engine is flight tested in the required range of a. in which the performance is best. 1) Dynamic pressure (q) 62±0.2 kpa 2) Angle of attack, a. 3) Aero dynamic Load on the vehicle, qa.< 5400 Pa-rad 117

8 1 2 where q = -pv 2 r, P - air-density, V r = relative velocity ii) Terminal constraint : Bum out Mach number =7 Where qa represents the aerodynamic load acting on the vehicle. The constraints on q and qa have to be satisfied to ensure controllability and maintain the structural integrity of the vehicle. The constraint on a is imposed to ensure that the air-breathing engine performance is assessed in the required range specified in the mission. The terminal condition imposed on the optimal trajectory is that, bum out Mach number should be 7. iii) Performance Index: Maximize payload. Optimization is carried out for maximizing the payload mass iv) Control Parameter: Thrust ABLV is similar to an aircraft, and can be controlled by controlling the thrust of the air-breathing engine. Optimal ascent phase trajectory parameters generated using NLP method are given in fig

9 H a q V Optimal Trajectory Parameters of ABLV Ascent Phase 'Q V-Vel(km/s) 2 o q-dyn.pres(kpa) a-ang.of.attack(deg) o H-Alt(km) Time(s) Fig Optimal Trajectory Parameters 7.7 ON-LINE GENERATION OF OPTIMAL TRAJECTORY TIlls section presents the problem of on-line trajectory optimization for a typical Air-Breathing Launch Vehicle (ABLV). The particular objective of fuel optimal ascent is examined for constant dynamic pressure ascent segment of an airbreathing Single Stage To Orbit (SSTO) mission. Generation of fuel optimal climb profile for constant dynamic pressure segment of air breathing launch vehicle flight path is addressed here. Functional dependence of air-breathing engine thrust on vehicle flight path (h-v) and angle of attack has major impact on the nature of fuel optimal ascent trajectories. The amount of uncertainties in aerodynamic parameters is significantly greater for ABLVs compared to expendable rocket launch vehicles. Due to above reasons guidance and control technology dependent on pre-launch, 119

10 predetermined trajectory as used in conventional launch vehicles is inadequate for ABLVs. The strategy of on-line generation of optimal trajectories using energy state approximation approach (A. J., Calise,. J.E., Corban, and G. A. Flandro.,1989) for flight of a point mass in a vertical plane over a spherical non-rotating Earth is described here. The algorithm proves to be computationally efficient and suitable for real time implementation. Performance of the algorithm is evaluated through extensive flight simulations for a generic ABLY with launch mass of 200T and lot payload. The air-breathing launch vehicle is similar to aircraft and thrust is taken as controlled force. The most adaptive approach to guidance problem requires no nominal trajectory and in fact continuously selects a new path for the remainder of the flight based only on the current state vector, the constraints to be met and the desired end conditions. There is only one part to the guidance algorithm and that is computed on-line with the feedback from the navigation system. Using this approach the guidance is not constrained to follow a trajectory computed prior to flight. Instead it can take advantage of dispersions that aid the flight and minimize the dispersions that detract from the flight. The onboard computational requirements to perform this function are more for this approach. But with present day flight computers these difficulties are disappearing. This work addresses the problem of on-line trajectory optimization based on the energy state approximation method. This method gives considerable insight into the nature of the optimal profiles and their relation to vehicle aerodynamic and propulsion characteristics. The vehicle model considered 120

11 includes a multi cycle propulsion system that works as Air Augmented Rocket, Ramjet, Scramjet and Rocket engines based on flight Mach numbers. The algorithm is found to give robust and accurate results even for off nominal performance of the air-breathing engine and dispersions in aerodynamic parameters Energy State Approximation The flight profile is optimized based on point mass energy state approximation model of the vehicle. The kinetic energy of an aircraft flying at supersonic speeds is comparable to its gravitational potential energy relative to the ground and thus it is possible to trade velocity for altitude or vice versa. In other words, in energy state approximation it is assumed that kinetic and potential energy can be traded back and forth in zero time without loss of total energy (Arthur. E. Bryson, 1969) Optimal Control Problem Formulation Total energy per unit weight- E is a function ofcurrent vehicle altitude -h and velocity-y and is given by Total energy per unit weight is given as y 2 E=-+h 2g (7.1) Which can re-written as v = ~2g(E-h) (7.2) The time rate of change ofenergy E is obtained by differentiating eqn (7.1) 121

12 E= v dv + dh g dt dt (7.3) Using equations of motion 6.5 to 6.8 to eliminate dv/dt and dh/dt, E=:!.- [Tcosa - D - Wsiny]+ Vsiny W E=~[Tcosa-D] W (7.4) (7.5) Where W is the weight. Since this work is concerned with minimum-fuel ascent to orbit the obvious performance index to be minimized is the fuel expenditure. For a vehicle with a fixed gross take-off weight this is equivalent to maximizing final weight. The ultimate purpose of the air-breathing ascent is to maximize payload/fuel mass fraction. For vehicles with fixed gross take-off weight, maximizing the bum out mass is equivalent to minimizing the fuel consumption. Minimum fuel trajectories are obtained by formally maximizing the bum out weight. Wlolalfuel = If fw fuel dt=fdwfuel (7.6) 1 0 Let the independent variable be converted from time to weight of fuel burned. Therefore (7.4) can be reduced to the single equation governing the energy change per weight -W of fuel burned. i) Minimum Fuel to Climb Problem Formulation Performance measure is written as 122

13 (7.7) Where lsp is the specific impulse and Wis the weight and Tis the thrust developed by the air-breathing engine. It is an established fact that such a vehicle should preferably fly within the atmosphere at constant dynamic pressure as long as possible and in order to achieve high propulsive efficiency, the Scramjet engine must be operated at a high dynamic pressure. However in the present study an appropriate bound is defined for the dynamic pressure in such away that the optimal solution converges for nominal and off nominal environments within 7 iterations and at the same time the propulsive efficiency is also not sacrificed. Dynamic pressure (q) is constrained as, 62k Pa< q < 64 k Pa pg(e - h) < 64kpa pg(e - h) > 62kpa (7.8) (7.9) Where pis the air density and g-acceleration due to gravity. ii) Definition of Hamiltonian Hamiltonian H is defined as follows (7.10) 123

14 H = [2g(E-h)J1' 2 [T c o ;-D Ipl ( t)-ii]+ P 2 (t{[64000-pg(e-h)] 2 II (-r 1 )]+ (7.11) P 2 (t{ (pg(e-h)-62000]2 II (-f 2 )] function. Where X1,X2 are the states, P1 & P2 are co-states, \\(-fi)are H eaviside step iii) State Equation The state equations are defined as X (t) = E(t)= ah = [2g(E- h)f 12 (Tcos a.-d) 1 a p w 1 (7.12) iv) Co-state Equation P1 and P2 are Lagrange multipliers, the co-state equation is defined as 1/2 1 t P (t)-- + ae 2 E-h W 1 T P. ( )- a H _ 1 [ 2g J (Tc o so.-d) [ Isp ] 2P 2 pg[[64000-pg(e-h)]ll(-f 1 ) (7.14) 2P 2 pg([pg(e-h)-6200qll(-f 2 )] (7.15) v) Control Equation Thrust is the control variable and the control equation is defined as follows 124

15 (7.16) vi) Boundary condition Assumed boundary condition is final time free and final state fixed, but for simplicity it is converted into a final time fixed and final state free problem by modifying the performance measure as _!_[ _ = ) 2 1 _ Vlsp ( Tcosa - D J 2 Edr Er + 1J ) T W lo (7.17) Therefore the boundary condition becomes (7.18) Here Er is achieved final energy and Ectr is the desired final energy. Er is treated as if it were free. The Two Point Boundary Value Problem (TPBVP) is solved numerically using the steepest descent algorithm (Bryson A. E. and Denham W. F, 1962), described in next section, to get the optimal thrust control history T* and corresponding optimal trajectory profile (h*-v*). The convergence of solution occurred in each computation cycle in less than 8 iterations. vii) Steepest Descent Algorithm The iterative procedure for solving nonlinear two point boundary value problem by using steepest descent method (Walter F. Denham, Arthur E. Bryson, 1964 ), for final state free and final time fixed is outlined as follows, 125

16 1. Select a discrete approximation to the nominal control history T C i) (t), t [to,tr] and store this in the memory of the computer. Let the iterative index i be zero. 2. Using this nominal control history,t C i), integrate the state equations from to to tr with initial condition X(to)=Eo and store the resulting trajectory x C i). 3. Using the value of p i (tr) from eqn (7.16) as initial condition and x C i ) stored in step 2, integrate co-state equation from tr to to. oh(il 4. Evaluate --, t [to,tr] and store this function. ot 5. If ah (') 1 ot <!l where n is a positive constant, terminate the iterative procedure and output the extremal state E* and control T*. 6. If the stopping criterion is not satisfied, generate a new control function as ah Ti+1 = T i - r at and follow step 2 to 4. viii) Termination Criterion The value used for termination constant n will depend on the problem being solved and the accuracy desired of the solution. table shows the value of norm in each iteration. Optimal trajectory generated off-line in section 7.6 using NLP approach and SQP solution is chosen as initial approximation to control history. 126

17 Table NORM VALUE IN EACH ITERATION No: of iterations Norm valuelloh/otii The stopping value of n was selected as after several trial runs of the problem. Thrust will be modified using a factor r as 2000 if the stopping criterion is not satisfied. ix) Optimization Results The initial conditions used for trajectory optimization and desired end conditions are given below. Desired end conditions are the target conditions achieved in off-line trajectory optimization method using NLP approach. Initial Conditions Altitude= 5.5km Velocity= 43 lm/s Mass = 176 T Flight path angle=

18 Mach no=l.3 Energy=14662m 2 /s 2 Desired End Conditions Altitude= 27.29km Velocity= 2119m/s Mass= T Flight path angle= 1.57 Mach no=7 Energy= m 2 /s 2 The optimization algorithm is validated for various types of off nominal conditions. The off nominal cases studied include air-breathing engine off nominal perfonnance as well as aerodynamic dispersions as described below. CaseO Nominal easel Thrust +5% Case2 Case3 Case4 Thrust-5% CL+20% CL-20% Case5 Co +20% Case6 Co-20% Case7 Thrust +2%, C 0-20%, CL+20% Case8 Thrust -2%, C 0 +20%, CL-20% Case9 Atmospheric density +2% 128

19 CaselO Atmospheric density -2% Dispersions in the achieved end conditions are given in table.?.7.2 TABLE DISPERSIONS IN ACHIEVED END CONDITIONS Cases Ahr AVr Ayr, mr km mis deg T

20 Maximum dispersion in achieved altitude is 2m. Maximum velocity dispersion is less than 0.4m/s and the maximum error in flight path angle was 0.45deg. This clearly establishes the efficiency and robustness of the on-line trajectory optimization algorithm in meeting the mission requirements and specifications. Reason for comparatively large dispersion in flight path angle for additive dispersion case is that in optimization, energy is the only state and that is a direct function of altitude and velocity only. Optimal thrust profiles for various cases are shown in fig c x x x10 6 A case 1 A case 2 V case 3 o case4 c nominal c.1se 2.0x x10 6 " ' Time (sec.) Fig Optimal Thrust Time Profiles 130

21 Dynamic pressure variation 1s restricted between 62 & 64kPa and is shown in fig o non-.inal case 1.o. c 11s e 2 v case 3 Cl c.::.se 4!50 L ! !50 Tilne (sec) Fig Dynamic Pressure Profiles Variation of specific energy, mass and Mach number are shown in fig V Sf>eclflc ene19y o 1nass C Mach numbe ::: "ii fw C eg l:., E :::e Time (sec) Fig Mass, Mach no, Specific Energy Profiles Figure shows the variation of co-states. 131

22 150 c 0 'P., 'C., > ~ 1;> b v " nolnlnal 100 ~ case 1 V case 2 o case 3 o case ' L..-...L.. """' Time (sec) Fig Co-state Variation Altitude and velocity profiles are shown in fig and respectively E ~ lit a.6 case 1 E c( 4 case 2 V C~lse 3 10 Cease.4 0 nolnin~ll o So Time (sec) Fig Optimal Altitude Profiles 132

23 ;; ,i;' g :: 1000 o nonllnal V case 1... case 2 V case 3 o case o~------_--._ _---i Thne (sec) Fig Optimal Velocity Profiles 7.8 SUMMARY This chapter presented the trajectory optimization for the air-breathing ascent phase of a typical Air- Breathing Launch Vehicle that performs a SSTO mission. The optimal trajectory generation was carried out using two approaches. First off-line trajectory optimization was carried using the SQP tool available in MATLAB. As a next step on-line generation of optimal trajectories is attempted. In particular, an energy state approximation approach is shown to be useful for rapid trajectory generation. The non-linear Two Point Boundary Value Problem is solved using the iterative numerical technique, the method of steepest descent. The (iteration) convergence ofsolution occurred in each computation cyclein less than 8 iterations. The optimization algorithm is found to be computationally efficient and 133

24 suitable for real time implementation. Robustness of the algorithm in meeting the mission specifications for dispersions in aerodynamic parameters and air-breathing engine performance is established through simulations. The algorithm ensures a fuel optimal climb profile. As a next step this scheme is used for real time generation of optimal trajectories on-board at longer intervals of time, and an adaptive guidance scheme is designed to follow this optimal profile in shorter intervals oftime. This is addressed in chapter