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https://ntrs.nasa.gov/search.jsp?R=19940020279 2020-07-10T14:57:50+00:00Z
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.
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