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NASA Contractor Report 4568

Approximate Optimal Guidance for the Advanced Launch System

T. S. Feeley The University Los Angeles,

and J. L. Speyer of California
California

at Los Angeles

Prepared for Langley Research Center under Grant NAG1-1090

National Aeronautics and Space Administration
Office of Management Scientific and Technical Information Program 1993

Abstract

A real-time guidance scheme for the problem of maximizing the pay-

load into orbit subject to the equations of motion for a rocket over a spheri-

cal, nonrotating

Earth is presented. An approximate

optimal launch guidance

law is developed based upon an asymptotic expansion of the Hamilton-Jacobi-

Bellman or dynamic programming

equation.

The expansion is performed in

terms of a small parameter, which is used to separate tile dynamics of the

problem into primary and perturbation

dynamics. For the zeroth-order prob-

lem the small parameter is set to zero and a closed-form solution to the zeroth-

order expansion term of the Hamilton-Jacobi-Bellman

equation is obtained.

Higher-order terms of the expansion include the effects of the neglected pertur-

bation dynamics. These higher-order terms are determined from the solution

of first-order linear partial differential equations requiring only the evaluation

of quadratures.

This technique is preferred as a real-time on-line guidance

scheme to alternative numerical iterative optimization

schemes because of the

unreliable convergence properties of these iterative guidance schemes and be-

cause the quadratures

needed for the approximate optimal guidance law can

be performed rapidly and by parallel processing. Even if the approximate solu-

tion is not nearly optimal, when using this technique the zeroth-order solution

iii

PI_A_OtNi; P_G[ 8(.ANK NOT FH.14ED

always provides a path which satisfies the terminal constraints.

Results for

two-degree-of-[reedom

simulations arc presented for the simplified problem o[

flight in the equatorial plane and compared to the guidance scheme generated

by the shooting method which is an iterative second-order technique.

iv

Table

of Contents

Abstract

iii

Table of Contents

V

List of Tables

viii

List of Figures

ix

List of Symbols

xi

1. Introduction

1

o The Peturbed

Hamilton-Jacobi-Bellman

Equation

5

2.1 Expansion of the H-J-B Equation .................

8

2.2 Solution by the Method of Characteristics

............

10

2.3 Determination

of the Optimal Control ..............

11

2.4 Determination

of the Forcing Functions ..............

12

1 Modelling

of the ALS Configuration

14

3.1 Equations of Motion for the Launch Problem ...........

16

3.2 Propulsion ..............................

18

3.3 Aerodynamics

............................

18

3.4 Mass Characteristics

........................

21

3.5 Gravitational

and Atmospheric Models ..............

22

V

3.6 Expansion Dynamics ........................

24

3.6.1 Two-Dimensional

Flight ..................

25

0 Zeroth-Order

Optimization

Problem

27

4.1 Optimization

Problem Statement

.................

27

4.2 Zeroth-Order

Coordinate Transformation

.............

29

4.3 Zeroth-Order

Analytic Solution in the Cartesian Frame .....

31

4.4 Linking the First and Second Stage Subarcs ...........

36

o First-Order

Corrections

40

5.1 Correction to the Lag-range Multipliers

..............

41

5.2 The First-Order

Forcing Function .................

41

5.3 Relating the Partial Derivatives of the Wind Axis Frame to the

Partial Derivatives of the Cartesian Frame ............

43

5.4 Partial Derivatives of the Analytic Solution ............

44

5.4.1 Partial Derivatives of Some Common Terms .......

44

5.4.2 5.4.3

Partial Derivatives of the Analytic States ......... Solution to the Linear System of Unknown Partials

45 . . 48

, Aerodynamic

Effect along the Zeroth-Order

Trajectory

52

6.1 Inclusion of an Aerodynamic Effect in the Zeroth-Ordcr

Problem

53

6.1.1 Zeroth-Order

Aerodynamic

Effect in the Rectangular

Co-

ordinate System .......................

56

6.1.2 First-Order Correction Terms ...............

59

Results for the Rectangular Pulse Punctions

...........

60

Aero Pulses in the Body-Axes Frame ...............

62

vi

7. Results

67

o The Relationship equation

between Calculus of Variations

and the HJB 83

8.1 Correction Terms to the Lagrange Multipliers

..........

83

8.2 Expansion of the Euler-Lagrange

Equations

...........

87

8.2.1 Expansion of the State Equations .............

88

8.2.2 Expansion of the Lagrange Multiplier Equations .....

89

8.3 Expansion of the Boundary Conditions

..............

91

8.3.1 Expansion of the Transversality

Conditions ........

92

8.4 Solution to the First-Order Problem ................

93

8.5 Solutions to First-Order Linear Partial Differential Equations..

95

8.6 Formulation of First-Order Correction Terms for the ALS Probleml00

8.7 Results ................................

105

9. Conclusions

114

A. Zeroth-Order

Solution for Three-Dimensional

A.1 Zeroth-0rder

Coordinate Transformation

Flight .............

B. Canonical

Transformations

C. Point Inequality

Constraints

D. Analytic

Partial Derivatives

BIBLIOGRAPHY

for Zeroth-Order

Solution

117 124 129 133 137 142

vii

List of Tables

3.1 Vehicle Mass Characteristics

....................

22

7.1 Comparison of Results .......................

72

7.2 Comparison of computation time .................

81

8.1 Comparison o[ open loop results ..................

106

8.2 Comparison of closed loop results .................

106

°o,
VIII

List of Figures

3.1 ALS Vehicle Configuration

.....................

15

3.2 Coordinate Axis Definition .....................

17

3.3 First Stage Drag Model .......................

19

3.4 First Stage Lift Model .......................

19

3.5 Second Stage Aerodynamic

Model .................

21

4.1 Transformation

of Coordinal_e Systems ..............

30

6.1 Coordinate frames for the aerodynamic pulse functions

6.2 Model for aerodynamic pulses in x-direction

...........

6.3 Model for aerodynamic

pulses in z-direction

...........

6.4 Open loop zeroth-order path for body-axes aerodynamic

.....

55

57

57

pulses . 66

7.1 Hamiltonian versus Angle-of-Attack first stage ..............................

at continuous points of the 69

7.2 First stage model for the drag coefficient .............

70

7.3 Comparison of the first stage and second stage aero models along

the vacuum path ..........................

71

7.4 Angle-0f-Attack

vs. Time .....................

74

7.5 Thrust Pitch Angle vs. Time ...................

75

7.6 Altitude vs. Time ..........................

76

7.7 Velocity vs. Time ..........................

76

7.8 Flight Path Angle vs. Time ....................

77

ix

7.9 Dynamic Pressure vs. Time ....................

78

7.10 Velocity Lagrange Multiplier vs. Time ..............

79

7.11 Flight Path Lag-range Multiplier vs. Time ............

80

8.1 Geometric Interpretation

of Integral Surface ...........

98

8.2 Open loop solution for Lagrange multipliers at staging conditions 108

8.3 Open loop solution for Lagrange multipliers at first stage initial

conditions ..............................

109

8.4 Closed loop solution for flight path angle Lagrange multipliers

110

8.5 Closed loop solution for velocity Lagrange multipliers

......

111

8.6 Closed loop solution for angle-of-attack

..............

112

List of Symbols

English Symbols

a, b, c
CD
CD_
C Dc.2 CDa3
CL CL_
C L_,2
cq
C_,,Cw
Cw
D
f(y,_,T) f,
f_

constants of the quadratic mass equation

drag coefficient

linear coefficient in the drag model

quadratic coefficient in the drag model

cubic coefficient lift coefficient

in the drag model

linear coemcient in the lift model

quadratic coefficient in tile lift model side force coefficient

constant terms associated with the Lagrange for the velocity components u, w

multipliers

constant term used to rewrite the Lagrange multipliers

in terms of mass, C_, = _--_,rmo+ C_ second stage value of Cw given first stage initial conditions drag force primary dynamics the i th term of the asymptotic expansion of the primary dynamics

partial derivative of the primary dynamics with respect to the control u

xi

g g_
C(y, u, t)
h hi hf.p_c he H
H Opt
[f w_nd
HLH
HI
H_,
Isp
J K(Q,P,t) L

perturbation

or sccondary dynamics

the i th term of the asymptotic expansion

of the

perturbation

dynamics

partial derivative of the perturbation

with respect to the control u

dynamics

gravity sea-level gravity scalar function of the augmented altitude final attained altitude

performance

index

specified final altitude atmospheric density scale height the Hamiltonian of the systcm

the optimal Hamiltonian the Hamiltonian of the wind axis system the Hamiltonian of the local horizon or Cartesian the Hamiltonian evaluated at the final time

system

first derivative of the Hamiltonian with respect to the control u second derivative of the Hamiltonian with respect to the

control u

specific impulse

performance index

Hamiltonian

for a new set, of variables Q and P

lift force

xii

L_L
rnf
17_s_ge t Yns_ge2
M N(y,t) p P P(x,t)
P= P,
Ptt
Q

Lagrangians used in Appendix B mass of the vehicle final mass

specified mass at end of first stage before staging

specified mass at beginning Mach number; M = rE_
303
number of engines

of second stage after staging

dynamic pressure equality constraint appears in Appendix C the partial of the dynamic pressure equality constraint generalizcd coordinate of old system in Appendix B

generalized coordinate of new system in Appendix B the optimal return function starting at the initial conditions the partial derivative of the optimal return function with respect to the initial state x

the partial derivative of the optimal return function with respect to the initial time t

i th term of the asymptotic expansion of the primary dynamics the partial derivative of the i Lh term of the expansion of the optimal return function with respect to the initial state x the partial derivativc of the i Lh term of the expansion of

the optimal return function with respect to the initial time t dynamic pressure

generalized coordinate of old system in Appendix B side force in Chapter 3 on ALS modelling

generalized coordinate of new system in Appendix B

.o0
XIU

T Te
80S
S S(q,Q,t) t, to tl
_s_ge
T
rl T_ T_
U,
V
vl
/'f a pec
X
X
(x,Y,Z)
Y

radial position of the vehicle: re q- h radius of the Earth

the forcing function associated with the i _a correction term

speed of sound

Cross-sectional

area of the combined vehicle

generating function defined in Appendix B initial time final time

stage time total thrust of the vehicle

value of the thrust for the first stage value of the thrust for the second stage

vacuum thrust per engine the i th term of the asymptotic

expansion

series of the control

velocity components velocity

associated with the inertial frame

final attained velocity specified final velocity initial states

downrange Position coordinates state vector

for the right-handed

inertial frame

xiv

Greek Symbols

/3 X 6(c, h) A Amst,,ge
"/f. pec

angle-of-attack;

control in the wind axis system

vehicle sideslip angle; control in the wind axis system

velocity heading angle

ratio of the atmospheric density to the small parameter

discriminant associated A = 4ac- b2

with the quadratic

mass equation

discontinuity in the mass at staging the small expansion parameter;

ratio of the atmospheric scale height to the radius of the Earth

the jth power of the small expansion parameter

flight path angle

final attained flight path angle

specified final flight path angle

Lagrange Lagrange

multiplier multipliers

associated associated

with the state y with the wind axis states

Lag'range multipliers Ah, Ax, Ay, A._

associated

with the Cartesian

states

#

velocity roll angle; control in the wind axis system

Lagrange multiplier associated with the terminal constraint

ft(y(tst_9e))
¢

constraint latitude

imposed by the staging condition of the rocket

¢2(q,p,t)
¢(yf, Ts)

new generating function equal to S(q, Q, t) scalar component of performance index

on y

Xv

_(_)
P p_ p_
O" T
0

vector of terminal constraints atmospheric density sea-level atmospheric density reference atmospheric density specific fuel consumption time longitude pitch angle; control in the Cartesian system

Miscellaneous
nm
sin
C08
tan sinh -i
_(m) a( ) _() _() _dT--_()
_()
_o) (

Symbols nautical mile sine function cosine function tangent function inverse hypcrbolic sine function argument of the inverse hyperbolic sine function the differential of ( ) the time-varying variation of ( ) the variation of ( ) with time held fixed denotes the time derivative of ( ) with respect to the independent variable time partial derivative of ( ) with respect to the independent variable mass partial derivative of ( ) with respect to the initial state x

xvi

_ot( ) ),
0
)s )o ),
lira

partial derivaLive of" ( ) with respect to the initial time t

prime superscript used for second stage values which are

linked to the initial conditions on the first stage subarc

subscript denotes the initial conditon of ( )

subscript denotes the final conditon of ( )

superscript denotes the optimal ( )

subscript denotes sea-level value;

subscript denotes the characteristic

direction in Chapter 2

limit operation

xvii

Chapter

1

Introduction

An approach to real-time optimal launch guidance is suggested here

based upon an expansion of the Hamilton-Jacobi-Bellman

or dynamic pro-

_amming equation. In the past, singular perturbation

theory has been used

in expansion techniques used to solve optimization

problems [1, 2, 3]. For

singular perturbation

methods the states are split up into a set of 'fast' and

'slow' variables. The solution is then sought in two separate regions; one re-

gion where the fast states are dominant and an outer region where the slow

states are determined.

A composite solution can then be determined by com-

bining the two solutions. Matching asymptotic expansions is one method for

obtaining the final solution. This research uses a regular asymptotic expansion

which is assumed valid over the entire trajectory of the launch optimization

problem. An example of a launch optimal control problem is to determine the

angle-of-attack

profile which maximizes the payload into orbit subject to the

dynamic constraints of a point mass model over a rotating spherical Earth.

The solution of this type of optimization

problem is obtained by an iterative

optimization

technique.

Since the convergence rate of iterative techniques is

difficult to quantify and convergence is difficult to prove, these schemes are not

suggested to be used as the basis for an on-line real-time guidance law.

In contrast, an approximation

approach is developed which is based

2

upon the physicsof the problem. Thrust and gravity are assumedto be the dominant forcesencounteredby the rocket while the angle-of-attackis usually kept small in order to minimize the effect of the aerodynamic forces acting on the vehicle. Numerical optimization studies [4] havebeen performed which support this assumption. These results also indicate that ignoring the aerodynamic pitching moment has a negligible effect on the performanceof the vehicle. Thus the launch problem would seemto lend itself to the useof perturbation theory. It is shownthat the forcesin the equationsof motion can be written as the sum of the dominant forces and the perturbation forceswhich are multiplied by a small parameter c, where ¢ is the ratio of the atmospheric
scale height to the radius of the Earth. The motivation for this decomposition is that for ¢ = 0, the problem of maximizing the payload into orbit subject to the dynamics of a rocket in a vacuum over a fiat Earth, is an integrable opti-

mal control problem. The perturbation

forcing terms in the dynamics producc

a nonintegrable

optimal control problem. However, since these perturbation

forces enter in with a small parameter, an expansion technique is suggested

based upon the Hamilton-Jacobi-Bellman

equation.

The expansion is made

about the zeroth-order

solution determined when c = 0. This zeroth-order

problem is now solved routinely in the generalized guidance law for the Space

Shuttle [5] with a predictor/corrcctor along the desired path.

scheme employed to guide the vehicle

The higher-order

terms of the expansion are determined

from the

solution of first-order linear partial differential equations which require only

integrations which are quadratures.

Quadratures are integrals in which the in-

tegrand is only a function of the independent variable. Previous so]ution meth-

3

ods applied to guidance problems have motivated the approach suggested here.

These include the explicit gnlidance laws, E-galidance, developed by George

Cherry [6] for the Apollo flight. By writing the dynamics strictly as functions

of the independent

variable a solution was obtained by quadrature integra-

tions. Past applications [7, 8] of the proposed scheme, have shown that very

close agreement with the numerical optimal path is obtained by including only

the first-order term. Because no iterative technique is required, this scheme is

suggested as a guidance law since the quadratures

can be performed rapidly.

Chapter 2 contains a general formulation of the perturbation

prob-

lem associated with the Hamilton-Jacobi-Bellman

partial differential equation

(HJB-PDE).

The technique for determining the higher-order expansion terms

due to the perturbation

forces caused by the atmosphere

Earth model is discussed. Lastly, the recursive relationship

and the spherical for the control is

presented. In Chapter 3, the characteristics

for the Advanced Launch System

(aka National Launch System) and the general equations of motion in terms of

the small parameter e, are given. For e = 0, a simplified optimal launch problem

in the equatorial plane is formulated, and its solution in terms of elementary

functions is given in Chapter 4. The coordinate system transformation

used

to obtain the analytic solution is included. Also discussed is the linking of the

trajectory subarc for the first stage to the subarc of the second stage. In Chap-

ter ,5 the first-order correction term to the control is determined.

Results are

presented in Chapter 6 and compared to the shooting method solution, which

is a numerical iterative second-order optimization

technique. It was found that

during much of the first stage the aerodynamics

are not small when flying the

optimal vacuum trajectory.

Chapter 7 presents a method for reshaping the

zeroth-order trajectory by including an aerodynamic effect. This effort centers on the useof constant aerodynamicpulse functions which are obtained by averaging the aerodynamicsalong the zeroth-order path during various time intervals. Lastly, Chapter 8 relates perturbation theory and the Calculus of Variations with the expansionof the Hamilton-Jacobi-Bellman equation. Tile equivalenceof the two solution methods is presented.

The Peturbed

Chapter

2

Hamilton-Jacobi-Bellman

Equation

The optimal control problem can be formulated as one which mini-

mizes a performance terminal constraints; Minimize

index subject to a set of nonlinear that is,

dynamics

and a set of

J=

(2.:)

with the dynamics

= f(y, u, r) + _9(y, u, r)

(2.2)

subject to the terminal constraints

qJ(yf, Tf) ---- 0

(2.3)

and the initial conditions

y(t) = x = given

(2.4)

Note that Y is an n-dimensional

state vector, u is an m-dimensional

control

vector, _ is a small parameter, r is the independent

variable, _) =a dy/d'r, t is

the initial value of the independent

variable, and x is the initial state at t.

namics.

Eq. (2.2) is separated into two portions: primary and secondary dyNote that the control appears in both parts. The primary dynamics

5

can be assumed to dominate over the secondary dynamics because the secondary dynamics are multiplied by the small parameter (e) and therefore have a small perturbing effect on the system.

The Hamilton-Jacobi-Bellman

(H-J-B) equation [9] is

- Pt = H °pt = min H = p_[/o_t + cgOpt]

(2.5)

uEbt

where/4 is the class of piecewise continuous bounded controls and u_t(x, P_., t)

is obtained from the optimality condition H_ = 0 and from the assumption

that the Legendre-Clebsch

condition is satisfied (H_,_, is positive definite). In

addition, fopt =_ f(x, uOpL, t) and gore _ g(x, uOpt,t). The Hamilton-Jacobi-

Bellman equation will be used to determine minimizes the cost criterion J.

the optimal control policy which

The function P(x, t) is called the optimal return function and is de-

fined as the optimal value of the performance index for a path starting at x and

t while satisfying the state equations (2.2) and the terminal constraints, i.e.,

P(x,t) = ¢(yl,r/)

at the hypersurface

Bellman partial differentional equation

_P(y/,'r/) = 0. The Hamilton-Jacobi-

(2.5) can be interpretated

[10] as the

derivative of the optimal return function P. The optimal return function is

a constant since it is dependent only on the terminal conditions and thus the

total derivative of the optimal return function along an extremal path must be

zero.

dP Pt + p_[fovt + cgOpt] 0 dt

Each point in space belonging to the optimal trajectory must give the same value to the optimal return function as the optimal P(x, t) since the trajectory

is considered optimal from thc initial conditions (x, t) to the terminal manifold.

Now, if a non-optimal control is chosen at any point in the trajectory, then the

resulting terminal state, as generated by' the system equations, must produce a

value for the optimal return function equal to or greater than the optimal value.

Thus the control that minimizes the cost is the control which at each point of the trajectory causes the derivative of the optimal return function to be zero.

This is the fundamental

notion represented by the Hamilton-Jacobi-Bellman

equation. Note that x and t can be either the initial or the current state and

time, respectively. In this context, it will be used to represent the current state

and time. Also note that ew._ry admissible constraints qJ(Yl, rl) = O.

trajectory

must satisfy the terminal

P(z, t) can be expanded ,as a series expansion in e as

,_'(_,t)= _ f',(_, t)_'
i=O

(2.6)

and the optimal control can also be expanded in a series expansion as

oo
_°_(_, &,t)= _ _,(_,t)_'
i=0

(2.7)

where u _t is obtained by substituting

Eq. (2.6) into Eq. (2.7) and expanding

the function. Therefore, it is possible to obtain the control law in feedback

form.

The zeroth-order

control, Uo, is the optimal control for the zeroth-

order problem where e = 0. If an analytic solution can be obtained for the

zeroth-order

problem then higher-order

solutions for the control can be ob-

tained by expanding the Hamilton-Jacobi-Bellman

equation

P, = Z P,,(_, 0 _'= -
i----O

F,_(_,t)_'

f,_' + _g,_'

i=O

i= 1

(9.8)

8

where the dynamics have been expressed as expansions of the form

OC
f°Pt(m, u °m, t) = _ f_(x, u, t)d
i=0

f"(x,

t)=
i=O

(2.9)
(2.1o)

Expanding Eq. (2.8) and collecting terms of equal powers in e, produces the

following set of linear, first-order, partial differential equations

Pit + P_zf_

t=
=

i-I

-_ j=o

Pjz(fi-j

_- gi-j-l)

R4(z,t, ei-l,...,Po)

i= i,2,...

(2. ii)

The expansion next section.

of the Hamilton-Jacobi-Bellman

equation will be detailed in the

2.1

Expansion

of the H-J-B

Equation

The solution to the optimal control problem requires the evaluation of the Lag-range multiplicr, P_. Note that the quantity P_ is the partial derivative of the optimal return function with respect to the state y at the initial time or the current time (since at r = t, y = x). The function P= is expanded in a series in the small paramcter e. The terms of this series expansion, P_=, are evaluated in terms of quadrature integrals which are functions of P_. Recall that the functions P_ require the previously evaluated terms Pj=, f,_j, and g__j_ l for j = 1,...,i - 1. The coefficients f, and gi are the i it' term in the series expansion of f and g given in Eqs. (2.9)-(2.10). Since f and g are assumed to be sufficiently differentiable, they are expressible in a power series in e in terms

9 of the conLrol. For a scalar control, this yields

g°Pt(x, It °pt , t) =

0U i

x,t,_=0

_ uje 3

(2.13)

The above equations

assume that the zeroth-order

control, uo, is the dominant

term in the series (Eq. (2.7)). This implies that the higher-order

correction

terms, 7zl, _z2, ..-, have a much smaller ef[cct on the optimal return flmction,

[_(x, l), than the zeroth-order

term. rFhe first ['our terms of f and g are obtained

by use of [']qs. (2.12) _n(i (2.13).

fo -- f°m(x, Tzo,t)= f(x,_zo, t)

fl =
&-
f3 --

utf_(x, uo, t)
zt 2
_f_(x, uo, t) +u2f_(x, uo,t)
tt 3 -j f_,_,_(x, Zto, t) + zt,Tz2f_,(:c, Uo, t)

+u_f_(_, _o,t)

(2.14) (2.15)
(2.16)
(2.17)

9o = 9°_(_, _,o,t) = 9(x, _o,t)
gl = ulg,,(:c,uo,t)
_ - 2 _""(_:'_o,t) + _9,,(x, uo,t)

g3 -

6 g,,_,(X, Uo, t) + Ulu2guu(X, +u39,,(x, uo, t)

Uo, t)

(2.1s) (2.19) (2.20)
(2.21)

lO
Note that in taking the partials with respect to u in Eqs. (2.12) and (2.13), the partial is taken first and then the partial is evaluated at x, t with c set equal to zero. In other words, the partials arc evaluated along the zeroth-order path.

2.2 Solution by the Method of Characteristics

The H-J-B equation (Eq. (2.5)) is a first-order partial differential equation. The expansion of the H-J-B equation results in the first-order differential equation for P_ stated in Eq. (2.11) with the boundary condition P_(xl,tl) = 0, for i = 1,.... Recall that f_t denotes the dynamics of the zeroth-order problem (e = 0) using the zeroth-order control u = u0. Recall also that the forcing term /_ is only a function of expansion terms of P of order less than i.

The method of charactcristics

is used to solve a set of linear or quasi-

linear partial differential equations. cation and solution of characteristics

This technique [11] requires the identifi-

curves. The characteristic

direction ds is

defined by the equation

Pi,(dT)s + P_,(dy), = (dP_).,

i= 1,o,, ..-

(2.22)

Eqs. (2.11) along with (2.22) can be put in the form

(ayL

= (aP, L

The characteristic

directions for Eq. (2.23) are given by the solution of the

differential equation that is obt'ained by setting the determinant of the matrix

given in Eq. (2.23) equal to zero, such that

(dy)s- fo(d'r)s = 0 ==_ (dy/dv), = fo

(2.24)

11

The subscript s denotes tile characteristic

direction.

Therefore, the charac-

teristic curves of the equations, zeroth-order optimal trajectory

for any order term of P/, are given by the

90 = f0

(2.25)

whose solution is denoted as yo(r; x, t). The solution for P/ is given by

P,(x, t) = - fit, R°dT where /_ is defined along the zeroth-order path as

(2.26)

R °= l_(yo,r, Pi__(yo,r),',Po(Yo,

r)),

i= 1,2,...

(2.27)

Thercfore, having already dctermincd P terms of order less than i, a solution for P, can be determined by integrating R4 from the current 'time' to the final 'time' along the zeroth-ordcr path.

2.3 Determination

of the Optimal

Control

Since the primary and secondary dynamics, f and g, are expanded

in terms of the control (Eqs. (2.12) and (2.13)), the control expansion terms

u0, ul, u2, ..-, need to bc determined.

The optimality condition provides the

necessary tool to obtain these control tcrms. It can be stated as

By expanding

Px[f_ + eg_] =

P,= ei

(fi_ + eg,.)e' = 0

--

i=0

(2.28)

and multiplying out the terms of the two power series and equat-

ing like powers of e, the following relations are obtained

e°

:

P0. £ = 0

(2.29)

12

12 +&.[9,, + u:f..] + P2.f. =0

(2.30) (2.31)

Note that uo, the optimal control for the zeroth-order problem, can be solved using Eq. (2.29). Similarly, ul can be solved using Eq. (2.30) and u2 can be solved using Eq. (2.31).

2.4 Determination

of the Forcing Functions

Eqs. (2.14)-(2.21)

and (2.29)-(2.31)

can be used to solve for the

forcing

functions

Ha where Eq. (2.11) can be restated as

i--I
Ha= - Z PJ_(f,-J + ._t,-,-,)
j=O

i = 1,2,...

Using the above equations, RI is

(2.32)

R, = - &.(f, + o0) = -&.(u,L + g)

(2.33)

With the use of the optimality condition of Eq. (2.29), R_ becomes
& = - &=go

(2.34)

Similarly, the equation for It2 is

R2 = -- Po.(f2 + gl) - Pl=(fl + go)

R2 simplifies to the following equation when Eqs. (2.14)-(2.21)

are substituted

into the previous equation.
u_ D
R2 = --_, o=L_, - Pl_go

(2.35)
and (2.29)-(2.30)
(2.36)

Finally, R3 can be expressed as
R3 = -Po.(f3 +g2) - P,.(f2 +gl) - P2.(fL +go)
This simplifies to

13 (2.37)

= ,,r:'.:o,+-ULU 2 _ 1 U 1 ,-,.U[l go+(2y.3I8).]

Using the expression for Ri, the expression ers, Pi., can be expressed as

for the Lagrange

multipli-

- OOPx, - fits O-_Pz_ dr + _lt_-Ot _1_, OOtxI

(2.39)

Once these P,, are determincd,

they can be used in the optimal control ex-

pansion (Eq. (2.7)). As made apparcnt in the above equations, the solution

becomes increasingly complex as thc higher-order correction terms rely on the

state information from the lower-order trajcctories.

Modelling

Chapter

3

of the ALS Configuration

This chapter presents the modelling characteristics

and the equations

of motion for the rocket. Included are sections on the properties of the propul-

sion, aerodynamics,

masses, gravity, and the atmosphere.

A small expansion

parameter, the ratio of the atmospheric scale hc'ight to the radius of the Earth,

is then used to separate the dynamics into the primary and perturbation

ef-

fects. Lastly, the equations of motion for the zeroth-order a vacuum over a flat Earth are presented.

problem of flight in

The Advanced Launch System (ALS) is designed to be an all-weather,

unmanned, two-stage launch vehicle for placing medium payloads into a low

Earth orbit. The spacecraft (fig. 3.1) consists of a liquid rocket booster with

seven engines and a core vehicle that contains three engines. All ten liquid

hydrogen/liquid

oxygen low cost engines are ignited at launch. Staging occurs

when the booster's seven engines have exhausted their propellant. The three

core engines burn continuously from launch until they are shut down at or-

bital insertion.

Launched in the equatorial plane and ending at the perigee

of a 80nm by 150nm transfer orbit, the flight occurs in two-dimensions

over a

nonrotating,

spherical Earth. Note, the booster is assumed to ride on top of

the core throughout the first stage trajectory.

14

15
3315.2

Liquid RocketBooster

2667.2 Core Vehicle

1737.2----- _l

1497.2._......_

_ "_

1683.2 1516.6

I

4

50.9

0.0 ....

ii

Stations Measured From Exit Plane in Inches

Figure 3.1: ALS Vehicle Configuration

16

3.1 Equations

of Motion for the Launch Problem

The general equations of motion for a launch vehicle modelled as a point mass over a spherical, nonrotating Earth are given for flight in threedimensions as

h

Vsin7

=

(T cos_ cos_ - D)
- g sin y

m

= [- (T cos a sin/3 - Q) sin # + (T sin a + L) cos/z]
mV

V

g

+[(To+hi _]cos7

[(Tcos_sinB-Q)cosl_

+ (T sin c_ + L) sin/_]

=

(mV cos_)

V tan ¢ cos y cos X

4

(re+h)

= (Vreco+s h"f)ccoos sX¢

_) = rh =

V cos ")"sin X (re +h)
-aT.,c

(3.1)
(3.2)
(3.3)
(3.4) (3.5) (3.6) (3.7)

The vehicle coordinate system is shown in figure 3.2. Note, the engines are not

gimbaled and the aerodynamic pitching moments are neglected. For a vertical

launch Eqs. (3.3)-(3.4) experience a singularity caused by the velocity being

zero and by a flight path angle of 90 degrees, respectively. Therefore, a pitch-

over maneuver must be made at launch and equations different coordinate frame must be used.

of motion written in a

17

L

V

Y

D mg
Figure 3.2: Coordinate Axis Definition

18

3.2

Propulsion

Thrust is assumed to act along the centerline of the booster-core vehicle configuration and to be the same constant value for each engine. The total thrust of the rocket changes after staging as the seven engines of the booster are discarded, leaving only the three engines of the core vehicle.

T = (T,_c - npA_)

T,,_ = n x 580, 110. lbs.

where T,,,c is the total value of the thrust when acting in a vacuum and the

number of engines is n = l0 for the first stage and n - 3 for the second stage. Notice the variation of the thrust due to the atmospheric pressure p is given

for an undcrcxpanded

nozzle and thus a conservative value for thrust is used.

The value of the engine nozzle exit area is A_ = 5814.8/144. sq ft. The specific

fuel consumption of the rocket is

=l

sea

I_p g_ ft

(3.8)

and the specific impulse I_p = 430. seconds. after staging occurs.

The value of a remains

the same

3.3

Aerodynamics

Since sideslip causes drag, the vehicle is assumed to fly at zero sideslip

angle, so that only the angle-of-attack

gives the orientation of the vehicle rel-

ative to the free stream. The direction of the lift vector is then controlled

through the velocity roll angle. With no sideslip, the side force Q is identically zero. Therefore,

19

:_

0

Alpha

o_---_ 0

8

0

Mach

Figure 3.3: First Stage Drag Model

I0

\

Alpha

Mach

Figure 3.4: First Stage Lift Model

20

L = Ct.qS, D = Ct)qS, Q = CQqS = O

(3.9)

where CL, Co, CO. are the lift, drag, and side force coefficients, respectively, S is

the cross-sectional

1

2

area of the combined vehicle (booster + core), and q = ipV

is the dynamic pressure. The cross-sectional area S is assumed to be the same

constant value before and after staging occurs.

The aerodynamic

data has been provided in tabular form [4] and is

modelled by polynomials in a with Mach-number-dependent

coefficients. For

the first stage, the aerodynamic coefficients arc written as

CD(M, ol) = Coo(M) + CD 2(M)ol 2 + CD 3(M)c_ 3 CL(M,o_) = CL_(M)c_

(3.10)

where the Mach-number-dependent

terms have been obtained from cubic-spline

curve fits of the tabular data. Three-dimensional

plots [12] of the first stage

drag and lift models are shown in Figmres 3.3 and 3.4. Note that the drag coefficient of this vehicle at supersonic and hypersonic speeds has a minimum

at a positive angle of attack as shown in Figure 3.3. This is caused by the

aerodynamic shielding of the booster by the flow field of the core.

After staging, the vehicle operates in the hypersonic

the aerodynamic

force coefficients are modelled as

flow regime and

CD(OI) ----- CDo Jr- CD,_ Ol -t- CDc, 2Ot 2 CL(a) = CL.a + CL _a 2

(3.11)

with constant coefficients CDo = .2011, CD,_ = 0.0, CD,_2 = .001811, CL_. =

21

0.4

I

w Ct_

0.35

0.3

0.25

0.2

-10

-5

!
0 ct (deg)

!

,I 0.5

.... Ct, ,

0.25

-0.25

-0.5

5

10

Figure 3.5: Second Stage Aerodynamic

Model

.039962, and CL2 = .00100272. vided in figure 3.5.

Tile aerodynamic

plot of CL and CD is pro-

3.4 Mass Characteristics

The inert weights of the booster and core, the weight of the propellant,

the payload and payload margin, and the weight of the payload fairing comprise

the ALS takeoff weight. The fairing encases the payload and is carried along by

the core vehicle until orbital insertion.

The vehicle mass and sea-level weight

characteristics

are shown in Table 3.1. The time at which staging is to occur is

obtained from the first stage mass flow rate and the propellant

of the booster

rr_-o_tt,_,,t = 153.54 sec.

tstage

_-

7aT,_c

22

Vehicle Stage

Vehicle Component

Core
Booster Core + Booster

Inert Mass Propellant Payload Payload Margin Payload Faring Total Core Inert Mass Propellanl: Total Booster Total at Take-off

Take-off Weight
(lbs.)
176,130.00 1,479,180.00
120,000.00 12,000.00 39,120.00
1,826,430.00
216,880.00 1,449,980.00 1,666,860.00 3,493,290.00

Table 3.1: Vehicle Mass Characteristics

where the vacuum thrust per engine is T_o_ = 580110. Once the stage time, tile total first stage mass flow rate, the takeoff
weight, and the inert weight of the booster are known, then the weight of the vehicle at the end of tile first stage and the initial weight in the second stage can be calculated. For this vehicle the values are
msao,1 = 1421890. lbs., mst_oc2 = 1250010. lbs., Amst_gc = 216880. lbs.

3.5 Gravitational and Atmospheric Models

The gravitational

acceleration is modelled as an altitude-varying

tion by the inverse square law,

r2
e
g = g"(re + h)2

func-

23

but will be assumed constant in the zeroth-order problem to facilitate obtaining an analytic solution. The constant values for gravity at sea-level and for tile radius of the Earth are

ft g_ = 32.174 --
see 2

re = 2.09256725 x 10 r ft.

The atmospheric density is expressed by the exponential function,

p = pre-(r¢+h)/ho

= pre-rJh,

e-h/h, = pse-h/ho

(3.12)

where he is the atmospheric scale height and ps is the sea-level reference density. The values for these parameters are

p, = .002377 slugs h., = 23,800. ft. ft 3

The form of the density is chosen to motivate the selection of a small

parameter to exclude chosen as

the aerodynamics

in the zeroth-order dynamics.

If e is

e = hs/rc

(3.13)

and defining
then by atmospheric isfies the requirement small, i.e.,

_5(e,h) = p(e,h) e

(3.14)

properties ¢5(e, h) > 0. Tile exponential density also sat-

[3] that the perturbation

term in the dynamics remains

lim 6(e, h) --+0
_---+0

(3.15)

Satisfaction of this property used in the launch problem.

will allow more general atmospheric

models to be

24

The atmospheric pressure is "also expressed as an exponential function,

p -- p_e -h/%

(3.16)

where hp is the atmospheric pressure scale height and p_ is the sea-level reference

pressure. The values for these parameters are

lbs Ps = 2116.24 f-_

hp = 23,200. ft.

The speed of sound can be obtained by thc relationship

SOS _ W_

with the specific heat ratio for air given as F = 1.4 .

The gravity can be rewritten as

g=g_-

gsh(2r_ + h) (r_+h) 2 =gs-

egsh(2r_ + h)r_ hs(r_ + h) 2

(3.17)

where the expansion parameter has formally been introduced and the second term is clearly small in comparison to the first term which is the value for gravity at sea-level, g_.

3.6 Expansion Dynamics

In terms of the small parameter are rewritten as

c, the full-order equations of motion

V sin 7

(3.18)

cos c_cos/3 - 9_ sin 7
m

npA.r_

g_h(2r_ + h)r_ sin 3'

+_

cos a cos f_ +

mh,

hs(r, + h) 2

P SV2CDre 2mhs

(3.19)
] ]

25

T"V (cos o_ sin L¢sin # - sin ol cos #)

.QsCOS-7 V

rzp Ae re

-- e m--m--_.' (cos a sin ,g sin # - sin a cos #)

+ pSVr_.,,., e--t_Q 2ruh,

sin # + Ct, cos #)

+e -re+h

+ .qs V(r_+h)

cos _] g

Tvac

_

(cos c_ sin _ cos # + sin oesin #)

mV cos 7

npA_r_

-emvh,

cos I' (cos a sin/3 cos # + sin c_ sin #)

(3.20)

J r pSVr_.

Vr_. tan 0 cos "7cos X]

+e Lm-_z,T;-.y7(ocs,. sin _ - CQcos ;_) + h,(re + h)

(3.21)

V croes-CTOcoSs0 _(1 - _/_) Vc°s-TsinX(lre - e/@.)

(3.22) (3.23)

Where the binomial formula has been used to rewrite (r_+h)-l and latitude since re >> h.

for the longitude

3.6.1 Two-Dimensional

Flight

In this section the three-dimensional

equations of motion are reduced

for flight in a great-circle plane (the X-Z plane) over a flat, nonrotating Earth.

If the vehicle is assumed to be restricted to fly in the equatorial plane then

the lift, thrust, and velocity vectors all lie in the same plane and the roll angle

(# = 0) is eliminated from the equations.

Under the previously mentioned

assumptions of no side force (Q = 0) and no sideslip (_ = 0), the zeroth-order

equations of motion representing flight in a vacuum over a flat Earth become

h = Vsin'7

(3.24)

26

9 -- TVQC cos a - g_ sin 7 m

_ Tt, ac

gs

mV sin a - _- cos 7

V cos 7
-
re

rh = -aT,_ _ m = mo - aT,,,,c(7- - To)

X = Xo = 0.0
¢ = ¢0 = 0.0

(3.25) (3.26) (3.27) (3.28)

These are the system dynamics used to obtain an analytic solution

zeroth-order

optimization

problem presented in the next chapter.

to the

Zeroth-Order

Chapter

4

Optimization

Problem

The solution to the zeroth-order

a coordinate transformation.

A canonical

optimization transformation

problem is derived by from the wind axis

to the rectangular

or local horizon coordinate frame allows the zeroth-order

problem to be solved analytically.

The solution is in closed form up to some

constants that can be determincd numerically to solve the two-point boundary

value problem. The conditions for connecting the second stage subarc to the

first stage subarc are then prcsented.

4.1 Optimization

Problem

Statement

In this section the zeroth-order optimization The problem is to maximize the payload into orbit

problem is presented.

J = -rrt$
subject to terminal constraints on the altitude, velocity, and flight path angle,

h/ = hl,,,,o , Vf = Vfop,_, "),I ="tlo_,

subject to the state discontinuity in the mass at a interior point where staging

Occurs j

7_stage2

_ ?T_staqel -- /_sta9

e

27

28

and subject to the equations of motion for flight in tile equatorial plane.

h. = Vsin7

= --cosa-9_sin'_

-
O-
£n =

T

9_

mvSina-_c°s'7

V cos 7

re

-aT

_

rrl = trio -- aT(T

-- TO)

(4.1) (4.2) (4.3) (4.4) (4.5)

Note, in this section and when discussing the zeroth-order trajectory, the total vacuum thrust will be represented by T and the subscript notation will be dropped.

The Hamiltonian for this system can then be expressed as

H= AhVsinT+Av(Tcosa-g.,sinT)+
m

T

9s

A-r(_---_ sin a - K cosT)

(4.6)

The zeroth-order

control law determined by the optimality conditon is

T

H_, = -TAmr sina + m-V'%cosa = 0

(4.7)

By the strengthened

Legendre-Clebesch

condition H_,_ > 0 choose

x,

tanol

--

VAv

COS _ ----

VAv
+

sin a =

X'r

(4.8)

+

Whereas the optimal control can be derived in terms of the states and Lagrange

multipliers, an analytic solution is not possible for the states and Lagrange

29
multipliers written in the wind axis frame. Therefore, a coordinate transformation into the Cartesian reference frame is presented in the next section. In section 4.3 an analytic solution is obtained using this transformation.

4.2

Zeroth-Order

Coordinate

Transformation

The analytic solution for the zeroth-order

problem can be found in

the Cartesian coordinate system but the equations of motion of the full sys-

tem which include the aerodynamic

forces are written in the wind axis system.

Therefore, to derive the zeroth-order control and the first-order correction to

the control the transformation

of coordinates and especially the transformation

of the Lagrange multipliers must be known. This can be accomplished

by a

canonical transformation

[see appendix B] from the (0, ¢, h) coordinates to the

right-handed

coordinate system (X, ]i, Z), where X is positive in an eastward

direction along the equator, Z is positive pointing is orthogonal to the X - Z plane. The relationship

towards the Earth, and Y between the two reference

frames (see figure 4.2) is X = re0, Y = re¢, and Z = -h. In two-dimensions,

the corresponding

velocity coordinates (u,w) are considered positive in the pos-

itive X and Z directions, respectively. A necessary and sufficient condition [13]

for a canonical transformation

is the equivalence of the Hamiltonians

in the two

reference frames.

HLH = AxdX + AvdY + Ahdh + A,_du + A_,dw Hw_,_ = AodO + A,d¢ + Ahdh + AvdV + A._d'y

(4.9) (4.10)

30

___[__!d_ I
w

_./....f

T Body Axis

Axis Local Horizon

=X
Inertial Reference Frame

Figure 4.1: Transformation

of Coordinate Systems

31

This equivalence is obtained through the Jacobian of the transformation. fore, the transformation

u = V cosT,

w = -Vsin7

There-
(4.11)

requires

and thus, This produces

A._

- V sin 7 - V cos 7

]Aw

the transformation

of tile Lagrange multipliers,

Av = A_cosT-Awsin7 A-r = -V(A,,sinT+A_cosT) Ao = T_Ax Ae = reAy

and the transformation

of the states,

V = v/u 2 +w 2

1//

sin7 -

V

(4.12) (4.13) (4.14) (4.15)
(4.16) (4.17)

4.3 Zeroth-Order Frame

Analytic

Solution

in the Cartesian

In this section an analytic solution will be derived for the zeroth-order

problem of maximum payload into orbit for flight in a vacuum over a fiat Earth.

This solution is made possible by the coordinate transformation

presented in

32

the previous section. The equations are

of motion in a Cartesian coordinate

fraxne

.]( _- u
? = o_Y=Yo=O

h = -_
T = -- cOS0p
7?2

iJ = _b -

O_v=vo=O
T sin 0p + g_
Tt2

rh = -aT ==_ m =mo

- aT(T

-- TO)

(4.18)
(4.19)
(4.20)
(4.21) (4.22)

The Hamiltonian is

H = Axu - AhW + A,_T cos0p + A,_( -T sin0p + 9_)

m

m

The zeroth-order control law is determined by the optimality conditon

(4.23)

Hop -

T A,, sin 0p - T)% cos 0p = 0

m

m

(4.24)

Therefore, comes

using the strengthened

Legendrc-Clebesch

condition the control be-

tan0p

-

COS _p

sin0p =

A,, A_
+
A,,
+

(4.25)

33
The Lagrange multipliers are obtained using J_y
£x = 0 i_ = 0 ;(. = -,Xx _" = X_
with the boundary conditions

where _x, r'h, _., v_ are unknown Lagrange multipliers associated with the ter-

minal constraints. For the unconstrained

downrange problem, the solutions to

the adjoint differential equations are

-_x = tl X = 0 Ah = r'h A_ = v,,=C,, A_ = C,_ + kh(T-- T0)

(4.26) (4.27) (4.28)

The equations of motion can be integrated by changing the independent

vari-

able from time to mass and using the mass equation (Eq. (4.5)) to substitute

mass for 7-. As a consequence, the Lagrange multipliers are rewritten as

._,_ = C,, m
_ + _ = c__+ _ + _

(4.29) (4.30)
(4.a)

34

where

c- (aT7

b-

2 AhC_,
aT

a = C +VL

--

mo

c,,, = C,,,+ Ah-j-_

(4.32) (4.33) (4.34) (4.35)

The derivatives of the states with respect to mass are

du

C,,

-

dm

amx/cTr_ 2 +bm + a

dw _

A,_

9s

dm

am_/cTn 2 +bm + a aT

dX

u

-

_

dm

aT

dh

w

- --

dm

aT

(4.36) (4.37) (4.38) (4.39)

Note that c > 0, a > 0, and the discriminant of the quadratic

A _=4ac-b 2>0since

4 A- (aT) 2 (AhC_) 2

mass equation (4.40)

From these differential equations the solution is found from standard integrals.

u = Uo av/'a sinh -l \ m_v/_ ] - sinh -l \ too v/._ ]J
= ,_o- _T(m- too)

(4.41)

aC%-/-'_a [sinh-' {<2_ma%_/__+b_n]]--sinh-I {k2a_+-bnmoo-]-]_ ]J

h

gs (m- too)_ + (m- too)

ho 2(aT)2

aT

Wo

(4.42)

35

ma(af_,/- E sinh-l \ m_/_ _ ] _ sinh- , \ 7-r__v_

C_'

[sinh-' (2crn + b) - sinh-' (2_/_+

(_- _o)

X

No --

//,0

aT

C,,

\ / \ 7 o /j [sinh-'( 2a+bm]

- sinh-' (2a+bm0'_]

(4.43)

a(crT)v/-C

\_

sinh-I

The equation for the altitude common terms.

can be manipulated

further to eliminate some

h
ho 2(efT) 2

(m - too)

-k

WO

aT

-mG(_r)2v_ sinh-1

a(#-_Vv_/-,d [Lsinh-' (\2_arav4/--_brn'_) sinh-I (\2a+7-bnmoov_ )]

G(_AT, )2c [_/Cm2o+bmo+a_x/cm2+bm+a

]

At the final time, H I = -1 by tile transversality

condition. Using the tlamil-

tonian and the three state equations u,w, and h, which have prescribed initial

and final values, the four unknown constants associated with the two-point

boundary value problem can be solved. For the problem of flight restricted to

a plane, the unknowns are mj,, C_,, C_,, and Ah. The analytic state equations (Eq. (4.41)-(4.43)) are nonlinear and thus no statement can be made about the

36

existence solutions

or uniqueness of the set of constants found. Therefore, if multiple

are found tile solution set which minimizes the Itamiltonian

would

be chosen. At the very least, the Legendre-Clebesch weak relative minimum must be satisfied.

condition, H,,,, _> 0, for a

4.4 Linking

the First and Second

Stage Subarcs

Of interest in this section is the linking of the two subarcs of the two-stage rocket. By the corner conditions, the Lagrange multipliers for all the states must be continuous.

(4.44)

The analytic solution previously presented is still valid for either subarc but

only by using this relationship between the Lagrangc multipliers can the sec-

ond stage be connected to the first stage subarc. Recall that the constant C_,

is associated with the initial condition of the Lagq'ange multiplier for the ver-

tical velocity component.

For a subarc with first stage initial conditions, the

equations become

A_,(t) = ),,o(t,t_g_)+ ,\a(t - t._t_g_) t _>t_t_v_+

(4.46)

Rewriting the Lagrange multipliers using the corner condition

replacing time as the independent

variable, results in

and with mass

Ah = r,h = constant

to < t < tf

A,, = u,, = C,, = constant

to<t <t I

(4.47) (4.48)

37

/\h (TYl0-
Am = C,_+c_7,--- _

fit)

tO < t < tstag e

Ah

Ah

x,o(t+) = c,o+ gT[(.,o - Tr_,.,) + ;-_ (,_,_,_,_- -0

(4.4!))
(.t.5o)

where Tl and T2 represent tile thrust tbr tile first and second stages, respectively. The equations of motion, written with rn_s as tile independent variable, which were previously presented are still valid but the constant coefficients of the quadratic equation are of a different form.

(4.51) (4.52)

b!

(4.53)

at

(4.54)

Cw

(4.55)

Therefore,

the state equations become

du
dm dw dm dX dm dh
dm

crrnCdm
amV'dm u
_ T2 w a 7"2

C_
'2 + b'm + a' £, 2 + b'm + a'

g.q
o-T2

The same standard integrals apply to the solut, ion of the problem a' > O, c' > 0 and the discriminant

because

A'=4a'c'-b

a=4\aT2]

C_>0.

(4.56)

38 The simplified form of the solution to the state equations (Eqs. (4.41)-(4.43)) is also still valid but with the first stage subarc used as the initial conditions of the second stage subarc.

u

uo-_---_

sinh-_

C_, Is

L _sinh -_ {2a' + b'ms_g_2_ 1

(4._7)

0,11

/2a' + b'rnstaqe2

-_2T2x/'d

k_---_)-

sin},- k-- _

= _ m2o) 9,(m _ -- = TrZsta_e2 )

h

ho +

2(_T,)2

2(_T_) 2

T/%W

m0"UJ0

(4.58)

e_ [sinh-' /2a+brn.,t,_v_,'_

{2a+bm°_l

Ah

x/Urn = + b'm + a' - _/d _t,_'2 + [tm,L_,v_2 + a'

-_ a(_T=)2d

-

rLstagc i

•

+-_(o:q)_c

39
These are the equations that result from linking the first stage subarc to the second stage subarc. These equations will be used to evaluate the states at a time after staging occurs when the initial time is before staging. The first-order correction terms will require the analytic solution for the states at any future time along the zeroth-order trajectory.

Chapter

5

First-Order

Corrections

The use of the asymptotic expansion of the dynamic programming

equation as discussed in Chapter 2 by the approximate optimal guidance scheme

is an improvement over past analytic techniques whose guidance laws were lim-

ited to operate in tile exoatmospheric

region [6, 14]. The higher-order correc-

tion terms of the HJB expansion can bc used to compensate for tile effects of

the atmospheric

forces neglected in tile exoatmospheric

mination of the first-order correction to tile zeroth-order

solution. The detercontrol is the subject

of this chapter. As noted before, tile solution to the first-order optimization

problem requires only the integration of quadratures,

which can be evaluated

quickly enough to permit this method to be implemented as a real-time guid-

ance scheme. The correction to the I,agrange multipliers and thus tile cor-

rection to the control is constructed

in the following sections. Also derived

are all the partial derivatives needed to evaluate the quadratures.

The partial

derivative chain rule is employed since the analytic solution is found in the

Cartesian frame while the first-order forcing function, Rl, used to evaluate tile

quadratures is expressed in the wind axis frame. Recall that the angle-of-attack is the control variable and tile aerodynarnic coefficients are modelled as func-

tions of the angle-of-attack.

For this reason tile perturbation

dynamics are left

expressed in the wind axes frame.

4o

41

5.1 Correction

to the Lagrange

Multipliers

The higher-order
in Eq. (2.26).

terms of the optimal return fimction were presented P, (:r, t) = - _tl ROdr

By taking the partial derivative of this integTal the correction Lagrange multiplier can be caleulatcd. Recall,

term to the

& 05 _
- Oz

ft "

OR<JT+
Oz

R_ [,O8tzz

-

R,I
_i

OOtz:

(2.39)

where the first-order forcing flmction was ]?,l = -Po,go.

The first-order correction term for tile Lagrange multipliers is used to determine the first-order expansion term of the control. By the first-order optimality condition, Eq. (2.30), the correction to the control is obtained.

u, = - (L_&.)-' [1_,.w+ Pl.L]

(5.1)

5.2 The First-Order

Forcing

Function

For the launch problcrn as formulated first-order forcing function is

in the wind axis frame, the

r__ _' D

r_.(2r_, + It)

npA,..

+ ( + g,

It) ) cos 3'

V rn

r

(r_ + h) 2

sin a] m

(5.2)

The Lagrange multiplier for the first-order term of the expansion series is found by integrating the partial derivative of R_ with respect to the initial state. For the launch problem, the optimal control depends on the Lagrange multipliers

42

for the velocity and flight path angle, i.e., x = [Vo, %]. Tile partial derivative of the first-order forcing function with respect to the initial state is

ORl

_x

Ox \ 2mh_ (AvCD-

pSV2re Av

+

-

CL

-t 2mh,

Ox

V Ox _

-g,

+ h) 2 Av cos _' - -_- sin'7 O-x

-g" h--2;+-oh)2 [ 0x sin_ + _

cos

O0x( g_ hh(.,(2r_r+oh+) h)r2o) (Ay sin _ + _--_- cos 7)

-t mh_ [-O-xz cos c_ + _

sin a

mh,hp

sin a + Au cos a _x

where

h,(r_ + h) \ Ox cos'y - X_ sin 7_ x
o0_(\h_(K;h) ) (a, cos_)

0 (pSV2r_ Ox k, 2mh_ ]

pSV2r_ 2mh_

20V V Ox

2g_r a Oh
h,(r_ + h) 30x

O°x( \ v-5 h,(re + h) )

OCD (M, a) •

Oz

-

OCt.(M, a)

Ox

--

h_(rV_r_+ h) [vOVOx

OCD OM

C_CD (90/.

OM Ox + O_ Oz

OCL OM OCL Oc_ OM Ox + Oa Ox

(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)

43

Tile partials of the wind axis states and Lagrange multipliers are related to

the partials of the analytic Cartesian states and Lagrange multipliers by the

canonical coordinate transformation.

These partial derivatives are presented

in subsequent sections.

5.3

Relating

the Partial

Derivatives

Frame

to the Partial

Derivatives

Frame

of the Wind

Axis

of the Cartesian

The canonical transformation

of section 4.2 provides all the infor-

rnation needed to relate the analytic solution of the zeroth-order

states and

Lagrange multipliers to the states and Lagrange multipliers in the wind axis

frame. Thus, the variations in the analytic Cartesian coordinates due to varia-

tions in the initial wind axis states can be determined and it was for this very

reason the canonical transformation

was necessary.

Using the relationships

obtained become

in section 4.2, the partial

derivatives

of the wind axis coordinates

OV Oz
O7 Oz OAv Ox OA_ Ox

V

Ou

&l"_

tan 3`

V 2 Ox V 2-_z

0A,,

0A_,

03'

Ox cos 3' - 0--7- sin 7 - (A,, sin 3' + A_, cos 3`) _zz

(0 o V \-0-_x sin 3` + --0-_--zcos 3` + [A_,cos3` - A_, sin 7] _z
OV Ox (A,_sin7 + A,o cos3`)

(5.10) (5.11 )
(5.12)

44 and from the zeroth-order control law Eq. (4.8)

OOax-C°sasina

( A1_0XOvx

A1v0AVOx

V10VO)x

(5.13)

Now that the partial derivatives for the wind axis coordinates are expressed

in tcrms of the partial derivatives of the Cartesian coordinates,

the partial

derivatives of the Cartesian coordinates with respect to the initial states are to

be derived along the analytic zcroth-order trajectory.

5.4

Partial

Derivatives

of the Analytic

Solution

In this section, the partial derivatives of the Cartesian coordinates arc

derived. The zeroth-order analytic trajectory is used to evaluate the integral of

the partial of the forcing fimction/_q from the initial time to the final time. For

the sake of notational brevity, derivatives are defined.

the following

common

terms and their partial

5.4.1 Partial Derivatives

of Some Common

Terms

The partial derivatives of the constants a,b,c, and C,. used to express the analytic state equations are

(5.14)

8c Ox 8Cw
Ox

2At, OAh

(_T) 2 az

OCw mo 8,\h

+

Oz

aT 8x

Recall that the function A = 4ac- b2, so tile partial derivative is

45
(5.16) (5.17)

OoA-7_- 4a_x + 4cO_a _ o_b Ob

Let the arguments of the inverse hyperbolic sine function be denoted

2crn + b 9,(m) -- _

2a +bm 92(m) - mv/_

Thus the partial derivatives of the arguments arc

(5.18) (5.19)

1 [2(._ a_g)t._ Oc + (1 + bv(_31'_&Ob - ,_.._cl_j Oa]

(5.20)

GgX

1 [ m4/- _ 2(1-m_ c9_)o-_x_+m(1 + bv"g-A,'+_-_oxb max,/9--A_O0zJc](5.21)

and by the partial derivative chain nile for a trignometric of the inverse hyperbolic sine functions are

function, the partials

Ox0 (sinh_tgt)

= _+1_lO0__,

o_0 (sin},-' 9_) = _ +1 _,_0O_2 x

(5.22)
(5.23)

5.4.2 Partial Derivatives

of the Analytic

States

The general form of the state equations in Eqs. (4.41)-(4.43) is used to derive the partial derivatives of the states with respect to the initial velocity or flight path angle. Using the terms defined in the previous section and

46

simplifying the equations, the partial derivatives arc

Ou

OX

DUOOx C_v_l [sinh- (._2(rrt) - sinh- _2(mo)] \ _

2a

c_ [_/ 1 o_(._)_

_1 o_(._O)]ox (5.24)

OW

Or,

, (o-c -C Oa) Owoox O'V_I [sinh- _2(rn) - sinh-' _2(rno)] _ Ox

5 _zz

_,_ v/_+ _(_) o:_ v/_+ _(._o) _ j

1 []-sinh-l_l(rrt)-sinh-Ic_'(rrt°)-\-_z

{ Oah a2c,,_xOc)

Oh

Owo(m - rno)

Ox

Ox

aT

(_.25)

_m

(c_TA)_h V'_ [k/

1

O_,O(mx) _

l

0_, O(xmo) ]

I

aTv/- d [sinh-' _2(rn) - sinh- _2(rr_)] \ Ox

C2.a O-_az)

-IC_,
-m_

1

O_(m) _

1

Oc_2(mo)

v/1+ -_(_) o_ vh + _(mo) Ox

+m2a(aT)2C3A/h_

Oc [sinh-' _ (m) - sinh-' _,(mo)] Oz

_(_Tr)n_ z'-i [sinh-' _,(rn) - sinh-' _,(mo)] OOAxa
[ 1( +_(_r)_c _/_ + V._+ _- V/_g + _._o+ a t, O_

X_ Oc )

_ 2a(aT)_c

_

,-5¥+mo_57 + --

[ m'_22oc +m--_Ob + a

rrv_/cOrnc_o + bmOob + ___]

(5.26)

47

The initial velocity components expressed in terms of the wind axis states are

Uo = t{) cos %,

u'o = - Vo sin 70

(_.27)

and therefore the partial derivatives with respect to the initial velocity and

flight path angle are

Gq?20
OVo -
0710
07o
D?IJ o
-
:)to
(_ llJ 0
07o

cos % -Vo sin 70
sin % - Vo cos 70

(5.28)

These partial derivatives are valid for a point during the first or second

stage of the trajectory with initial condition corresponding to that subarc. For a

point on the second subarc with first stage initial conditions, the state equations

which link the two subarcs must be used. Note also that these equations all

depend on the partial derivatives of the constants, At,,C_,, C,,, and m I which

are unknown. Tile partial derivatives of the constants are dependent on the

initial and final conditions of tile two-point boundary value problem. Using

the transversality

condition

H z = -Mwf+c_T2 --cos01 mf

+ A,_(tI)(-T2

sinOf + g,) = -1

mf

the partial derivative of tile Ilamiltonian at the final time is

(5.29)

OO_ z = 0 =-wf-_z 0/_ h + g, (\(_COxw

(Tnf c:-T- 2Trio) Oc_3zh

c_:hT2 OT0_zf)
I

+

T2 (bmf + a)

oqmI

2m}V/.m+ : +

T2 | :/:k_m/_+--

["7_2 0c --

Ob

(5.30)
_]

48
These results produce a system of four equations (o____oo==, -_-o,=o_____oli=ne:ar
in the four unknown partial derivatives: 0_h _ 0C__%and 0-2/- The partial derivatives of the four constants are determined by the solution of this linear system.

5.4.3

Solution to the Linear System of Unknown

Partials

For the second stage subarc, the solution to the linear system of four

unknown partial derivatives in the partial derivatives of the four transcendental

equations is determined by the matrix equation

0

_ owo ('hi-'no)

Ox

aT

OuQ_ Oz.

Owo Oz

The eoeffiecients of the matrix

O ,kh
O),h
O_h
ow
O),h
are

oA_z o__ o_qz
OC. OC_ Orn y
°__b_L°___ °_b.Z_
OC,, OC,,, OrnI

Ox
oc__G,
O:r

OCt, OC,,, Ore/

O:r

OC,, OC,, Orn I

O:r

(5.31)

OH:
OAh
OH: OC,,

(m: - too) (ms - _)[c_ + _(_o - "_s)]

w/- 9_ aT

+

crrn/_/c'm2/ + brn I + a

C_T

(5.32)
(5.33)

OH:

[C,_ + _-_T(rno - rn/)]T

OC,,, - g"-

+ bin:+

OH:
- Orni
OUf
Oz _

T
av_l [sinh_ 1 _2(rn/)

Ah ),h[C+,_.(rn.-om:)l

o'rn I _/crn} + bin: + a

(oco

- sinh -I _2(rno)] \ Ox

C2a,, -O_az)

(5.34)
(5.35)

C, [

1

O_2(mi)

ov_ [_/1 + _(m:) 0_

1

09_(mo)

_/1 + a_(mo)

(5.36)

49

OWl

I [sinh -t _2(ml) - sin},-' _2(rno)] ( OOUx_,

-)C_, Oa
2a -Ox

-[C,_ .

1

0_2 (rrzf)

_,/-a v/, + __-_(_s) &

1
v/: + _(,_o) O_(mO)]O:r.

a=T1v _ [sinh-' a,(my) - sinh-' _,(mo)]

0OAxh

a2Tx/_

1 -+- ..%:_(mf) 0x

J1 -+- _(rr_)

Ohl

.x,,[ , o.%(_s)

, o_,(_o)]

Ox

o'Tv_

( -) C,. Oa Ox 2a

-Cw [ +mf 2a(crT)2ACha/2

I

Oc.22(ml)

I

[sinh-' _,(m/-) - sinh -I _,(mo)]

oq_2(mo)]
Oc _xx

<_(:T)2v/_-

_, (m/)- sinh-'

Ox

-_

_} + v_ + _ - v/_o_+ b_ + a

+

Ah

2a(aT)2c

F 20c Iraf-_x

--

Ob

Oa

_20c

-t- rrlf ;--_z + _ _ moT_z

Ob

Oa

+ moT_z + _

(5.38)

where the equations _ ' _Oa: ' and °Ohxl are the samc a.sderived forthe analytic

state partials but are derived with respect to the constant parameters,

i.e.

x = {Ah, C,,, C_,}. All these terms thus depend on the partial derivatives of

the common terms a, b, c with respect to the constant parameters.

So,

m
OCt, mo
o_/, = 77

--
OC,,,
oc_ - :

m
0C_,
oc_ -0

OOFa ,, - 2C_, + _-c-_-O£UC,_7

5O

0a
0C_
Oa
m
O_n
Ob _=0
oC_ Ob
O_h Oc
OAh

OC_

Ob

2 OC_

aT '_
2A_ Oc -0
(aT) 2 OC_

_=O0c
OC_

Remember that the variation of the terms with respect to the final

mass is also needed. For the arguments of the inverse hyperbolic sine functions,

the partial derivatives with respect to the final mass become

09,(ml)

- 2c

ore: - -_'

092(mi)
Ore:

2a
,_v}_

The partial derivatives of the analytic states with respect to the final mass are

Ou:

C,,

1

0_32(m:)

Ows
Om S
Oh/ Orn I

g, -C,. 1

092(ms)

aT

av/'d-_/1 + 9_(ms)

/)ms

Ah

1

O_l(rns)

a2Tv/-C _(m/)

Ore!

gs

WO

(_)= (ml- too)+

Xa

I

091(mI)

51

Ah
4

(2cvnI + b)

All these relationships

are used to determine the coefficient terms of

the algebraic set of equations.

The variations in the constant parameters of

the zeroth-order two-point boundary value problem with respect to variations

in the initial states can subscqucntly

be determined.

These variations are

embedded in the quadraturcs

used to calculate the first-order correction to

the Lag-range multipliers and determine }low a change in the initial conditions

changes the path while flying along a path which will satisfy the terminal

boundary conditions.

For the situation where the vehicle has not yet staged, the partial derivatives are similar to those shown above but the equations of section 4.4 which link the two subarcs of the trajectory are used.

Aerodynamic

Chapter

6

Effect along the Zeroth-Order

Trajectory

Previously the problem of minimizing the fuel into orbit for the flight

of a rocket in a vacuum over a fiat nonrotating

Earth was the zeroth-order

problem, i.e., e = 0. It was found that this zeroth-order trajectory deviated

significantly from the optimal trajectory and the resulting correction terms were

not small as was assumed in deriving the expansion method. To compensate for

this problem the zeroth-order trajectory needs to be reshaped in order to keep

the assumed perturbing effects small. One method that might work is to include

a constraint on the control which will limit the zeroth-order angle-of-attack

and

thus the aerodynamics

generated along the zcroth-order path. The problem in

implementing

such a constraint is that the zeroth-order solution must still be

analytic. Since the analytic solution was found in the local horizon coordinate

system the control was the pitch angle. From the standpoint of the physics

of the problem, there is no logical constraint which can be imposed on the

pitch angle. Limiting the angle-of-attack

would create a mixed constraint in

the local horizon coordinate frame involving the state and the control and this

type of constraint is difficult to solve. A practical and necessary constraint for

launching a rocket is a dynamic pressure limit. How such a constraint may be

incorporated

theoretically

in the HJB-PDE expansion technique is presented

in appendix[C].

But a dynamic pressure constraint arc also does not allow

an analytic solution to the zeroth-order problem. Therefore, the zeroth-order

52

53

trajectory was modulated by including aerodynamic

terms in the zeroth-order

problem formulation. This process involved averaging the aerodynamics

along

the vacuum trajectory and solving anew the zeroth-order

two-point boundary

value problem. This technique was suggested by the successive approximation

method used in [15]. By modelling tile aerodynamics

as constant terms, closed

form solutions are still available. This chapter presents the details of includ-

ing aerodynamic

pulse functions

coordinate systems.

averaged

in the local horizon and body axes

6.1 Inclusion

of an Aerodynamic

Order Problem

Effect in the Zeroth-

Instead of assuming tlight in a vacuum, tile zeroth-order problem is now formulated to include aerodynamic terms. Then if e = 0 the equations of motion for the zeroth-ordcr problem, valid over both subarcs, become

h = V sin "7

_, ":'/ --

T cosa - g, siny + --

rr/

77z

T

g,

Z2

rnV sin (_ - _ cos "_ rnV

V cos 7

7"e

rn = -aT_rn=m0-aT(r-r0)

(6.1)

where

Z) = (A °cosy-A

°siny)

Z: = (A°siny + A °cosy)

(6.2)

54

are the assumed lift and drag forces along the zeroth-order

trajectory.

The

constant terms A °, A ° are the averaged aerodynamic

forces in the x- and

z-directions.

For a vacuum zeroth-order trajectory these terms would be iden-

tically zero. Nonzero values will be used in order to improve the zeroth-order

trajectory and keep the perturbation

effect due to the neglected aerodynamics

relatively small compared to tile effects due to thrust and gravity. Since these

terms are added to the zeroth-ordcr dynamics, identical terms of opposite sign

are included in the perturbation

dynamics. Thus their effect is identically zero

in the full-order system of equations.

The variational Hamiltonian is altered by tile inclusion of these terms,
e.g.

H = -AhV sin7 + Av(T-- c.os cz - g_ sin 7 + _/9_)

m

m

)_, (T +--_- mSina-

/2 .q.,sin'y - m)

(6.3)

Notice since the pulse functions used in the aerodynamic terms are constants,

the zeroth-order

control law determined by the optimality condition is not

changed from the solution obtained for vacuum ttight.

tan c, -

(6.4)

VAv

Once again the analytic solution to the zeroth-order found in the Cartesian coordinate system.

problem will be

55

lift-direction

axialbodya.xes-direction

Lift Force C

Thrust

A-aForce

normal body a.xes-direction

Velocity wind axis-direction

x-direction

Drag Force D

A-z Force

z-direction

Figure 6.1: Coordinate frames for the aerodynamic

pulse functions

56 6.1.1

Zeroth-Order

Aerodynamic

dinate System

Effect in the Rectangular

Coor-

The equations of motion in a Cartesian coordinate frame become

h

_

=

_i -

mw

T

A0

--- sin 0), + gs + --

m

771

r

A°

cos 0p + --

m

m

(6.5)

where the control variable for this problem becomes the pitch attitude 0p = + 3`. The terms A ° and A ° represent the constant assumed aerodynamic
forces along the zeroth-order trajectory in the x- and z-directions, respectively.

Ao - ' /"+'A.e,- _ /"+'

ti -- ti+l

Jti

ti -- ti+l

Jr(

Ao _ _ /"+'A,d-,= _ f"+'

(6.6)

ti -- ti+l J_

ti - ti+l Jt_

Figures (6.2-6.3) show the aerodynamics intervals or subarcs.

averaged over a different number of

The zeroth-order

Hamiltonian is

H = - Ahw + A,,,(- -T-sin0p+gs+,4°=)+A,_( 0

T cos0p+m) A °

(6.7)

m

m

where Ah, A_, and A_ are Lagrange multipliers. These Lagrange multipliers are

propagated by the Euler-Lagrange

differential equation Ay = -H T. Thus

L, = o, ;\,,=0, _,_= _

(6.8)

with boundary conditions

Ah(rt) = uh, A.(rl) = u,.,, A_(rl)= u_

(6.9)

57

Fit of Aerodynamic

I

i

Forces in X-direction

[

I

[

I

5"0 lOS I

0.0 I0 ° _--

/

-S.O IO s --

I
-1.0 I0 G --

-1.5 10 6 --
-2.0 10 6 -t3

-- multiple pulses -- -single pulse - - -force in x-direction

-2.5 10 6 r

i

I

1

I

_

[

25 75 125 175 225 275 325 375

time

Figure 6.2: Model for aerodynamic

pulses in x-direction

Fit of Aerodynamic Forces in Z-direction

1.0 1 0 6

I

_

[

_

I

I

[

--multiple

pulses

]
J

5.0 10 s _

-- -single pulse

L
- - -force in z-direction

0.0 I0 °-
5
-5.0 10

/'

-1.0 10 G

-1.5 Figure

10 6

I

25 75

6.3: Model

I

I

I

125 175 225

time for aerodynamic

i

t

275 325 375

pulses in z-direction

58
where vh, v_,, and u,_ are unknown Lagrange multipliers associated withthe terminal constraints. Since the aerodynamic effect is added as a constant term there is no change in the solution to the Lagrange multipliers or to the control from the solution found for a vacuum zeroth-order trajectory. Therefore, the zeroth-order analytic state equations become

u = Uo-_ln

C,, [sinh_, ( 2a, + b,m _

( 2a, + birno

w = wo- g (m-m_oT) ,

aAOT,`. In (m-_0)m

h= +

C-"_, [sinh_, (2ai + b,m'_

f 2a, + b, mo

Ah [

• , -i,2cim+bi)

a2T_x/_[smn

t _4,_

_ sinh-l(2ci_+bi)]

(m- .%) (m- too)_

ho + Wo

gs

(aT,)

2(GT,)2

Am a(aT,)2c i [(cim 2 + bim + ai) '/2 - (airn_ + bimo + a,) '/2]

a(aCr,%)v_,m [Lsinh_\l rnJ-N, )-sinh-' (\ 2a7n,+o_bm, 0]]3]

sinh -I

_

1( 2cimo +

) sinh-

_

)

where

(aT,)2 m In

- m + mo

(6.10)

(6.11)

and the subscript i refers to the current subarc. More pulse functions could be

59
usedto model the aerodynamicsin an attempt to capture the effect of the aerodynamicsin the closedform solution and thus the path would be brokenup into smaller subarcs. Note that becausethe assumedaerodynamicsare only constant terms their effect is an accumulativeone. The zeroth-order trajectory is altered sincethe boundary conditions can not be satisfied flying the samepath as the path flown in a vacuum. The vehicle doesnot modified its orientation instantaneouslyin order to reducethe aerodynamicsthat it will encounter, i.e. the vehicle cannot predict the aerodynamic effect on the vehicle by its choice of angle-of-attack. Thus any changeis in the total energy of the system and the vehicle is not penalized for flying at large angles-of-attackand for incurring large drag forces. This can be seenin the new open loop zeroth-order trajectory in that the vehicle initially pitches over more than in the vacuum solution. Bui: over the entire courseof the trajectory the vehicle remains at lower angles-of-attackand doesnot lift up as much in the secondstage. If more pulsesare addedthe aerodynamicsbecomelarger over certain intervals and the vehiclereacts accordingly to theseregionsof large aerodynamicforces.

6.1.2 FIRST-ORDER CORRECTION TERMS The correction terms to the zeroth-order problem can be calculated
by the quadratures representedin (2.39). Therefore, for the launch problem

Ri = hr-__{ Av [D+Dm

9, h((r2er_++h)h) _ sin3,+_cosanpmAe

A_ L+/2

V

rn

V2

h(2r_+h),

+ ((re + h) + 9s (-_ _

)cos'),

(6.12)

npAe

]1

Jf m sin(_

The first-order term of the optimal return function evaluated along the zeroth-

order trajectory with initial conditions before staging is written as in (2.26),

60

but separatedinto two integrals. Only the velocity and flight path angle state equationscontain the control. Thus, the first-order terms in the expansionof the Lagrangemultipliers associatedwith the velocity and flight path angleare the only co-stateexpansionterms neededto construct the first-order correction to the zeroth-order control. Tile partials of P_ with respect to the arbitrary

current conditions, x = (V0, 70), become

Pl_ _ oP, - r - / "'°'_

Ox

.,t

d,
Oz

,,,°_,_ Oz

Ox

Because aerodynamic pulses were added to the zeroth-order dynamics

the opposite terms are added to the perturbation

dynamics such that the over-

all system equations are unaltered. If the zeroth-order trajectory is the vacuum

trajectory then the assumed aerodynamic terms (7?,/2) are zero. For nonzero

assumed aerodynamic forces the new perturbing aerodynamic effect is the dif-

ference between the actual drag and the assumed drag along the zeroth-order

path. It is necessary to keep this new perturbing aerodynamic effect small in

order to accurately approximate the optimal solution. That is the entire reason

for the inclusion of the aerodynamic pulse functions. The next sections present

the results for various assumed aerodynamic pulses.

6.2 Results for the Rectangular

Pulse Functions

It was found that the more pulses used the closer the first-order cor-

rected solution came to the first-order solution obtained using a vacuum zeroth-

order trajectory. The best solution for the approximated

control was obtained

61
by using one pulse per stage. This seemedto keep the perturbing aerodynamic effect small over a larger span of the trajectory. The convergenceof the Lagrangemultipliers up to a first-order approximation using the onepulse aerodynamic functions for the zeroth-order problem is demonstrated by the plots presentedin the Resultschapter. Iteration of the zeroth-order trajectory for the assumedpulse functions wasattempted but it was found that the firstorder correction terms alternated back and forth betweenthe optimal values and the solution basedupon the vacuum zeroth-order path. This was a consequenceof the assumedaerodynamicsswitching betweenlarge and small values on successiveiterations. If large forces were assumedon a particular iteration than the actual aerodynamic forcesalong the new zeroth-order trajectory would becomesmall and thus on the next iteration the assumedaerodynamic pulseswould revert to smaller valuesand thereforethe first-order correctionsresembledthe solutions obtained using a vacuumzeroth-order path. Attempts to averagethe iterations alsoprovedunsatisfactory. For multiple pulsesper stage, the averagediterations did not adequatebring the assumedaerodynamic pulse functions closer to the actual forces along the new zeroth-order path. For a one pulse per stage solution the iterations could not improve on the solution obtained from the first iteration and thus were not worth the computational time and effort. In general,assumingmore than onepulse per stage and more than one iteration causedthe first-order corrections to go towards the values obtained assumingno aerodynamicforcesalong the zeroth-order trajectory. In a final attempt to lift the vehicleup and keepthe vehicle from trying to pitch over, aerodynamicpulse functions weremodelled as constantsin the body-axes frame. The next sectionbriefly describesthat effort and the results.

Aero Pulses in the Body-Axes

Frame

Because the use of aerodynamic pulses modelled as constant terms in

the local horizon coordinate system the vehicle did not respond in an instan-

taneous fashion to the aerodynamics

it encountered along a particular flight

path. To remedy this situation tile aerodynamic pulses were modelled as constant terms in the body-axes frame. Thus there are aerodynamic components tangent to and normal to the thrust. Rotation of these forces into the local

horizon coordinate frame still allows an analytic solution to the zeroth-order

problem but now the control law becomes a function of the aerodynamic ef-

fect assumed during a particular interval. This was not the case in using the

aerodynamic pulses in the local horizon system as presented in the previous section. Because of the reliance of the zeroth-order control upon the aerody-

namic pulses used, the control becomes discontinuous

along the zeroth-order

trajectory. Since the aerodynamic intervals are chosen as functions of a fixed

time interval the Hamiltonian is also discontinuous across these intervals. The

integrand used to derive the first-order correction to the Optimal Return Pune-

tion and to the Lagrange multipliers is thus discontinuous

and the integration

of these terms along the zeroth-order path must be broken up according to the

aerodynamic intervals. The equations of motion in rectangular coordinates for body-axes aerodynamic pulses are

h_
@/L -

--W

T

A°

A°

sin0p+g_---sin0p-mcOS0p

m

m

m

T cos 0p + -A-_ cos 0p - -A-_ sin 0p

m

m

m

(6.14)

63

Tile terms A_ and A,°v represent tile constant assumed aerodynamic

forces

along the zeroth-order trajectory in the axial and normal body-axes directions,

respectively. Tile zeroth-order variational Hamiltonian is

H = -.Xhw+)_,o( -T- + A°a sin@+9,
m

-A°a- cos @) +,_,_ ( r + A° cos Op-'A°xN sin 0p)

772

772

gr_

(6.15)

The solution for the Lagrange multipliers does not change from the solution to the vacuum zeroth-order problem and the multipliers are continuous across subare times, as are the states, since these times are considered fixed. The

first-order optimality condition produces the following result.

A_,(T + A °) + A,,A ° tan@ = AwAO _ A,,(T + A_)

(6.16)

Using this new control relationship in the state equations the closed form solution can still be obtained and the states are written as

_ ZL0
aTC-7,,-¢---v/-a[L-s7ink-' /\'-2_ai + birn )-sink-'

(\2ai mo+v_ birno )]

w = w0-g,

c_7'/

arCiV_,/,-Td7 [sinh_l {lv2ai_-rt_+/_b, rn ) -sink-I

(I2Va,_-no+v/_b, rno )]

(aT-75g-v/__ sink-'[

_

) - sink-'(

_

)

(m- too)

- too)

+ <,(Agh-2r?-j)3[q(c_rn 2 + b, rn + ai) 112 - (cirn2o + bimo + ai) 112]

e_'iern

[sink-' (2ai +birn)

(aT_--_-_v/'fi]"[

IV m_"-_ ]-sink-'

(2ai +birno'_]

IV 7r_/'_-

]J

64

(aTAzh--T-_rn/-_ [ sinh-1 (,2cam_ +bi )-sinh-

1( 2cam_0+bi ) ]

where

ca

= c_aTi J _

bi =-2_e_,•,

2
ai =c_ +i2_,

-C,,, -- c,,, + a ,,_-_,, -C_,_= c,,, + ,xh(''° -'_'', ''_ + ),h _

Ai

= 4aica - b2 = 4 \ aTi ,,I

i _- 1,2

(6.17)
(6.18)

and the effective thrust ¢ = _/(T + A°) 2 + (A°) 2 is the magnitude of the sum of the thrust and assumed aerodynamic forces. A typical open loop zerothorder trajectory is shown in figalre (6.4). While the initial pitch over action was curtailed compared to tile previous results, the trajectory still deviated from the optimal trajectory sharply especially in the regions of high dynamic pressure.

equation

Corrections

to the Lagrange Multipliers are made by the familiar

_

Ot/

P'"

--

OPlox

i--_l JfQ+tQ 0_1 pt()yO_x

dT -- ]_l (y_pt (if))

OX

(6.19)

for n aerodynamic intervals and where

R1 = hr_-_ Av

m

g, (re+h) 2 sin "y + -- m cos a

(6.20)

-

--+((r__h)

+gs

)cosy

m sina

The assumed drag and lift terms are the transformation

of the body-axes

dynamic forces into the wind axes coordinate system, that is,

aero-

79b = (A_cosa-A_vsin@ £b = (A°sina+A°cos@

(6.21)

6S

Tile correction terms to tile LagTange multipliers based upon the

zeroth-order

trajectory using body-axes aerodynamic

pulses did not give any

improvement

over tile use of local horizon aerodynamic

pulses. If anything

the solutions obtained were worse since the trajectory was strongly influenced

(as were the pulse functions) by the regions of high dynamic pressure and thus

the perturbation

aerodynamic effect remained large. Tile results from iterating

with the averaged aerodynamic pulses and from averaging the iterations of the

averaged pulses exhibited tile same pattern as the local horizon case. Thus

one pulse averaged over the first stage came closest to producing agreement

with the optimal solution. The one positive effect of the body-axes approach

when used in feedback to generate a trajectory was the elimination of the dis-

continuities in the control previously found when minimizing the ltamiltonian

using the first stage aerodynamic model. Unfortunately,

the path generated

did not match as closely the optimal path as the results using the second stage

aerodynamic model matched.

66

Multi-Subarc 60

Body-Axes Pulse Functions

50 flight path angle

- angle-of-attack

(z

40

--_-- pitch angle 0

30
bD

20

hO

<

I0

,

I"

LI

•

*"

time
Figure 6.4: Open loop zeroth-order path for body-axes aerodynamic pulses

Chapter

7

Results

In this chapter the approximate

optimal solution is compared to an

optimal solution for the launch of a vchicle in the equatorial plane. While

previous results for flight in the exoatmospheric

regions [16] showed excellent

matching of the approximate solution with tile optimal, problems arose during

the first stage. First, even at high altitudes where the aerodynamics are indeed

perturbing effects to the vacuum trajectory, it was found that the linear control

law derived for the first-order correction to the control (5.1) was in greater error

than the error in the first-order corrected Lagrange multipliers.

As a remedy

the control was calculated by minimizing the Hamiltonian of the entire system

using the Lagrange multipliers approximated

to first-order. This produced the

desired effect and the control profile converged to the solution obtained by the

shooting method.

The next difficulty encountered

was due to the first stage aerody-

namic model. This model seemed to produce an irregular Hamiltonian.

The

Hamiltonian

was badly behaved and exhibited discontinuities

in the control

at various points along the trajcctory.

The asymmetric configuration

for the

rocket and the cubic spline functions used to fit the aerodynamic

the Hamiltonian

to take on almost identical values for different

data caused values of the

angle-of-attack.

This can be seen in figure (7.1) which are plots of the Hamilto-

67

68
nian versusthe angle-of-attackat two consecutivepoints in the trajectory. Tile sequenceshowstile Hamiltonian exchangingthe location of the minimum betweenpositive and negativeangles-of-attack.Part of the problem can be seenif the drag model is shown for larger angles-of-attackthan waspresentedin chapter 3.3. Figure (7.2) showsthe drag coefficientfor different angles-of-attackand Mach numbers than would be encounteredalong the optimal trajectory. Remember the first-order correction terms are basedon the aerodynamicsalong the vacuum path but the aerodynamicsare not modelled adequately for these regions. The drag model of figure (7.2) showsthe peculiar nature of the aerodynamics that would be usedat the larger angles-of-attackof the zeroth-order trajectory. The smooth curve usedto model the secondstageaerodynamicswas substituted into the algorithm to eliminate this strange behavior and remove the discontinuities in the control. This would prove successful. Figure (7.3) comparesthe drag and lift forcesalong the first stageof the open loop vacuum trajectory using the first and tile secondstageaerodynamic models. Another advantageof using the secondstage aerodynamic model can be seenin that the drag has been reducedwhile the lift along the trajectory remains roughly the same.
Overcoming thesedifficulties still left a problem. The first-order correction exhibited a boundary layer type effect near the initial conditions. This would occur evenif the problem wasstarted at variouspoints in the first stage. When the approximation method was used in feedback,this effect would diminish during the trajectory and the solution would convergeto the optimal solution. In order to eliminate the initial over-correctionsof the first-order approximation, the zeroth-orderproblem wasreformulated to include an aero-

69

-5o

l

1

I

-55
o_ ©
a= -60

Time= 84.79

-65

-5

0

5

lO

15

Angle-of-Attack a (deg)

°_
©
.,_a
E

Angle-of-Attack

ot (deg)

Figure 7.1: Hamiltonian first stage

versus Angle-of-Attack

at continuous points of the

70

Alpha i0 20
N
6j 4
Cd 2,
0 -2
Mach

i
8

Figure 7.2: First stage model for the drag coefficient

71

Drag Force along V_uum

Trajectory

7.01°6 t___first_ stage aeroJ model I

J

]

6"01067

-sec°ndstage_

I

5.0 10 6

I

4.0 10 6 t 3.0106
2.0 106
!
1.0106_
!
o_
25

/_

_ NN

50

75

100

125

150

time

2.5 106

Lift Force
I

along; Vacuum

[

I

Trajectory
1

2.0 106

1.5 106

1.0 106

5.0 105

0
7 -5.0 105

--

first stage aero model

-- -second stage aero

-1.0 106

I

l

1

I

I

25

50

75

100

125

150

time

Figure

7.3:

the vacuum

Comparison path

of the first stage and second

stage aero models

along

72

Method
zeroth order first order first pulse shooting

final time
(see.)
371.50

final weight (lbs.)
322861.

B.C. error

7 deg I h ft

-0.24

35.

369.91

329293.

.03 -.002

369.59

330576.

.0001 .0007

369.57

330678.

Table 7.1: Comparison of Results

dynamic effect. This technique was presented in chapter 6. In this chapter the results will be presented along with the results of the zeroth-order solution, the first-order solution without the aerodynamic effect in the zeroth-order problem, and the shooting method [17, 18].

Tile trajectories generated by the zeroth-order,

the first-order with

and without zeroth-order aerodynamic pulse functions, and the shooting method

are shown in figures (7.4-7.9). Also plotted are the Lagrange multipliers for the

closed loop trajectory, mainframe computer.

figures (7.10-7.11). Each technique ran on a IBM 3090

Integration w_ done by an eighth-order

Runge-Kutta

method for the shooting method. The approximate optimal guidance schemes

employed a fourth-order

Runge-Kutta

integrator.

The approximate

method

used a fixed number of integration steps in the first and second stages with the

control held fixed over each step. Four hundred steps were used in both the

first and second stages. The gime-to-stage was fixed at 153.54 seconds.

73

All tile methods were started at the sameinitial conditions: to = 35

see., ho = 660. ft., V0 = 9406. ft/s, % = 58. deg., rn0 = 3021107.44/bs.,

00 = -79.0 deg., and X = _b = 0.0 degrees.

The terminal constraints

to

be satisfied are h/ = 486080. ft., V/ = 25770. ft/s, and 7/ = 0.0 degrees. The results are compared in Table (7.1). The solution shows the approximate

optimal guidance law using tile first-order correction term matches the control

and state trajectories

of tile shooting method.

Initially only the first-order

correction with the aerodynamic

pulse generates a nearly optimal trajectory.

The cost obtained by tile two techniques is nearly identical. The final weight

using the shooting method was 330678. lbs. at a final time of 369.57 seconds.

The final weight was 3305r6. lbs at a final time of 369.59 seconds when using the

first-order approximation.

The zeroth-order solution shows a greater variation

in the control from the optimal control. The final weight obtained was 322861.

lbs. at a final time of 371.5 seconds. The zeroth-order

solution also does

not satisfy all the boundary conditions as closely as the optimal and first-

order solutions, with an error in the final flight path angle of -.24 degrees and an error in the final altitude of +40 feet. Because of this error in the

terminal constraints,

large angles-of-attack

can be seen in fig. (7.4) for the

zeroth-order solution in attempting to meet the terminal constraints. order correction picked up most of the deviation of the zeroth-order

The firsttrajectory

from the optimal trajectory and as a result the boundary conditions are met more closely with a better behaved control. The most important aspect in

obtaining good results is the convergence of the Lagrange multipliers to the

optimal Lagrange multipliers.

With the use of the aerodynamic

pulses the

flight path angle Lagrange multiplier approximated

to first-order shows good

74

3O

I

2O I
_0
I I
10
Io
.-M ¢,9

0.0

-/

O

!
_9

bD

-10

l

I

I

J

I

---

shooting method

.....

zeroth-order

- - - first-order

.....

first-order w/pulse

•. ,

• ".

°'

t

-2O 25

I

I

I

l

J

I

75

125

175

225

275

325

375

time

Figure 7.4: Angle-0f-Attack

vs. Time

'/5

80.0

1

\

60.0

\

t_0

¢19

_''

40.0

¢:D

t_0 20.0
¢.9

I

I

I

I

I

--

shooting method

.....

zeroth-order

- - - first-order

- - - first-order w/pulse

0.0

-20.0 25

I

I

I

I

1

I

75

125

175

225

275

325

375

time

Figure 7.5: Thrust Pitch Angle vs. Time

76

500000 400000 -'-- 300000

l

I

i

l

I

. . • -. -

-

.

200000 "x7

q

100000

J

i ._../_

/

- - -first-order .... -- zfierrsot-tohr-doerdr er w/pulse

r

T

I

I

I

I

25 75 125 175 225 275 325 375

timc

Figure 7.6: Altitude vs. Time

30000
25000 "7o,
20000
r_
"-* 15000
"_ 10000 0 > 5000

-- shooting method .... zeroth-order - - -first-order

L
/_ _

T

I

I

I

25 75 125 175 225 275 325 375

time

Figure 7.7: Velocity vs. Time

77

6O

I

J

J

]

I

I

5O

\

t_0 _9
:_,\

4O

'_ \',\

, _) 3O

--

shooting method

.....

zeroth-order

- - - first-order

.....

first-order w/pulse

2O

10

0.0 25

I

I

I

I

I

I

75

125

175

225

275

325

375

time Figure 7.8: Flight Path Angle vs. Time

78

1500.0

I
1000.0

v

¢9

oO ¢D
¢2, _9
°_,,_

500.0

I

I

1

I

1

I

--shooting

method

.....

zeroth-order

- - - first-order

.....

first-order w/ pulse

\

N_\

'

0.0

I

I

I

I

I

I

25

50

75

100

125

150

175

200

time

Figure 7.9: Dynamic Pressure vs. Time

79

0.0

I

f

I

I

I

I

-0.5 "7
t.)
-1.0

hO

J
,!

I

-1.5

I

>

I

I

J

-2.0

I

I

J

shooting method

.....

zeroth-order

- - - first-order

.....

first-order w/pulse

-2.5 25

i

I

I

i

I

I

75

125

175

225

275

325

375

time

Figure 7.10: Velocity Lag-range Multiplier vs. Time

80

3000 2000 1000

I

I

I

I

I

I

J

•e

• "/_

9t

J-,
¢9
hD -1000
-2000
-3000
-4000 25

shooting method

.....

zeroth-order

- - - first-order

.....

first-order w/pulse

I

I

J

I

I

I

75

125

175

225

275

325

375

time Figure 7.11: Flight Path Lag'range Multiplier

vs. Time

81

Method
CPU time
(see)

zeroth order
49.

first vacuum
304.

first pulse

shooting

344.

426.

Table 7.2: Comparison of computation

time

agreement with the optimal solution. A last point about these result is that the

inclusion of the rotation of the Earth in the problem is expected to continue to

reduce the time of flight and consequently orbital insertion.

increase the final weight available at

The convergence of the asymptotic expansion is indicated by the re-

sult of the first-order solution in comparison with the shooting method so-

lution, thereby precluding the need to include higher-order correction terms.

This convergence is tentative since it took the inclusion of the aerodynamic

pulse functions in the zeroth-order problem to achieve the best results. Alas

the convergence properties when using these pulses cannot be guaranteed or

even quantified. Finally, since this algorithm is being proposed as a real-time

guidance scheme the computational

time that was needed to generate the entire

trajectory by each method is presented in Table 7.2. While none of the codes

have been optimized for computational

efficiency, the use of quadratures

does

decrease the time needed to solve the launch problem in comparison to the

shooting method. It should be noted that the flight time is approximately

the

same as the cpu time for the first-order approximation

methods and that the

shooting method was given a good initial guess (nearly converged) of the un-

knowns. As expected, the zeroth-order analytic solution was found extremely

82

quickly. The introduction of the aerodynamic caused a modest increase in the computation

pulse functions into the method time.