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Trajectory Optimization and Guidance of an Air Breathing Hypersonic Vehicle

Brinda.V* Vikram Sarabhai Space Centre,Indian Space Research Organisation, India

Dr.S.Dasgupta† Mohandas College of Engineering. , India

and

Madan Lal‡ Vikram Sarabhai Space Centre, Indian Space Research Organisation, India

This paper addresses the problem of trajectory optimization and adaptive guidance law synthesis for constant dynamic pressure ascent phase of a typical Air Breathing Launch Vehicle (ABLV). Characteristics of air-breathing engines, such as fuel flow rate and specific impulse depend upon altitude, Mach number, angle of attack and consequently, a high sensitivity of performance to the flight path exists. Hence the generation of optimal trajectory and development of guidance law for following the prescribed optimal trajectory for ABLV is a challenging task. In the present study, minimum fuel ascent trajectory for constant dynamic ascent phase is first solved using Sequential Quadratic Programming (SQP) technique. Then an adaptive guidance law is developed that controls the angle of attack using a feed back loop based on a second order rate controller for angle of attack (αααα). The Mission selected is that of a Single Stage To Orbit (SSTO). The algorithm is validated through extensive simulation studies for a conceptual ABLV with launch mass of 200t and 10t payload. Robustness of the developed adaptive guidance algorithm is established through extensive simulation studies. The guidance algorithm ensures that the mission specifications are met accurately for off nominal performance of air-breathing engine as well as aerodynamic parameter dispersions.

Nomenclature T = Thrust Isp = Specific Impulse L = Lift D = Drag M = Mach number q = dynamic pressure Sref = reference area s = laplace transform operator CL = Lift coefficient CD = Drag coefficient r = radial distance from the center of the Earth r0 = equatorial radius of Earth v = inertial velocity Vr = relative velocity

* Deputy Project Director, Mission, RLV-TD Project, v_brinda@vssc.gov.in † HOD, Electrical & Electronics Department, Mohandas College of Engg.,India, santanudsgpt@yahoo.co.in ‡ Project Director, RLV-TD Project, madanlal@vssc.gov.in

14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference AIAA 2006-7997

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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R = horizontal range α = angle of attack φ = bank angle γ = flight path angle ρ = air density m = mass g = acceleration due to gravity h = altitude hact = actual altitude hcom = commanded altitude K = adaptive gain parameter k = control system gain τ = control system time constant CLα = slope of CL curve with respect to α Gc = controller transfer function Gv = vehicle transfer function

I. Introduction N air-breathing Launch Vehicle (ABLV) is a relatively new concept in space transportation system that can operate from conventional runways like an aircraft, yet reach hypersonic velocities. ABLVs are being

considered as promising candidates for the future low cost space transportation systems. The anticipated advantages are the operational flexibility of horizontal take-off, the operational simplicity of a single stage and the propellant mass reduction that takes place from using air-breathing engines. Potential missions of these vehicles include large payload transportation to LEO, intercontinental passenger transportation and a wide range of defense missions. ABLV technology program emphasizes the goal of full reusability because of its pronounced impact on cost and operational flexibility, and trajectory optimization and better guidance strategy plays a major role in the performance and reliability of ABLVs.

The flight path of an ABLV is substantially different from that of a rocket powered expendable launch 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 air-breathing propulsion is more efficient. The thrust developed by an air-breathing engine depends on altitude, Mach number and angle of attack and that emphasizes the dependence of vehicle performance on flight path and atmospheric parameters. Due to above mentioned reasons we need to develop an adaptive guidance algorithm to generate the steering commands on-line to reshape the trajectory depending upon the current performance of the vehicle. Much research is being vigorously pursued in the area of ascent optimal trajectory synthesis and development of closed loop guidance law for following the prescribed optimal trajectory. A J. Calise et al. are using the energy state approximation method for trajectory optimization and the guidance law is developed using singular perturbation theory1-3. Ping Lu proposes inverse dynamic approach for solving the ascent problem for a hypersonic vehicle4. A trajectory tracking guidance law was developed to meet the constraints and to track a reference altitude verses velocity path in Ref 5-6. C. R Hargraces et al. obtains optimal solution by Nonlinear Programming and collocation method7. A real time optimal guidance law was developed to trace the maximum dynamic pressure trajectory using a Proportional Integral Derivative (PID) controller by Hirokazu et. al for an air breathing vehicle8. A nonlinear predictive control law was used for tracking a reference optimal profile for an ABLV in Ref 9.

In the present study following strategy is followed. The optimal control problem is solved using SQP method, to determine the fuel optimal ascent trajectory for ABLV10-11. The guidance module includes a gain adaptation algorithm that varies the feedback gains on altitude and altitude rate errors, depending on the current estimate of errors from Navigation. This real time optimal guidance law controls the flight path using a second order rate controller for α. The guidance law is found to be robust over a wide range of operating conditions, even with large variations in propulsion, aerodynamic parameters.

II. Vehicle Modeling and Mission Description A typical flight path of an ABLV is shown in fig.1.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 for getting the

A

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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 scram-rocket 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 segment (Phase 4) of ABLV flight path. Typical Flight path of ABLV is shown below.

Figure.1 Typical Flight Path of SSTO ABLV

A. Equations of Motion In the present approach the vehicle is treated as a point mass model performing planar motion over spherical,

non rotating Earth. The governing equations are expressed in a flying coordinate system with the origin at the centre of gravity of the vehicle and the x-axis along the direction of velocity vector. The y-axis is perpendicular to x-axis and lies in the plane of motion, positive to the right when looking into the direction of positive x-axis. The z-axis is perpendicular to the motion plane, thus completing the right handed coordinate system. The equations of motion are then expressed as

v g sin(Tcos D)

m

.= − +

−γ

α (1)

γ γα. v

rgv

cos(L Tsin )

mv= −�

��

�

�� +

+ (2)

r v sin.

= γ (3)

R v cos.

= γ (4) Where v -is the vehicle velocity, g- acceleration due to gravity, r- radial distance of the vehicle from the center of Earth, m- vehicle mass, γ- flight path angle, α- angle of attack. Thrust, lift and drag forces are given by T, L and D.

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Gravity is computed using the expression g = µ/r 2 , where µ is the gravitational parameter, ro is the radius of earth . These equations are solved simultaneously at each time step. In order to obtain a feasible flight envelope, constraints are imposed along with numerical integration. These constraints are taken into account in the simulation algorithm

and in the trajectory optimization scheme. The vehicle constraints are limits on α,.

α . The constraints on the trajectory are limits on q and aerodynamic load qα. These constraints are derived from aerodynamics, structures and control surface requirements.

B. Atmospheric Model Indian Standard Atmosphere is used where atmospheric properties such as density, pressure and temperature are

stored as a function of altitude. Density (ρ) will be used for computation of dynamic pressure q (q = ½ ρVr2), which

will be required, by lift and drag equations. Vr is the relative velocity. Pressure will be used for thrust correction and temperature for the computation of speed of sound.

C. Aerodynamic Model Lift and drag are the aerodynamic forces acting on the vehicle. The lift and drag forces are related to the lift

coefficient CL and drag coefficient CD as follows

L = qSrefCLα (5)

D = qSrefCD (6)

Sref is the reference area. Aerodynamic coefficients CL and CD are functions of Mach number and angle of attack (α ).

CL =A0+A1 α+ A2 α2 (7)

CD =B0+B1 α+ B2 α2 (8)

Where A0 , A1 , A2 and B0 , B1, B2 are functions of Mach number.

D. Propulsion Model A combination of engines needs to be used in order to achieve best efficiency through out the flight. So a rocket

based combined cycle propulsion system will offer maximum payload fractions. This engine will operate in rocket, ramjet and scramjet modes depending on the flight conditions. Fig.2 shows operation of such an engine.

Figure.2 Propulsion Model of ABLV During take off and at low Mach numbers the engine operates in Air Augmented Rocket mode. For flight Mach

numbers between 1.2 and 6, the vehicle operates in Ramjet mode. During this mode of operation the fuel is injected in to the Ramjet combustor and it is a complex angled structure that rams the onrushing air to supersonic speeds for combustion within the engine. Above Mach number 5, the combustion of the fuel takes place in a supersonic stream within the combustor. The fuel is injected in to the combustor and the length of the combustor should be sufficient enough for mixing and spontaneous combustion of fuel and air. Above Mach number 8 the Scramjet fuel supply is cut off and the engine geometry suitably modified to operate in rocket mode.

M=0 M=6 M=1.5 M=10

Rocket

Ramjet

Scram jet

Rocket

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III. Trajectory Optimization As the case is with conventional rocket launch vehicles, an ABLV must ascent along a near minimum fuel

trajectory, for maximizing its payload capability. 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 air-breathing propulsion is more efficient. The ascent performance of an ABLV depends heavily on the operating dynamic pressure and the air-breathing engine performance. The entire optimization of the system is quite different from that of conventional rockets, because air-breathing engines operate within a narrow atmospheric flight corridor from low to hypersonic Mach numbers. Trajectory optimization of an ABLV is a challenging task, since the performance of the engine is highly sensitive to flight path.

A. Problem Formulation 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. There are two general methods available for solving an optimal control problem, direct and indirect. 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. 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.

B. Performance Index and Constraints Trajectory Optimization Problems are constrained optimization problems. In constrained optimization, the

general 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 offline using NLP method. The NLP problem is solved using Sequential Quadratic Programming (SQP) 10-11 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 ascent phase trajectory parameters are given in fig.3.

5 0 1 5 0 2 5 0 3 5 00

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5V

5 7

5 9

6 1

6 3

6 5

6 7q

1

2

3

4

5αααα

0

1 0

2 0

3 0H

V - V e l ( k m / s )q - D y n . P r e s ( k P a )αααα - A n g . o f . a t t a c k ( d e g )H - A l t ( k m )

T i m e ( s )

O p t i m a l T r a j e c t o r y P a r a m e t e r s o f A B L V A s c e n t P h a s e

Figure.3 Optimal Trajectory Parameters for ABLV ascent phase

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Following constrained optimization problem is solved using Optimization tool box in MATLAB package for generating the optimal trajectory parameters. The trajectory is optimized for maximizing the payload satisfying the following constraints q = 62 ± 0.2 kPa (9)

°≤α≤° 61 (10)

q 5400 Pa . radα ≤ (11) Where qα represents the aerodynamic load acting on the vehicle. The constraints on q and qα have to be

satisfied to ensure controllability and maintain the structural integrity of the vehicle. The constraint on α 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, burn out Mach number should be 7. Since ABLV is similar to an aircraft, angle of attack is taken as the control parameter.

IV. Guidance and Control Algorithm Development Guidance and Control algorithms are needed for the vehicle to follow the prescribed optimal trajectory. Though

the control system parameters do not contribute to the computation of optimal trajectory, it is essential to have a robust control system that can follow the guidance commands faithfully. The control system used in this work is briefly described here in order to discuss the tools used in the research work. Several choices were explored to determine the variable to be controlled. It was decided to command altitude as a function of the current velocity. The reason for this choice is that the investigator will usually have some insight into the (h-V) (altitude-velocity) flight corridor of the vehicle and mission. The adaptive guidance law is then developed that controls the flight path through a second order rate controller for angle of attack. The over all block diagram of control and guidance schemes is shown in fig.4 below.

Figure. 4 Block Diagram of Guidance and Control Schemes

A. Guidance Law Development From the guidance law, a set of command equations are developed to meet the vehicle constraints and to follow

a desired altitude versus velocity path. This is accomplished by setting thrust to the maximum value to maximize the payload. Flight path control is achieved through a second order rate controller on the angle of attack. The feed back controller has an adaptive gain, K that is computed on line depending upon the current state/performance of the vehicle. τ is the control system time constant that gives a stable control system performance. The guidance law commands the altitude as a function of current velocity.

Control Law

Vehicle

Optimal Trajectory

Sensors

h,v ,α

hopt,Vopt hc αc Guidance Law

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)..h

2

2.hh(Kcom

.∆

τ+∆τ+∆=α∆ (12)

acthcomhh −=∆ (13)

)v(comhcomh = (14)

.v

vcomh

com.h �

�

���

�

∂

∂= (15)

..

com2.

2com

2

com

..

vv

hv

vh

h ��

���

�

∂∂

+���

����

�

∂∂

= (16)

γ= sinvact.h (17)

γγ+γ= cos.

vsin.vact

..h (18)

act.hcom

.h

.h −=∆ (19)

act..hcom

..h

..h −=∆ (20)

B. Feedback Controller Design The optimal ascent trajectory is computed off line and stored “on board” as a reference h-v profile. The thrust

level of ABLV is directly controllable through throttle CT ,(T= CT Tmax) whereas the aerodynamic forces level and orientation are controllable through angle of attack α, and bank angle φ. Present study assumes that optimum thrust is maintained at the maximum value (throttle CT=1), the vehicle performs only planar motion (φ=0) and only angle of attack is the control variable. The control is accomplished by means of a feedback loop as shown in fig.5. It is based on a second order rate controller for α. ABLV configuration is similar to an aircraft. In present day aircrafts the pitch attitude and pitch rate are controlled through angular accelerations about the Y-body axis. Since we are

dealing with two degree of freedom, point mass analysis, the relationship between .

α and vehicle pitch rate is linear. Hence a rate controller is selected instead of a displacement controller for α.

Figure. 5 Vehicle Control System

Gc

Gv

H=1

hact hcom ∆h com

.

α∆ +

−

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The control system controls the altitude through a commanded.

α . The optimized flight path defines the commanded altitude at each time interval as function of current velocity. To proceed with the task, it is necessary to obtain the transfer function for the vehicle (Gv) and the controller (Gc). The transfer function for the vehicle is derived using linear theory and over most of the flight path it may be approximated as follows.

3Lref

v msTCqS

G+

= α (21)

As stated previously, a second order rate controller for α is adopted. It may be expressed as follows.

)..h

2

2.hh(Kcom

.∆

τ+∆τ+∆=α∆ (22)

Where ∆h= hact-hcom and τ is the ∆t value providing a stable control system and the adaptive gain K was specified as

)TLCrefqS(3

mk4K

+ατ= (23)

Expressing the controller transfer function by means of the Laplace Tranforms produces

)2

2s2s1(KcG

τ+τ+= (24)

The goal of the controller is to select the values of the adaptive gain (K) and the time constant (τ) to accurately follow the commanded altitude without burning excess fuel. It is also important that the controller is stable and robust over a variety of flight conditions.

Referring back to the feedback loop the over all transfer function may be expressed as follows

)s(vG)s(cG)s(H1

)s(vG)s(cG

comhacth

+= (25)

The characteristic equation for the control system that determines the stability features is therefore

0)s(G)s(G)s(H1 vc =+ (26)

Let H(s)=1, 0)2s

s1)(TCqS(mK

s22

Lref3 =τ+τ+++ α (27)

Next important step is to investigate the values of k and τ to optimize the system performance and ensure good tracking and stability. A τ value between 1 and 10 and a k value between 0.25 and 1.5 produces same relative payload. Root locus technique is used to find the range of values of k and τ for ensuring required performance of the system. Root locus of the system is shown in fig.6.

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-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Root Locus(Tow=5)

Real Axis

Imag

inar

y A

xis

Figure. 6 Root Locus of the Open Loop system

Location of closed loop poles in the s-plane gives insight into the stability of the feed back loop and the transient

response of the system. For, k=0.5, two poles lie on the imaginary axis and the response of the system is oscillatory as can be seen in the step response shown in fig.7. For k=1 & 1.5 the control system is stable with all three roots in the left half of the s-plane.

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2k=0.5k=1k=1.5

Step Response

Time (sec)

Am

plitu

de

Figure. 7 Closed Loop System Step Response

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The following Table-1 summarizes the results of step response. Transient response is best for k=1.5 and it is selected for further studies.

Table.1 Step Response Results

Gain, k Tr Ts Mp

s s %

0.5 *

*

*

0.75 2.57 107 60 1.0 2.19 53.2 50 1.5 1.72 25.1 38

* - oscillatory response Tr- Rise time, Ts- Settling time, Mp- Peak over shoot Now the effect of variation of time constant τ (1 to 10), for the selected value of k=1.5, is studied. It was

observed that smaller values of τ provide faster rise time and settling time (seen from step response), but gain and phase margins remain same with only a change in the frequency at which peak gain or phase margin occurs (seen from nyquist plot). τ=10 was eliminated since it was resulting in large rise time and settling time. As a next step long period simulations were carried out for case 8 (worst under performance case) with τ=1 and 5. The results were analysed from the point of view of (i)final burn out mass, (ii) final states achieved and (iii) control command - α profile. τ=5 was selected as it was giving better payload mass, smooth α command and dispersions on achieved end conditions were also minimum. Nyquist plots for k=0.25, 0.5, 0.75, 1&1.5 and τ=5 is shown in fig.8. The system is unstable for gain k< 0.75.

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Nyquist Diagram(k=0.25 to 1.5, tow=5)

Real Axis

Imag

inar

y A

xis

Sys1

Sys2

Sys3

Sys4

Sys5

Figure. 8 Nyquist plots for Open Loop compensated System

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Following Table-2 highlights the available stability margins for the variation of gain, k for selected tow=5. From Nyquist plots it can be seen that the system has enough stability margins. From the investigation it is clear that k=1.5 and τ=5 ensures the performance requirements.

Table.2 Stability Margins

Gain, k PM GM GRM Stability deg dB dB

1.5

50.4

∝

-9.54

Yes 1.0 32 ∝ -6.02 Yes

0.75 18 ∝ -3.52 Yes 0.5 0 0 0 No

0.25 -23.5 6 -- No

PM- Phase Margin, GM- Positive Gain Margin, GRM- -ve Gain Margin or Gain Reduction Margin

V. Evaluation of the Guidance Algorithm Flight simulations with off nominal performance of air-breathing engine and dispersions in aerodynamic

coefficients are carried out to establish the robustness of the guidance algorithm. Analysis of simulation results shows that the mission requirements are achieved accurately, ensuring fuel optimality also. The case descriptions and analysis of simulation results are presented in subsequent sections.

A. Initial Conditions for Guidance Altitude = 5.450 km Velocity = 428.8 m/sec Flight path angle = 28.77 ° Mass = 176 t Pitch angle = 31 °

B. Desired end condition Altitude h f = 27.337 km Velocity Vf = 2116.2 m/sec Flight path angle γf = 1.906 ° Mach number Mf = 7.000 Mass mf = 161.33t

C. Off nominal cases studied The adaptive guidance algorithm is validated for air-breathing engine off nominal performances as well as

aerodynamic parameter uncertainties. Following cases are simulated Case0- Nominal Case1- Thrust 5% more Case2- Thrust 5% less Case3- 20% more Cl Case4- 20% less Cl

Case5- 20% more Cd Case6- 20% less Cd Case7- Thrust 5% more, 20% more Cl,, 20% less Cd Case8- Thrust 5% less, 20% less Cl,, 20% more Cd

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D. Analysis of simulation results

Achieved terminal conditions are given in Table.3 Guidance command to shut off the engine is issued once the target Mach number of 7 is reached. Maximum dispersion in achieved altitude is less than 0.005km. Maximum velocity dispersion is 1m/s and the maximum error in flight path angle is 0.2deg. Achieved end conditions in altitude, velocity, flight path angle, Mach number and mass for all the above cases are well within the mission specifications. This clearly establishes the robustness of the closed loop guidance algorithm to meet the desired end conditions accurately without burning excess fuel.

Table.3 Achieved end Conditions

Dispersion cases h f

km Vf

ms-1 γf

deg Mf mf

T Case0 27.337 2116.2 1.9 7.00 161.33 Case1 27.332 2115.2 1.7 7.00 161.60 Case2 27.333 2115.4 1.9 7.00 161.09 Case3 27.336 2116.0 2.0 7.00 161.07 Case4 27.335 2115.6 1.6 7.00 161.53 Case5 27.336 2116.0 1.9 7.00 160.20 Case6 27.335 2115.6 1.7 7.00 162.37 Case7 27.333 2115.4 1.7 7.00 161.38 Case8 27.336 2115.9 1.9 7.00 160.10

Altitude and velocity profiles for the above cases are shown in fig.9.

0

10

20

30

50 150 250 3500

500

1000

1500

2000

2500

VelocityCase0 to 8

AltitudeCase0 to 8

Vel(m/s)

Time(s)

Alti

tude

(km

)

Altitude & Velocity Profiles (CLG)

Figure.9 Altitude and Velocity Profiles

Figure .10 gives the angle of attack steering commands that are well within the α limits.

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0

2

4

6

5 0 1 5 0 2 5 0 3 5 0T i m e ( s )

αα αα(d

e)C o m m a n d e d αααα f o r C L G s t e e r i n g m o d e

Figure.10 Angle of attack Profiles

Effectiveness of the guidance law is tested in the event when actual flight condtions are very much different

from nominal ones. For the dispersion cases shown in fig.11, the flight path angle error after lift off was > 4deg. This change in flight path (kick) angle alters the trajectory very much if open loop guidance is used and the vehicle impacts on ground soon after take off as depicted in fig.11. This highlights the importance of appropriate angle of attack profile for meeting the mission requirements for ABLV. If closed loop guidance is employed, the feedback nature of the guidance law, enables it to adapt to the trajectory deviations and precise end conditions are achieved even for all the dispersion cases studied.

5

6

7

8

9

1 0

9 0 1 0 0 1 1 0 1 2 0

C a s e 4 - 2 0 % l e s s C lC a s e 1 - T h r u s t 5 % m o r eC a s e 2 - T h r u s t 5 % l e s sC a s e 5 - 2 0 % m o r e C dN o m i n a l

T i m e ( s )

Alt(

km)

A l t i t u d e p r o f i l e s f o r O L G m o d e o f s t e e r i n g

Figure.11 Vehicle Impacts on ground- Open Loop Guidance

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VI. Conclusions This paper presents the trajectory optimization and guidance law development for a typical Air Breathing Launch Vehicle during its air-breathing ascent phase. In the present study, reference optimal trajectory is generated offline using NLP method. The NLP problem is solved using Sequential Quadratic Programming (SQP) method. A real time guidance law is developed based on the strategy that the guidance law controls the angle of attack using a second order rate controller for angle of attack (α). The feedback controller has adaptive gain that is computed on -line based on the current performance of the vehicle. The developed guidance algorithm is found to be robust over a wide range of operating conditions. Extensive simulation studies are carried out to validate the guidance law. Maximum dispersions in altitude, velocity and flight path angle are 0.005km, 1m/s and 0.2° respectively. Guidance law also ensures that the constraints on angle of attack, dynamic pressure and aerodynamic load are always met without affecting the fuel optimality.

Acknowledgments Authors acknowledge the encouragement given by Dr.BN.Suresh, Director, Vikram Sarabhai Space Centre.

Authors also wish to express their sincere thanks to the reviewers for the comments and observations.

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10Paul T Boggs, Jon W Tolle., “Sequential Quadratic programming”, Acta Numerica, 1996 11Fletcher.R., “Practical Methods of Optimization”, Vol. 2, Constrained Optimization, John Wiley and Sons, New York,

1985 12John T. Betts., Practical Methods for Optimal Control using Nonlinear Programming, SIAM, 2001