/
A
HIGH THRUST OPTIMIZATION
THROUGH
HAMILTON -
JACOBI THEORY
by
ALAN ROBERT MITCHELL
S.B., Massachusetts Institute of Technology
(1965)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June,
Signature of
1966
Author_Signature
redacted
L
Department of Aeronautics
and Astronai;tics, June 1966
Certified
by_
Signature redacted
Thesis SupQrvisor
Accepted by
Signature redacted
Chairman, Departmental
Graduate Committee
038
ii
HIGH THRUST OPTIMIZATION
THROUGH
HAMILTON -
JACOBI THEORY
by
Alan R. Mitchell
Submitted to the Department of Aeronautics
Astronautics on
20 May 1966
and
in partial fulfillment
of the requirements for the degree of Master of Science.
ABSTRACT
This thesis begins a sequence of closed form
approximations to minimum time, high thrust space vehicle
transfers in an inverse-square, central gravity field.
A Hamilton-Jacobi perturbation procedure is used.
The
familiar zero gravity and constant gravity closed form
solutions are derived with the Hamilton-Jacobi formalism,
and the solutions are written in a more convenient form,
for the ensuing perturbations, than the forms available
in the literature.
The overall goal of a perturbation
solution or of a family of perturbation solutions is to
approximate minimum time solutions as they would exist
in a gravity field comprised of the first n terms in a
Taylor's series, position expansion of gravity about the
initial point. The constant gravity solution precisely
expresses the effect of the first term in the gravity
series.
Then a family of perturbations, (which are shown
to essentially number two), are required to approximate
the effect of the second, linear term in the gravity series.
The first member of the linear gravity approximations,
-id
iii
relative to the precise linear gravity solution, produces
the well known costate, the position's homogeneous
solution, the position's particular solution due to the
constant gravity term, and, in some sense, produces an
approximation of the position's particular solution due
to the optimal linear field thrust program.
The linear
gravity family of approximations must then improve the
position's particular solution due to the thrust, and a
procedure for accomplishing this in one step is outlined.
A precise example trajectory,
within the context of the
problem's model, is solved on a computer, and the important
results are graphed.
Only a brief analysis of the numerical
results is made to study the second member of the linear
gravity approximations.
The second member of the linear
gravity approximations and later solutions for a higher
order gravity series are left for future research.
Thesis Supervisor:
Title:
James E.
Potter
Assistant Professor of
Aeronautics and Astronautics
iv
ACKNOWLEDGEMENTS
The author is gratefully indebted to
William E. Miner of the NASA-Electronics Research
Center for inspiring and continually assisting the
development of this thesis.
Professor James E.
Department,
To his thesis advisor,
Potter of the M.I.T. Aeronautics
the author expresses his gratitude for
invaluable criticism and support of this thesis.
For many fruitful discussions,
the author extends
a thank you to Theodore N. Edelbaum of Analytical
Mechanics Associates and David Schmieder of NASAElectronics Research Center.
Marilyn2 Wisowati, who aided the computer
programming,
and Linda Miles, who typed the manuscript,
also earned the author's gratitude.
Most essential was the wonderful encouragement
of the author's family,
to whom the thesis is dedicated.
v
TABLE OF CONTENTS
Page No.
Chapter No.
1
Introduction
2
The Formulation
7
The Cost Function and the
2.1
Differential Constraints
2.2
The Optimal Control and
the Canonical Equations
Canonical Transformations
2.3
and the Hamilton-Jacobi Equation
2.4
The Perturbation Procedure
The Gravity Expansion
2.5
3
The Base Solution
3.1
3.2
17
The Results
The Generating Function of
Zero Gravity
3.3
The Ordinary Integrals
3.4
The Canonical Transformation
for Zero Gravity
3.5
The Canonical Transformation
for Constant Gravity
4
5
The Numerical Example
4.1
The Chosen Problem
4.2
4.3
4.4
The Differential Equations
The Transversality Conditions
The Numerical Output
The Linear Field Problem
5.1
The Definition of an
Accurate Solution
5.2
The Third Perturbation and
the Results
5.3
The Derivation of the
Third Transformation
5.4
The Ensuing Perturbations as
One Perturbation
5.5
A Brief Analysis of
32
44
vi
6
Summary and Conclusion
6.1
6.2
64
A Summary of the Results
Concluding Remarks
Appendices
A
The Constant Field Parameters
69
B
Schmieder's Sweeping Parameters
77
Figures
1
Example
r(t)
_
2
Example
e(t,
'6 f+
3
True
and Fitted Linear Field>,,
4
9,&A,
83
6
01) 02,C13)c 4 1 15.
85
7
b,, b2, b3
86
_
(t1
80
81
82
5f,84
References
b4
87,88
vii
NOTATION
General
it
;
one sub bar for a column vector
N
;
two sub bars for a matrix
;
super-T for transpose
S or
;
(if iL is a defined vector),
component label
or
;
(if m is not defined),
time label
;
row vector gradient
9;/bv
ag/b6
; matrix of rows
m
;
total time derivative
;
as supermarks,
;
enclosing a submark, variable's
or L
variable's label
label
Defined Vectors
r
position from dominant c.g.
Lr
r
, ie. velocity
gravity,
-Af
r
the state of
the costate
and A,,,
the
R V~i*M
of
,
_! and mass M
Atri
ihparameter (vector parameter)
; the first, second and third
segments of X
viii
T
(vector
;
the i" coparameter
coparameter)
;
the first, spcond and third
segments of..X
;
thrust
-IM~
IT)
Defined Scalars
t
;
m
; mass
0'
; mass flux
C
;
exhaust velocity
'A
;
constant
9
r
;jri
U11-
Mf
time
gr2
~I~1
;;h canonical generating
function
H <;>
l"
total Hamiltonian afterit"
perturbation
;
segment of
H "i
orti+i)
perturbation solution
H (")*
;H
H."'+
CHAPTER I
Introduction
It
is
desired
to approximate
the minimum time
flight of a high thrust space vehicle in closed form.
This
requires a perturbation
method,
(H-J) perturbations are chosen.
and Hamilton-Jacobi
Historically, in this
investigation the perturbation method searched for
meaningful problems that it could solve, one of which
is the high thrust space vehicle transfer of minimum time.
If but a single optimal trajectory and perhaps
the field of neighboring extremals were sought, then one
would numerically solve sets of ordinary differential
equations, like the canonical and the Riccati equations,
in a two point boundary value problem.
A large field
or bundle of optimal trajectories may theoretically be
generated through the solution of a first order, partial
differential equation, the H-J equation.
The numerical
approach to the H-J equation, dynamic programming,
is
virtually impossible for systems of high dimension, viz.
the dimensions of present interest.'
In any case,
dynamic
-2-
programming is difficult to apply unless the end state
that one desires is fixed.
Closed form approximations
to the H-J solution may result from an H-J perturbation
method when the system differential constraints are well
behaved, closed form functions, as for the gravity field
of space.'*8
H-J perturbation theory might better be called
canonical variation of parameters
theory.
The minor
problem that constitutes a perturbation is solved from
the canonical or the H-J equations of the minor problem
such that one obtains new variables which are canonical
constants of the motion in the minor problem and canonical
variables in the total problem.
A set of variables is
canonical in a problem if the set segments in half,
(into
what are to be called
parameters),
the parameters
and the co-
such that there exists a Hamiltonian
of the problem with respect to which the variables obey
the canonical differential equations:
just as the state
and co-state are canonical with respect to the original
Hamiltonian of the total problem.
A transformation of
variables is canonical if canonical variables are
transformed to canonical variables.
Each perturbation
solution will take the form of canonical constants of
a previous minor problem transforming to canonical
constants of the latest minor problem.
Each perturbation
-3is initiated by an expansion of the latest total
Hamiltonian, which may be regarded as the scalar
embodiment of the remaining dynamics.
Ideally,
the
minor problems on the average converge to the total
problem in the sense that the latest canonical variables
tend to become virtual constants in the total problem
too.
When one is convinced that he has an excellent
approximation to the total solution then he may regard
the latest variables as
canonical constants of the
total
problem which, with time, through all of the transformations, implicitly define the closed form, general
solution of the original canonical equations for the
state
and co-state.
At the expense of retaining canonical form,
one gains a well defined perturbation method where the
differential equations for each perturbation's set of
variables is immediately known in the canonical form,
and the next perturbation is always in terms of a scalar
function, the Hamiltonian.
Also, the canonical form is
compatible with the requirements of the digital
computer
and thus saves considerable effort in numerical investigations.
The unfortunate aspects of H-J perturbations
include the immense effort involved in retaining canonical
form, the ease with which one is lost in minor problems
of abstruse physical significance, and the impossibility
-4of knowing a priori the form of the solutions beyond
a quite early
stage.
In the present investigation the total problem
is modeled as a deterministic flight of a single stage
high thrust vehicle in an inverse
square central
gravitational field.
the minimum time
One desires
flight from a fixed initial state to a manifold in
state-time
space.
The first
and second perturbations
are for the field free and flat earth problems, and
together they are considered the
"base" solution.
The
third perturbation begins a family of perturbations to
include
the effect of the linear term in a Taylor's
series expansion of gravity about the initial point.
This family of perturbations will barely be started at
the close of this thesis,
for more numerical investigation
is required before one may ernestly begin the family.
However,
the third perturbation is singularly important.
It immediately produces the well known guidance law of
the entire linear field problem, and this will remain
invariant:
the co-state
co-parameters
will
be related
to the
and time in an invariant form.
Only the
parameters will change with each perturbation,
always be
related to the
so-called
and they will
state and time and invariant
functional forms of the invariant co-state.
with the third perturbation
Therefore,
the complexity of including
the linear field problem is cut in half.
-5-
Because an H-J perturbation solution can have
sufficient accuracy in a large but limited region and
because one requires numerical studies to support the
many analytical decisions in forming perturbations, it
is necessary to have an example optimal trajectory that one
solved on a computer.
The
example trajectory is chosen
to be similar to a Saturn V, second stage transfer
to near earth circular orbit.
H-J perturbation theory for the calculus of
variations has not yet produced a good solution for
a meaningful space trajectory problem.
H-J perturbation
theory has been applied in analogous mechanics' problems
(uncontrolled).
Delaunay, without the benefit of the
computer and after twenty years of work, produced a
successful lunar theory through H-J perturbations. S
Of course, Delaunay could see the true solution.
There
exists a vast literature from celestial mechanics on
H-J perturbation theory for long term, almost periodic
motion and disturbances;
eg.
the von Zeipel method.
This literature is of no use in the present problem.
However,
the power series
expansions
in radius and
time or mass that are useful here are much simpler to
work with than the forms common to celestial mechanics,
like
Fourier-Bessel
series,
for the lack of references.
and this
is
some solace
-6-
Notable in the present research for H-J
perturbation
solutions
of optimal
space transfers is
work of William E. Miner at the NASA Electronics
Center.
Mr. Miner is most
thrust solutions,
the
Research
intently working on low
but his larger,
long term interests
have made him an invaluable advisor for this high thrust
investigation.
H-J theory for the calculus
of variations
may
seem to result from purely the first order necessary
conditions,
but a more rigorous
development than found
here will show the theory to be intimately dependent
upon second order, sufficiency conditions like
controllability and the absence
ie.
of conjugate points,
a simply covered region of extremals.
Therefore,
the following development is prefaced with the assumption
that one operates
in a "field"
in the sense of Bliss.
-7-
CHAPTER 2
The Formulation
2.1
The Cost Function and the Differential Constraints
The minimum time problem is written in a Mayer
form, so the cost is simply the final time,
at.
In this
early stage of research one must confine the problem
to a single stage,
continuous thrust trajectory.
The
latter condition is insured in formulating a minimum
time rather than a minimum fuel problem.9
It is assumed
that the change in radius over the initial radius and
that the transfer time over some local circular orbit
period are small ratios.
Position, . ,
is written in nonrotating,
Cartesian coordinates from the dominant center of gravity.
Velocity,y = _'
, and mass,m , combine with r
to form
the state, x.
_
rM
(2.1.1)
Gravity, #,
_ -
r
is defined
(2.1.2)
-
-8
Thrust,T,
is defined
CT)(2.1.3)
T=
constant max. mass flux
constant max.
The control,
_1
= _1)
exhaust velocity
(2.1.4)
(2.1.5)
, is a 3 dimensional unit vector
of arbitrary direction.
It is desired to take a vehicle
initial state,X (t,) = X,
t,= 0 ,
,
at a fixed
from a fixed
initial time,
to a manifold in state-time space in minimum time,
subject to the differential constraints
Ol.= F( 2u)
+
24.2
t
(2.1.6)
-rI
The Optimal Control and the Canonical Equations
If one introduces
the costate,
,
where
costate of C
-
costate
of i
(2.2.1)
--
costate of m
and then defines the Hamiltonian,H , where
)
H AT F(X
-
Ari T+AE49.+
atm
: ryT ]
-
M(2.2.2)
-9-
then it follows that the optimum control,
maximizes
(or minimizes, if one prefers)
,
L:o,
H
(2.2.3)
=
One may then write
H '>= __F(XU( ,))
Ar+and it is necessary that X
N
respect to
+
and
\ be canonical with
on an optimal trajectory.6
_
2.3
(2.2.4)
C
(2.2.5)
9r
Canonical Transformations and the Hamilton-Jacobi
Equation
A set of variables is canonical in a problem
if it segments in half,
coparameters,
into the parameters and
such that there exists a Hamiltonian of the
problem with respect to which the variables are canonical.
Mathematically,
parameters,
X
,
there exists a Hamiltonian,
,
and the coparameters,
by the canonical equations:
A
,
of the
related
-10-
H
A transformation
(2.3.1)
of variables
XA
(2.3.2)
(2.3.3)
is canonical
the
if
jn variables and the
For every canonical
are canonical.
(I1)
variables
transformation
there exists a generating function of a form like
(i
5
5
t)(2.3.4)
where det.
(2.3.5)
X
and
-A
x
The function
Eq.
(2.3.6)
(2.3.2) and
Eq.
-.of
(2.3.6)
_
S
is
a generating function because
implicitly
defines the transformation
(2.3.3), and any such function obeying
(2.3.5) may so generate a canonical
transformation.
It then follows that the new Hamiltonian is
(H<'S
yS
+
-X
(2.3.7)
.A1[:
-11-
Actually,
some transformations must be related to a
sequence of transformations,
S
where
is
a sum of
generating functions obeying the group property.
The transformation
(2.3.2)
general solution of Eq.
new variables
(2.3.1)
are constant.
H '
0
(2.3.3) is the
if and only if the
From the canonical
equations, the new variables
if and only if
and
are identically constant
.
Eq.
(2.3.7) is then called
the H-J partial differential equation.
0
If Eq.
=H (-X)
(2.3.8)
,)
)
+,
if-s---
(2.3.8)
Saa
S
can be solved in closed form then
produces the closed form solution to the problem described
by Eq.
(2.3.1).
S'by
One may either attempt to find
separation of variables or by the characteristic equations.
The characteristic equations are canonical equations
(2.3.1)
plus
(2.3.9)
If._.A_ appears
Eq.
(2.3.%)
tell one that
If one can solve Eq.
form when
H
linearly in
S
*
then Eq.
(2.3.9)
with
is a constant of the motion.
(2.3.1) or Eq.
(2.3.8)
in closed
is the total Hamiltonian of a problem then
one can solve the total problem in closed form.
In that
-12-
case it makes no sense to work backwards
characteristic equations to
form
from the
S611, for that amounts
to solving the total problem in two interdependent ways.
However,
one generally cannot solve the total problem in
closed form,
and so a closed form approximation through
a pefturbation method is
solution.
Such is
the only way to obtain a literal
the case
Because H-J perturbations
work backwards
5 "i
(j i)
2.4
,
where
"
in the present problem.
are chosen,
it makes sense to
from the characteristic equations
S
to form
relates to the solution for the
perturbation.
The Perturbation Procedure
If one can find a significant part of
that would lead to a closed form total solution if it
were the total problem's Hamiltonian then it may be
considered for the
(i + o)stperturbation.
H <=H
=H.
0
Assume that Eq.
H
+
+
(2.4.2)
' S
is
+(2.4.1)
(2.4.2)
solvable in closed form with
or without the aid of the characteristic equations.
H0 ('
is considered to be significant may result
from physical,
intuitive reasoning or from numerical
That
H#I
;+1_(2.4.3)
-13-
studies that compare the canonical equations
to
H"'
.
The derivatives of the variables due
H, o'i+)need not be greater than those due toH"'
,
and
for H.
but the former rates should at least be of the same order
of magnitude as the latter.
of many perturbations
And on the average,
the effect
should be to make the latest variables
more nearly constants of the
total motion.
One may regard a perturbative H-J equation as
a way of getting a canonical generating function that
transforms the constants of the ih perturbative problem
into
the constants
of the
(i+1)"
where the latter problem is
true problem.
perturbative
problem,
ideally closer to the
The canonical constants of a perturbative
problem are canonical variables of the total problem.
From Eq.
(2.3.7), the new total Hamiltonian is
H
I
It is intuitively clear from Eq.
(2.4.3) that
the new total Hamiltonian is the embodiment of the
remaining dynamics.
It is convenient to write each
Hamiltonian without the explicit appearance of
For example, if
(i4')(2.4.4)
time.
-14-
then the canonical equations or Eq.
,
that
problem.
Therefore, time is separable in
(2.4.5)
or the parameters number
Applying the generating Eq.
; 41
(minor)
S
+5"(1
We assume the coparameters
.
tell one
is a constant of its perturbative
A'=t
Ii
(2.4.2)
(2.3.6)
S
(2.4.6)
As
(2.4.7)
S
t='64(2.4.8)
Assume that
H
(I
in total was written without time.
The above equations tell one that time will always
appear as (:,,-t)
in the transformation,
the new total Hamiltonian,
H "'.
and
thus in
If one wants the
new total Hamiltonian to be a constant of the total
problem then one may use a time transformation to
starred variables:
S+
jn-. +(2.4.9)
-15-
As
a result,
the starred and unstarred variables
are
the same except for
=* X- *
--
(2.3.7)
From Eqs.
(2.4.10)
(2.4.3)...
and
H
~'(2.4.11)
4-H
It happens
that in the present problem the
time transformation may always take the above
5
so we will assume that after every
an
Therefore,
.
H(j41)will
form,
there follows
mean
and the
stak-is only retained on
In summary, one has a total, time invarient
Hamiltonian of the in variables, H® .
portion, H,
A significant
,that one can solve in the H-J equation
is used to define the generating function of the(i+1)S
problem,
.
The generating rules are applied and
one hopes to explicitly define the old variables in terms
of the ( j*O5s
problem.
by
set of variables-constants of their minor
Time will appear as
.
(x,, -
,
which is replaced
The new total Hamiltonian is therefore constant,
and is found by Eq.
(2.4.11).
-162.5
The Gravity Expansion
The trajectories with which we deal are
characterized by a small change in
initial
It therefore makes physical sense to
r(t,).
H< )with
initiate the expansion of
expansion of
r
r campared to the
a Taylor's series
about its initial value.
r
(2.5.1)
+,23/(2.5.2)
+
2-2
7,1N(+0
)
(2.5.3)
where
7(2.5.4)
_
N
L -3(2.5.5)
Placing Eq.
(2.5.2)
forA into the total
Hamiltonian of the state and costate,
H"o'of Eq.
will allow the perturbative process to begin.
H0
(2.2.4),
will
represent the field free case, and H. will represent
the constant field case,
(flat earth)
.
The constant field
case is the base solution.
3
Then H.,will
be an important first member of a family of minor problems
in
.
designed to incorporate the effect of the linear term
-17-
CHAPTER 3
The Base Solution
3.1
The Results
The closed form integrals of the canonical
equations for the problems of
zero gravity and constant
6,10
gravity are well known in one form or another.'
In a
variation of parameters study, Ross has written the
differential equations of the total problem for the
constant g
integration
"constants", producing better
equations for numerical integration'.
more:
way.
We desire much
improving closed form approximations all the
The initial conditions
for the state and costate
in these simple problems could be used as a second set
of canonical variables, but a much simpler form for the
first canonical transformation results from the following
somewhat arduous development.
To the author's knowledge,
no one has produced a simpler constant 9
transformation than,
for example, Ross's,
canonical
but a simpler
form is desirable for a perturbation theory.
It will be shown that the following
transformation is the general solution of the constant
-18-
problem.
Segment the parameters and coparameters:
2
V
(3.1.1)
A4*
(R I
(3.1.2)
=
=
(3.1.3)
a
(3.1.4)
-LL(R)
2
=~3I
+
M(RA
(3.1.5)
The state is then given by
a
r=fA-
(3.1.6)
+
tr <V -M *g,
cf
/ -M
if "')JI
-7C
-
R
(3.1.7)
3j
&
)m
where
X=T
(3.1. 8)
(3.1.9)
In~
ll
-19-
r'
(.&LS
+
-&(F0-ALLVI
(3.1 .10)
+
(3.1.11)
tAI-Ati
problem are the
same, except for..A(m,
I
2A
-
The coparameters of the zero
+
r0
,
The parameters of the zero $
to
iM
9
AR
I3 .1.12)
problem are simply related
Y>
M *,
am
3.2
(3.1.13)
The Generating Function for Zero Gravity
Using Eq.
(2.2.4)
the zero-
Hamiltonian is
defined.
(3.2.1)
Therefore,
,
-
(3.2.2)
,
H'"= K;, +Iot' rn'
the first H-J equation is
0 = H,"_X')+S(,it)
(3.2.3)
-20-
J.
If the first coparameter is
, there is a zeroeth
to relate the notation to chapter two,
coparameter,
this is clearly the costate,
Similarly,
_.
For brevity,
is the zeroeth parameter,X.
and
the state,X
the
parameters or parameter vector is termed simply "parameter",
and so forth for the term "coparameter".
One defines
of Eq.
(3.2.3)
JL
from the separability constants
or from the characteristic equations'
constants of integration.
ThenX. is defined by the
generating equation (2.3.6).
When first developing
a perturbative generating function, like
S
, whether
by separation of variables or from the characteristic
equations, one usually will not define the coparameters
until the complete integrals are written and the most
convenient definitions are evident.
The following
development will, however, employ post facto rationale
in defining
-A-
From equation
separates in t and
r.
(3.2.3)
it is evident that
S
Instead of proceeding from this
factwe will first write the canonical equations for
H
,
()
and
and there will result a
_
reduced problem involving
A(m,
.
From the
canonical equations
Ho
H'= 0
(3.2.3)
-21-
t
d
it
y-_rH0
O1
=-
AUr)
The coparameter
A .is
" =QO'-
HO
=
-
01
j
,
(3.2.3)
(3.2.5)
segmented as
1\_(M) }
A
and Eqs.
(3.2.4)
(3.2.6)
(3.2.4) and (3.2.5) may be integrated
to yield
A (,
HA
A (r) =
-
A
At)
'(
R)
.1
ii
consian
(3.2.7)
COfnStdnt
(3.2.8)
(3.2.9)
i(o
If one writes the time-mass relation
-t = 6oIm,-m)
(3.2.9) may be recast as
=1
A(RI
(V)
)
then Eq.
(3.2.10)
(3.2.11)
-22-
The coparameterj\. thus relates simply to
1r)
for
,
'S
A
and
, Eqs.
.
Using the generating equations
(3.2.7),
to yield most of
(3.2.8), and (3.2.11)
S
b ()=
b
can be used
H
t
(3.2.12)
1
-//0 =(M)
(3.2.13)
-
A -r
-A- (V) 4.-rn
-(tr)
S
~ = {..A~Mt
+
..
Alai
VLA
)+
4-
r
(i)
RI V
(3.2.15
into Eq.
I
(3.2.15)
(3.2.3) produces
)
Inserting Eq.
(3.2.14)
A(R)
?) S
=
.
a reduced H-J equation for
O4JL(M)
+CmIJ
However, f rom Eq.
(3.2.15)
bm
-
(r
--
S
I
0 ' I
d
A
(3.2.16)
and therefore Eq.
S
mAtR)j
(3.2.16)
(3.2.17)
c(mS
may be rewritten
(3.2.18)
~~(M)
-C rT~j
+Mj\
-23-
The integration of Eq.
(3.2.18)
S=~~
{#J.~r
~ '(I.v
In
The terms Ie and 6
I -A-W)l
is straight forward.
mL
AtI
A
(3.2.19)
are defined
+
ITl-
~
-AV~n~~(l
fl[JL(i+
The entire generating function
I~
(3.2.20)
I--tV)I
Th
2. 21)
-~(3.
S''
is thus
~icr
+J
+cr1A,)M
3.3
V)
'r ~'(LY
4- eC
lJ~+ m~
The Ordinary Integrals
In forming
S
only the canonical equations
for the costate were used in conjunction with the H-J
equation.
Also,
the Eqs.
(3.2.7),
(3.2.8), and
(3.2.11)
represent one-half of the desired zero field canonical
transformation.
The rest of the transformation results
from the generating equation for
However,
applied to
it proves useful to anticipate the
_
trans-
formation by integrating the canonical equations for the
-24-
state, forming a more common description of optimal,
zero field flight in terms of the costate and the
initial values of the state and costate.
=
=
dt
I
SCm
(3.3.5)
(3.2.10) and (3.2.11) one may reform
From Eqs.
Eq.
-r-
A
(1)
(3.3.5) to
a
-"-
- C
-_
_
~
(3.3.6)
-cm
.L
It,, I
kil
Eqs.
(3.2.20) and
subscript
(3.3.7)
indefinite integral of (
(3.3.7), is called
Using
and W , where the
(3.2.21) describe t
' in Eq.
.
WI&1i
) (M) =
refers tot=
t,.
as expressed in Eq.
_
_.
S
of Eq.
s'"={(t
in Eq.
(3.1.8)
(3.2.22) as
+ m) +A(', r 4 0- C
ts
M
One may easily check that
the original H-J Eq.
(3.3.8)
- V(m,)
M()m)
Lt(m)
+
_V(M)and the definition for-JX'
one may rewrite
The
S
of Eq.
(V)
V
()
(3.
9)
(3.3.9) satisfies
(3.2.3), so this final verification
-25-
is left for the interested reader.
Finally, the equation for r
is written,
(3.3.10)
d-t
Ma
rn
The integral of Eq.
(3.3.11)
(3.3.11) is most easily evaluated
by parts.t
r
-dm
3.3.11)
-M
fn
+i
t
m=r',-o'r
m
F
--
E2
+ x)
3(3.3.12)
(3.3.13)
The next section will show that the canonical
transformation for
X
(3.3.7) and (3.3.13).
results in equations like Eqs.
The difference is that
will
tA suggestion of William E. Miner's saving
considerable effort.
-26-
comprise all of the lower limits of integration.
The
reason for first presenting the above results is that
the prior knowledge enables one to spot unessential
,
terms of the transformation, viz. functions of
which may be systematically eliminated.
3.4
The Canonical Transformation for Zero Gravity
With Eq.
(3.2.22) and
6- 5
(3.4.1)
one may define the complete zero -
canonical trans-
formation.
The cumbersome mechanics of applying
Eq.
is in Appendix A;
(3.4.1)
the results of Eq.
and some additional alterations of
Sect.
3.1.
(3.4.1)
were given in
There are two details that one must describe
to explain the results in Sect.
3.1,
and this explanation
will now be given in a qualitative form.
Applying Eq.
equation
,
S (as
defined in
(3.2.22) results in a transformation which,
when compared to Eqs.
r
(3.4.1) to
(3.3.7)
and
(3.3.13)
for
If
seems to have many unnecessary, horrendous
and
terms.
One is soon convinced that the unnecessary terms are
functions of only_
.
If one can find a scalar function
of.A whose gradient with respect toA produces the
-27-
unwanted terms of the original transformation then one
should subtract
of I
the scalar function from the first form
From Eq.
.
(3.4.1), the new form of
represent the zero -
S
will still
canonical transformation,
but
without producing the unwanted terms.
Secondly, there is a more general reason to
S
further alter the form of
:
the differential equations for
homogeneous,
scaling.
Acriand
Notice that
Aig.are
and that, because the optimal steering law
only depends on
the direction of Air')and not its
one has that the trajectory
initial conditions of
r)and
by a nonzero constant,
versality condition,
r(tlis uneffected
length,
if the
are all multiplied
_i
k .f Assuming
that the trans-
(which is discussed in chapter 4),
does not impose a natural scaling condition, one
concludes
that the costate is scalable in one's boundary
value problem such that the state's
invariant.
that
From Eqs.
-Ais also
scalable.
effected by scaling'
to render
and
(3.2.11)
one wants
it is seen
Nevertheless,X
may be
One has the freedom to alterS
invariant under scaling.
desirable because
tIf
(3.2.8)
trajectory remains
is associated with
k
to be negative,
This may be
X
, which is
then a minus
.
sign should be written in front of Eq. (2.2.3), which
amounts to minimizing
H"" with respect to
t-r
-28-
invariant under scaling.
Also,
keeping
a variant
of scaling gives one numerical problems as one attempts
to use the freedom to one's benefit,
and insodoing
one may make the perturbations much too dependent on the
example trajectory from which numbers are obtained.
The transformations involving
found in Sect. 3.1
are chosen to be invariant under scaling.
In closing, it is mentioned that if one wishes
to change to doubly starred variables such that
-A
(3.4.2)
h
and
(3.4.3)
then from the generating Eq.
(2.3.6) one requires the
generating function
(3.4.4)
where
(3.4.5)
The above result is useful in following Appendix A.
3.5
The Canonical Transformation for Constant Gravity
From Eq.
(2.4.11) and Eqs.
(3.2.1) and (3.2.2)
one may write the total Hamiltonian of the zero-field
canonical variables
H-A
i-
--_\
(3.5.1)
-29-
A
H
into its proper variables one would replace
through Eq.
qr)
in its
(3.2.11) and write
proper form through Eqs.
(2.1.2) and
(3.1.7)
.
To put
The perturbation for the constant field
problem is obvious, where we employ the gravity expansion
(2.5.3).
=Ho1
HO
(3.5.2)
Hj./(21
( Aui1VA4
L/A(MI
(3.5.3)
(R)
The second H-J equation is therefore
+M*67521)
'b
-I-
'r
(3.5.4)
a
5
=S21z
(3.5.5)
The canonical equations of
and
-.
S
tell
tAi are still constant.
M
,
(A-
HO- A
(3.5.4)
cons-t ant
(3.5.6)
COS-tCnt
(3.5.7)
const ant
(3.5.8)
/
A(V)
or Eq.
/
one that
H0a
J(MtA-KJ.()V
~ctj
*
where,.
) (3.5.9)
-30-
Using Eq.
(3.5.9)
in Eq.
(3.5.4) one reduces
(
2) =jL
Eq.
(3.5.10)
M
+
)
the H-J equation to
(3.5.10)
integrates to give
IkeM W2 2 r#
2I
(3.5.11)
(3.5.9) and applying
into Eq.
Inserting
the generating equations one obtains:
0
(3.5.12)
Y-A
I
r
M
0
+
R)'*g1
(3.5.13)
,
Of course, just as-J
a
and
Jk
are segmented
similarly, it is convenient to segmentX and
E
M
similarly.
a
X
In writing
X
{(3.5.14)
as XL, and X,, , being the last component,
we have assumed that the time transformation has occurred
as described in Sect. 2.4.
-31-
The above results are easily obtained from the
field free answer in a generally similar form without
proceeding through so much formalism, but the formalism
is necessary to retain canonical
form.
Hopefully,
the
formalism will return great benefits as we continue,
no
for
longer are closed form solutions or approximations
well covered in the literature.
ferred to Sect.
Again,
the reader is re-
3.1 for the full transformation--an
elementary change from the field free answer.
-32-
CHAPTER 4
The Numerical Example
4.1
The Chosen Problem
It is desired to investigate the
or zero g result of Sect.
3.1,
constant
and later transformations
too, on the basis of an example optimum trajectory,
solved on a computer.
No longer is physical reasoning
adequate to choose the next perturbations,
possible exception of
with the
of Chapter 5, and therefore
one must expand the latest Hamiltonians on the basis
of numerical evidence.
The chosen problem is
similar to a Saturn V,
second stage going into near-earth,
circular orbit.12
The resulting trajectory will burn for over nine
minutes, will
subtend a central angle greater than
twenty degrees, but it experiences a change in radius
ratio of only 1:64.
Although the problem is planar,
the evidence is useful for near-planar problems.
The following information describes the
chosen problem in MKS units:
mllmq
T
-33-
C =4,,/0
sa
PH YSICAL
CONSTANTS
+
S=4.5v/0 33
c'=/
m,
7.5
10 k
6.44c/O6}
r=
0m
(4.1.1)
B.5,v/0a
={2,6/O}
S
lra=6.5V/O'm
|Lia|= (g r '(t.)a'*
Tg = 0
rf
FINAL
7.8 5 v/O
Y"-
BOUNDARY
CONiDITIONS
i
4.2
The Differential Equations
The differential equation for the costate is
actually found from the canonical equations for
and
yL
and
,
(3.2.11).
in conjunction with definitions
i,
or
(3.2.7)
This is coupled with the differential
equation for the state,
_
__
such that giving
_/..(defines
course, Yfln() is known,
X(t.) and
a unique trajectory.
and it
Of
will be shown that A(mi)
is not important for minimum time problems.
-34-
(3.5.1)
From Eq.
(4.2.1)
where
=
(4.2.2)
- (= 2
one uses Eq.
,
equation for
cj
(3.1.7)
Writing the canonical
:
__
<~I>
I
LPr
-
r
(4.2.3)
.
and
For
+M*/L,
'JLv
Note that it makes no difference in Eq.
_1
(4.2.4)
(4.2.4)
one
if
deals with the second set of perturbative variables
first set,
I
From Eq.
3c3 LAAIT MATRIX
(3.1.7) :
(4.2.5)
(4.2.3) :
From Eq.
(-- A
o
as shown.
/
-
rather than the
f
(,R)
rK]r
Er,r
(4.2.6)
(iE7) 31~
(4.2.7)
Similarly,
d IL-r
- 4H <'>-2
_
1
(4.2.8)
From Eq.
(3.1.7)
br
-SM
v
(4.2.9)
0)
.
one notes from Eqs.
(4.2.7)
(3.2.7)
()4 )'
(1)
and
and
and
(4.2.10)
(3.2.11) or
A
to
(4.2.2)
that
(4.2.11)
.
To relate Eqs.
(4.2.12)
tri
To close the loop one merely writes the
equations for
it
and
r
(4.2.13)
and
dre
dt -
(4.2.14)
-
(4.2.1
(R
V)
d
-35-
The following flow diagram illustrates the computational
process, where the numbers appearing within the boxes
refer to equations of this
section.
-36(4.Z-7)
i1
(0
Diagram 4.1
4.3
The Transversality Conditions
Given
and excluding
A1l0
If
unique trajectory.
*a
one requires
JA(O)to
0) and
the four parameters
Ai,(0),
is open,
define
a
then one effectively
has five degrees of freedom in the integral of
X
X R.), (f.))
(4.3.1)
Three degrees of freedom are necessary for the final
boundary conditions in
(4.1.1).
The apparent surplus of
two degrees of freedom is consumed by the transversality
and the so-called scalability conditions.
The analysis that produces the optimal control
and the canonical equation, as given in Sect. 2.2,
furnishes another necessary condition,
condition.,
d r (t,)
and
6
There is
dft
of the problem.
al
,
a condition relating
if
also
the transversality
todf 2 ,
dma,
they are allowed by the statement
If the statement of the problem inter-
relates some of the given differentials, then the
-37-
problem's requirements and the transversality condition
must be satisfied simultaneously.
Imagine a minimum time problem that,
academic reasons,
allows Or
0 sQ<M
just as
1(T,)
and zero.
also contains
H
must maximize
O'
(C
Cr'(t)
,
C'
on an optimal trajectory.
,,,
-
>0
0
CM-
0'
and
,
(4.3.2)
U
Then the control
to vary between C'm
for
(4.3.3)
The transversality condition is then
\,,(td 20
H dt,
((+
(4.3.4)
dt2
The differentials
to
d L,
drf
and
,
dt,
coasting arcs,
(
>
dm,
(4.3.4)
H
Am)
d
M.
are not constrained
and, because we have allowed
and
then argue from Eq.
and
A
"
are independent.
One can
that
=
-l(4.3.5)
O(4.3.6)
Ur
=0
(4.3.7)
-38-
Eq.
(4.3.7) must be satisfied with the requirements of
the boundary value problem:
Tdr
=0
(4.3.8)
J~~a=0
(4.3.9)
r =0
Tqd~ -4-(JTd
First,
consider the "switching" condition
(4.3.3) in the light of Eqs.
then has that
(4. 3.10)
(2.2.5), and (4.3.6).
,\rr'O for I '' f . The switching
, except possibly
function is thus positive for t<tz
for a finite number of points;
Next, consider Eq.
ergo no switching.
(4.3.5).
(4.3.11)
4C
Because,
as already mentioned,
scalable,
One
_
and
are
_
(ie. can be multipled by a nonzero constant X
without changing f(t)), one has that condition
(4.3.11)
can always be met and therefore is trivial and need not
be obeyed.t Scalability is the first result needed to
reconcile the surplus in degrees of freedom.
One can,
for example, arbitrarily define
I
t
required if
---(T)1)1 =
See Sect.
h <0.
)
(4.3.12
3.4 for the alteration that is
rr
-39-
Finally, Eqs.
(4.3.7),
(4.3.8),
and
(4.3.9)
(4.3.10) are nontrivially satisfied only if
OT
T'
OTT
rQT
det.
(4.3.13)
LEq.
is perpendicular to
Because
(4.3.
13)
reduces to
Fortunately,
the
\_
left side of Eq.
)x
0
(4.3.14)
is a
(4.3.14)
constant of the motion and can be incorporated at
t,
.
-4+
That this is true can be discovered by formulating the
problem in polar coordinates,
and,
after realizing that
the costate to the central angle is identically zero,
transforming the costates of the polar formulation
to those of the Cartesian formulation.
~~
(4.3.15)
.T.
X
k4-
Schmieder's sweeping parameters were found
to be a convenient pair of initial variables with
conditions
(4.3.12) and (4.3.15), for determining the
correct initial conditions of the boundary value problem
(4.1.1)./
Computer
t2
was defined by the attainment of
-40-
_LT(ta),
the desired
and
and the4r(ta)
twenty meters
r2
_l)conditions
~ta
Without a machine
were met by sweeping the parameters.
iteration scheme,
r
was easily satisfied to within
and the angle between
and
r,.
was only off by a few minutes of arc.
_e-
Schmieder's
results are given in Appendix B in a form commensurate
with the present problem.
Then,
in a separate part of the machine program,
and many important functions ofJL were calculated
from the algebraic results of Sect.
The Numerical Output
Fig.
in
Many of the
and will now be presented.
results have been graphed,
4.4
3.1.
r(t)and
1 illustrates
transfers requiring a
is characteristic of
rLta
The hump
_r(tI.
huge change in velocity but relatively little change
in radius.
Fig.
2 illustrates
(t
from direction one, and i3
direction two.
Transfers
,
C.C.W. angle of
C .W.angle ofT from
to circular orbit that
experience a hump in r(+) apparently have
->
(1
r
-
is most perpendicular to
r
t
,
and,
r
of course,
0
.
r- 0
This
so that
as
is because
is
almost
adequate to keep it so, as in a circular orbit, while
up"Mm
-41-
the basic job for the thrust is
to raise the magnitude
of
in the last few seconds
to the required value;
ie,
'''-0
r
and
.
Thus,
as
t-'
and
approaches 900
T
athe angle between
t-*O.
If
there were
a
large radius to be gained then the end condition would
occur at the
t 0.
to
r
at
Fig.
t
1T T/= 0.
()
first stationary point of
,
2 shows that
T
,
where
is almost perpendicular
and in a perfect run one would expect
Such facts, however,
are only of passing
interest because one hopes to approximate a large
bundle of trajectories.
Fig.
3 is of great interest here, for it
shows that the true
by the
,C
_
i (OI)and
/I)
trajectory is well approximated
predicted in the linear field model, where
A(Olwere
matched.
the more striking if one
The close
fit is
all
fits the model curves to the
true curves to obtain minimum integral-square difference.
The fit of Fig. 3 has a maximum error at
in terms of pUi) ,
t
which,
represents four degrees of arc.
A
minimum integral-square error fit will have a maximum
error of less than one degree of arc.
Therefore,
an
H-J solution that merely approximates the linear field
model might prove to be satisfactory for a wide class of
trajectories; if used with sampleddata procedures,
the
,."
-'/mI
-42-
overall accuracy of such an approximation would improve.
Indeed,
even the flat earth solutions of Sect. 3.1,
similar forms,
or
have been actually applied in sampled
'I
A~f
data operations.'
The reader is referred to Chapter 5
for a derivation of the linear field
A.
Because of the form of Eqs.
p,
(3.1.7) it was decided to graph
9a
in Fig. 4, and
and
(3.1.6) and
andR.-mV +
a),= A 2 in Fig.
(R -
)=A.
5.t In
Chapter 5 it will be shown that these terms of
are
uniquely important and interesting in their significance
for the linear field problem, and,
used in defining
in fact,
are to be
for the third perturbative
solution.
Fig.
6 illustrates the following terms:
/f/j....(
(4.4.1)
d,4
I
-.
X
=
00
Fig. 7 illustrates:
05
=
b,
2
6,= ~Ofci
,
6?=
R,i
tHere, ,
az
(4.4.2)
is that found from Eq.
(3.1.7) minus
-43-
Figs.
perturbation.
4 and 5 are useful for the third
Figs.
6 and 7 are useful for the
the linear term
of 9
.
ensuing perturbations to incorporate the effect of
-44-
CHAPTER 5
The Linear Field Problem
5.1
The Definition of an Accurate Solution
The present family of perturbations are
designed to incorporate the effect of the linear term
of the gravity expansion.
If higher order terms of
are never to be included, then an accurate closed
form approximation of the linear field minimum time
solution may be defined as one whose difference from
the linear field solution is in some sense of the same
order of magnitude as the difference between the
linear field solution and the inverse-square field
minimum time result.
For example,
the trajectory of
the linear field model using the same initial (
and
A
,
Chapter 4 may be compared with a computer solution for
which in turn may be compared with one's literal solution
using the same initial
X
and
A
.
The quantities of
comparison might be the components of r
at every
ti
If one planned, for example, to later incorporate the
quadratic terms of 4
in the Hamiltonian, then one's
tz
-45-
closed form approximation of the linear field solution
must be more accurate:
perhaps as accurate as the
quadratic field's minimum time solution appears when
compared with the inverse-square solution.
Now, with
a rough goal in mind, the linear field perturbations
may be started.
5.2
The Third Perturbation and the Result
From Eqs.
(2.4.11),
(3.5.1) and (2.5.3) one
has that the total Hamiltonian of the constant field
variables may be written
H
One puts
Eq.
i>
-A,,,(Mo
QIf/(
(92)
(5.2.1)
in terms of its proper variables with
(3.1.5) for
incidentally, Eq.
_jr) , Eq.
(3.1.7) for
A
(2.5.5) for
.
,
and,
The total Hamiltonian
for the linear field problem of the constant field
:
variables is called
<2>
= -A
It is
J
2
-
a
)
/
W,N
> that is of present concern,
to approximate the dynamics expressed by
region of accuracy, as outlined in Sect.
otherwise stated,
(5.2.2)
and it is desired
:4 -(>to
5.1.
some
Unless
the "problem" refers in this chapter
__"WNW
-iloo,
-46-
to the linear field problem embodied by
From the problem's canonical equations,
....
is clear that the rates of change of
R
of Eq.
appear in the term
2.t
-
_
bAH
e
jt-.AIR=
dt
Using Eqs.
-L
(5.2.3)
(3.1.5),
and
V
as
they
(5.2.2).
=
9%,) N
FV<a>Zt
(V)
and
d
2
a
with respect to time are caused by
it
at
H=R)
(5.2.3)
(5.2.4)
and (5.2.4):
=L2
\
j49
(5.2.5)
It is convenient to define
r
0
(5.2.6)
so that
0 0
(5.2.7)
*
'
N =0*
.0O0
1
The three scalar differential equations of Eq.
are thus uncoupled,
I, b ,cde,)
(5.2.5)
hence introducing the constants
one has
that
-47-
2/t}
+
+
b sinb
a sin-vt
+
d coshb4'2)t
4 sinVJt
{eCosvt
Ccos Vt
=
From Eq.
(5.2.3)
(5.2.8)
r>\
and
(5.2.8):
Cisinb4vt+bcosh4 yt) -M'*(cos2Vt*+b sinI429t)
a_
(.~
-,s'rnvt+fcos vt)
(5.2.9)
-M*le cosyt -4r sinyt)
from Eq.
Therefore,
(3.1.5) one has that
r( sinh4At +bcosh4- 2't)
sin Yt - d cos Z/
-fcos
_
Clearly,
if
~ -
'S
{$'}
i/f(5.2.10)
were approximated by
2 +
,
~c
(5.2.11)
then one would still obtain the linear fields'
the optimal guidance law,
form for
although the literal equation
-48-
for the state would be different.
Fig.
4 shows that
for the chosen problem of Chapter 4 the difference
A)
-
is of similar shape to
9, with
generally of greater absolute value than
is not particularly close to
the difference
(
?a -Aa is,
?,
.
to first
derivatives,nz.i,
of Eq.
(5.2.11)
; so A,
approximation,
?a
a
AR'S
; so
are somewhat approximate to ,'s
All things considered,
derivatives.
?,
Fig. 5 shows that
constant, albeit large compared to
nth
a magnitude
n+h
the approximation
seems to be a reasonable beginning.
From Eq.
(5.2.1) or Eq.
(5.2.2)
and Eq.
(5.2.11)
one has that the third perturbative Hamiltonian is defined
2M
'
4
0-M
1
(5.2.12)
It will be shown that the canonical transformation to
the constants of the third pertubative problem,
N,
embodied by
may be defined as follows.
.3
as
Segment
2
and
as
JL
with the superscript
and
"3".
were segmented,
One has
for
J\.:
33
A
=
A,,cosjt
(R/ Li~ o(5. 2.13)
+Y-A-1voisn/
A.(.) cos/t -
T s
A
hit
but
-49-
+( ircoshri vt + v d'M*sinhrFyt).A.vI.]
4- ( ca coSvt
-
crVM ts in
E(:Isinyt - 6M'cos
*
3
'tJA.
t - r'VM*s invt)
4-( dcs
3
(5.2.14)
3
-A,1 COsh ljt -4 -- AI ISinbri 'vt
.3
ie
-L
cos it
A
-
3
Cos vt
-CV
2Nr
VJ- L
3
in Yt
se)
3
VrA-A(V)s
(5.2.15)
in ;'t
N(Nd7' 2
(5.2.16)
+- a)
One then has
=
v
form:
ashI)t
cr'c
0
0
0
C'l
0
-VTAsinh-Rt A
0
j
L{
+
0
0
R
sinvt
0
J
0
0
-dI
3
a'-'Cos
0
+
9)
-
6
~sinj
(5.2.17)
i
-W. IIIs MINII
..au--.--.
-50-
2
c
++M*cos;hfri't)
-vrsinh-r?/t
0
ti
v('ir~7/t+MCOs2
0
I
0
coshzvt+4r avm*s inhbt)
0
(cos/t-vM*sinvt)
cos,/i.M~sihVt)
0
L
0
0
+0
(5.2.18)
(5.2.19)
M=x n
5.3
The Derivation of the Third Transformation
The H-J equation of the third perturbation is
(0
o0
With the results
of time in
(5.3.1)
,),,LH.(9
37A.
t
(5.2.8),
(5.2.9)
and the separation
one has almost solved Eq.
S
(5.3.1).
First, define
(3)
HO
3
=-/AMl,,
(5.3.2)
cornstdn+
3
Next,
use
to obtain -.
(a,b,c,d,e,f)
IR)
=
c
J\-t.
3
and.-A-Lv
(5.3.3)
-51-
(5.3.4)
One then has the results
(5.2.13)
and
(5.2.14).
The
2
generating equations for -.
A.
_
(3)=
t+[
V
Of course,
one
substitutes
.
3
and integrated to define most of
as
(
always,
are then applied,
(3
3
5.3.5)
(M-1V4)for t in Eqs.
(5.2.13)
and (5.2.14) in order to get the forms indicated in
Eq.
(5.3.5).
The entire process
already taken in forming
H-J equation
S
for
back into Eq.
is quite similar to that
.
S
One may write
by using the results obtained
(5.3.1).
PR- oM*V+ Z- -go)
and Eq.
(5.3.6)
a-
(5.2.15)
(5..3.5)
..
for
(5.2.12)
for
-
(5.3.6) use is made of Eq.
)
In writing Eq.
Eq.
a reduced
.
From
2T
~M~it,
-
2
6_A
(5.3.7)
-52-
However,
the operator
-
ergo Eqs.
to
rewrite Eq.
is,
(5.2.3) and
(5.3.7), equivalent
(5.2.4)
allow one to
(5.3.7) as
A N(-(M) (5.3.8)
b 5 'S=(S
Using Eq.
in Eq.
(5.3.8)
in
(5.3.6)
one has
' (5.3.9)
3(M1
Using Eq.
for
riti
one may rewrite
(5.3.9)
s!~
-"-(MI
Integrating Eq.
(5.3.10) for
may then use Eq.
4-
S
(5.3.5) to form
S'"= /AM~
M'
+V2M
(5.3.10)
is elementary, and one
'
Eq.
(5.2.6)
+ [LRJR
2
ES2
3
3
3
Sihh~rZJt)M*
Z-'os-Vt-AII
5 inhrJ2'it+ -A4tRj C(5..11
414~~ftj
The square brackets around
... /(~and
-.
J1L~ indicate that
3
they are to be written as functions of
-A-,k
and
-53-
Applying the
Eq.
generating equations to
(5.3.11)
,
----s
S'
i'=
(5.3.12)
2
and solving Eq.
(5.2.18) and
(5.3.12) for
(5.2.19)t
results in (5.2.17),
When these answers are combined
with the constant field transformation of Sect.
3.1
one obtains an interesting result.
3
.3
V/cosh "dt
+F
V R, S inA-P
cos COSI - P'2 Sin I/t +
(5.3. 13)
33
R cosht + j-V, sinY+Tt+
Rcost j-gvsiVt
=
where
(/
CO~,.,~J.~
3
2
(5.3.14)
)f
(5.3.15)
t This step is straight forward, and is left for
the interested reader.
-54-
and
-'
=
-4-Co'jj(-~
-
From Eqs.
and
due to
_T
~R)
(3.1.6) and
are the
RI
_
.A.ftV
,0(5. 3.16)
(3.1.7) one has that
particular solutions
,
in the constant
problem,
but with _A_
3
put in terms of..,
blem),
(a constant of the linear field pro-
and time through the linear field guidance law
(5.2.13), (5.2.14) and (5.2.15).
It will be shown
in the next section that all the remaining perturbations
of the linear field problem leave
-A-
might expect,
are invariant forms.
so that
Furthermore,
_V
and
constant,
as one
because the guidance law does
not depend explicitly on position and because gravity
is linear, one's differential equation for
4
L takes the
linear form in the linear field:
+
+
='
(5.3.18)
a 4-
4
Ttf(5.3.17)
+
Eq.
(5.3.18) means that the solution of Eq.
a superposition of the homogeneous
particular solutions due to
9,
,
(5.3.17) is
solution and the two
and T
.
Now,
referring
-7--7
AT
-55-
again to Eqs.
(5.3.18) and
that the additions
to
)
(5.3.19), one discovers
are
in
and that the additions to
precisejy
i+h
transformation, '..4
for
.
In
If the assumption
,of the
linear field family:
3
SV
delay,
4h))
one then expects the following form for
a
5.4
-+4!
9')but only expressed approximately for
conclusion,
an
(
are in facth).t
tMexpressed
One therefore has
and
fact
4-
(5.3.19)
tal
i
(5.3.19)
is unclear one should not
for the next section will discuss it further.
The Ensuing Perturbations as One Perturbation
The results of the third perturbation are
now analysed,
and a general form for all the ensuing
perturbations of the linear problem is developed.
is shown that Eqs.
It
(5.3.19) apply, and that the net
gration with respect to M* or
Using Eq.
the results of Sect.
(5.2.2)
t
.
result of all the perturbations is essentially one inte-
for
5.3 one has that
,
rule
(2.4.11)
and
-.. Mom
ON-
mr-I
-56-
<3>)
-V 'J(4JM
=-A
__NN
(5.4.1)
It is assumed that the ensuing perturbations
in time orM
on expansions of
the part of
for the(*+3)
rd
series.
are based
Call
4
.
H
perturbational Hamiltonian.
-4
-4h
(\
We have assumed that
A n(5.4.2)
,
(ie. -- '2 -(R)
v
),
-L-(V)
R
does not depend on
never changes form.
Because
3
3
3
A
JLRand
one has that
or V
are
still constant
regardless of the definition of
-AMw-
H[ 4
[
=t S'H+ ('
Clearly,
the integration
Am(N5.4.3)
A'"
(5.4.4)
constants
.4
4
-- 6-
(Rand .wA
(V)
3
of the H-J equation
3
--7LA
V)
(5.4.4)
may be defined as
3
.A...c)
and
From the generating rule for A
3
-r 3
4)
(5.4.5)
and the definitions4
(5.4.6)
(R
I(R\
3
4
-57-
one has
that
t4)
3
A4
A(+)
S
4
v
4~
(5.4.7)
(5.4.4) may be reduced to
4
i
MA~
~(4)
LiM )
+1S
Therefore, the H-J equation
S
(4)(4
-(5.4.8)
A4
J-cMI/I
Mz~\1I~A
VJLijVo0V
(5.4.9)
'I'd M*i
The square brackets in Eq.
(5.4.9) mean that the quantity
is to be written as a function of A
form of (5.4.9) depends on (5.2.7) for
So,
finding S
4
M
and
.
The second
NV
amounts to integrating Eq.
(5.4.9)
(4)
for
over
S
,
's
and because one has complete control,
ideally,
expansion then one assumes that a significant
can be defined such that Eq.
(5.4.9) is integrable.
Applying the generating equations for
to
one has
R)
4 (4)
(5.4.1
4
'r
V
A (41(5.4.11)
~~
S
(
))
)
that
-58-
Definition
(5.4.10) and
(5.4.6) and results
(5.4.11)
thus verify for f=,4 the assumptions of (5.3.19) for
the general form of the perturbation.
It is clear that
the total Hamiltonian of the linear problem for the
4th variables is, from Eqs.
HSo,
(5.4.1) and (5.4.2),
-- yi'
J-(M)
jA 4
(5.4.12)
the pattern may be repeated for all the perturbations,
for again and again R
and V will be absent from
H
One may write the general problem as:
3
MR
A
-
I
(5.4.13)
(RI ~A(9);
3
-A tv)
A
i
i-1
R=P
M
M
+-(
)
'/J
]iVT
)-S) /tw
"
S
(5.4.14)
(V
^M
~d M*(5.4.15)
(5.4.16)
(5.4.17)
However, thinking of many perturbations i.4
is not the clearest point of view.
for LJ
and
,
One has the results
(5.3.13) and (5.3.14), and the only
-59-
3
3
i
change at perturbation
(5.3.19).
prescribed in
4
3
= A
R
-4- (
R
is in
V
, as
(5.4.16) :
From Eq.
S
b
6A(RI
and
(IT
4
R +(
S
(5.4.18)
= R -(
.3
.'.
+(4
R
S
")
n
T
(5.4.19)
-
If
Z5
Similarly
3-
Using Eq.
v--rn
(5.4.15)
and Eq.
(5.4.20)
(5.4.2),
where we forget
due to the gradients of Eqs.
the terms
and
C(; ) -r
(5.4.19)
(5.4.20), one obtains
Z~~=
2J
T
V
I~
E 3~Ym
-o
(5.4.21)
Therefore, rather than thinking of many P
'S
and many
one can ask for
an entire, truncated
expansion of
~4%
4where
+4"
(5.4.22)
(5.4.23)
-60-
such that Cin
Eq.
(5.4.21)
is said to result in a
closed form.
3
In summary,
(5.4.24)
V(5.4.25)
expanding
to obtain a truncated
series
such that
is obtainable in closed form
via Eq.
(5.4.21) produces
the linear field approximate
solution.
The solution is had explicitly by putting
into Eqs.
resultsin Eqs.
(5.4.24)
(5.3.13)
One no
form or partial
a single,
and
(5.4.25),
and then these
(5.3.14).
longer cares
explicitly for canonical
differential equations.
clear goal:
This is not
and
define
There remains
and integrate Eq.
something that may be done uniquely nor,
any case, simply.
Nevertheless,
perturbational process
(5.4.21).
in
one has simplified the
to a well defined point that requires
one more, huge effort for the completion of the problem.
Of course,
if the term
later point then
formalism,
_(
is
to be included at a
one would later revert to the H-J
and one may then expect a far worse experience
than he has had before:
the uncoupled linearities
typifying the transformations of the linear problem would
be but fond memories.
-61-
A Brief Analysis of
5.5
Before closing the chapter, a brief analysis
is made on the basis of Figures 4 through 7.
of
From Sects.
3.1 and 5.2:
2
I-I(5.5.1)
-L4
For the purpose of expanding
through Eqs.
A
,
it is written as
(3.1.8),
(3.1.10) and
A
(3.1.11) for
-V7rand 3
and
for.A..(,,.A.) and
(5.2.15)
a
*
A,
and Eqs.
(5.2.13),
(5.2.14)
Notice that
Aj
Figure 4 shows that
determining
(5.5.2)
is quite significant in
and it is not easily approximated by a
low order polynomial in time.
is significant in determining
approximated by a constant or,
Figure 5 shows that
P
,
but it may be
better,
a low order
polynomial in time.
Figures 6 and 7 are basically concerned with
,A
A
andCy.
'I
S
=
c+, + a
+= (4(5.5.3)
,+
+
(5.5.4)
----- ~----
U ~-.-
-~
-~-
-
-
-62-
d;
6 illustrates that none of the
Fig.
t
insignificant in the determination of
The terms
0,
and as
terms are
and
-6
might be approximated by constants
by taking their first terms in their Taylor's series
about
t"0.
Also:
Q4
Fig.
=
-+#- +
62(5.5.5)
o
+ b +b 4
7 shows that
b3z -b4,
approximates
(5.5.6)
so that the constant that
Cy may be called unity.
first half of the flight
b,%
For about the
-b2 , but overall
C4
is
not justifiably approximated by unity.
In summary,
expanded in powers of
t
and
f+nhi27/#
,IA-/\4
similarly for
appear in Eq.
if /h
IY
0
are,
and
and
for example,
* 0 n/t
_
,
and
as they
(5.5.1), then one might have a quickly
may lead to a closed
convergent scheme, such that
form for
The expansions, to reiterate,
the basis of
of
2
i
V
._./I
,i(R)
and
,
_Al
as functions
written
and time, as if
take place on
....
and
were truely constant in the inverse squard field problem.
-
I -~
-63-
The actual definition of
requires much more thought
and experimentation, but the problem reduction of Sect. 5.4
at last brings one to a well defined perttrbational problem.
-64-
CHAPTER 6
Summary and Conclusions
6.1
A Summary of the Results
A
summary of
best given in terms of
the results
of this
thesis is
approximating minimum time
transfers in a linear gravity field.
The approximating
perturbations may be classed into four steps:
constant gravity;
"homogeneous"
"approximate" linear gravity.
gravity
zero gravity;
linear gravity;
and
By "homogeneous" linear
is meant a gravity field that varies with time
during the flight as if there were no thrust and merely
a linear gravity series in postion.
after the first two,
linear gravity
has been solved.
This third problem,
By
"approximate"
is meant a gravity field that varies
with time during the flight in the manner of one's best
and final approximation of the linear field problem.
This fourth problem has been shown to consist of an
expansion of a scalar function of time such that one
defines an integrable part with an ignorable remainder.
Note that although the third problem's
approximation of the gravity time series does not reflect
-65-
the effect of thrust, the resulting solution for the
state does approximate the effect of the thrust and
It is
the resulting solution for the costate is precise.
interesting that the half of the third problem's solution
for the position that remains invariant upon the fourth
perturbation is that part of the Hamiltonian whose effect
The
remains to be included in the fourth perturbation.
vector function in question,
,
is the third problem's
approximation of the position's particular solution due
to the optimal linear field thrust program.
6.2
Concluding Remarks
In performing the fourth perturbational
solution, it will be useful to see if any parts of
can be included without a series expansion.
Also,
a
second example trajectory should be solved on a computer
in order to better segregate numerical properties that
are typical of a rather large class of high thrust
transfers.
One might,
for example,
require that a slight
plane change and a relatively large change in radius
occur.
The troubles that would be encountered in
incorporating the higher order gravity terms are not
clear in the author's mind, but certainly they would be
-66-
significantly greater than those experienced so far.
Transfers that require many terms in the gravity series
for a model, ie medium or low thrust transfers, might
better be handled with a zero thrust,
inverse-square
field base soltuion rather than a full thrust,
field or constant field base solution.
zero
The former base
solution has been accomplished by MinerP
Actually,
a Taylor's series expansion of gravity will not work for
most low thrust transfers,
mechanics'
and in this case the celestial
literature on the von Zeipel method and so
forth will prove quite useful.
Finally,
if one achieves an adequate literal
solution then there remains an implicit function difficulty
in doing guidance.
That is,
one desires an explicit guidance
law like
A
(6.2.1)
but one actually has, for example,
=\R)-V~)
1 \/.A A(6.2.3)
___
Lt
_
_6-.2.4)
(6.2.2)
-67-
S1
1(6.2.5)
For illustration the form of E is chosen to relect
stationary differential constraints and stationary
boundary conditions, hence
(_
and
0
.
The "constants"
are implicitly defined at t
)Y()
L_(t)
_-E=
,
by
M.t)
(given a defined final boundary value-
minimization problem), and thus >(V)is implicitly
defined by
()
and %_(t).
One would probably determine
the constants numerically at discrete moments, a method
commonly called sampled data feedback guidance.
Of
course, one requires some feedback control because one's
solution is imperfect and one's knowledge of the state
and the control is uncertain.
At any rate, if the
numerical difficulties in obtaining the constants are
too great then,
from a practical standpoint,
this spells
the ultimate doom of one's closed form solution.
This
possibility must be objectively tested along with
questions of accuracy that ignore the guidance problem.
For example, expanding ~_
is necessary for the fourth
perturbation, and it may again be required to apply the
tIn between sample points,
Eq.
(6.2.4) represents
an explicit guidance law, albeit open loop.
-
-M64 -
_M
_
-68-
solutions in a guidance scheme.
It must be remembered
that this thesis merely investigates one of many
approaches to obtaining a literal approximation of the
minimum time,
high thrust transfer,
and the state-of-
the-art of doing guidance is so young that no one method,
existant or anticipated,
has a clear superiority.
-69-
The Constant Field Parameter
APPENDIX A
It is desired to generate
by Eqs.
(3.4.1)
and
(3.3.9).
as described
,
The superscripts
"i
"
are
First, we generate V
dropped for convenience.
T
V-
c+T
-
6LAW
(RI~- +I T VV)
M~ m-A
._
Cofstari vector
from Eqs.
One expects that the excess
constant,
knows that
ie
a function of
V
and
(3.3.8)
o
-
(A-2)
(3.3.7)
(A-i)
terms of Eq.
and
-AM
(A-i) are a
-A(V),
while one
is a constant.
fC&
a
If one can find .
)c
-'-A
(MJ'
)
(A-3)
and a scalar function
S
whose gradient with respect to JLY)produces A
can augment S with (-S)
AI(f, A-v
then one
and rid the transformation of L
.
However,
-Y)
M.M
-70-
From Eq.
V(M)
V) -AL V
,
(3.2.20)
From Eqs.
I
2f
(3.3.7)
)f
JA
and
-AR
(A-4)
(3.2.21)
-b
4
( A -x)
--
_-t
Tvt
ALVI
kA-6)
(T4
3
If one uses the above results in the left side of Eq.
(A-3)
and writes the result in terms of the orthogonal vectors
-.M
and
(A(
1
then one obtains
LA<adVf
4l~d'rV1
A,,)'(CR4Y
AN
trivially.
efficients of -A(R) and
proof and not a proof using
A
A
To prove that the co-
1-tmt A(vi
(which, because one desires
4
Tvlr
(A-7)
)v AR)
Of course, the question mark refers to whether
really contains y
J
',
LAfY
/tjcontain
,
m trivially,
means an algebraic
), one can probably
-71-
find simplifying relations among t , I
However,
the author used
"brute force"
and
0
algebra to
prove that
A6 =fci+
-
V
(A-8)
4+
v
LI~m
LA-1
-
(
ZA
(A-9)
Before attempting to find
to generate
R
6
it is desirable
, which also suffers from unnecessary
terms.
=
=
-
S
-
R
+ m
+(g
i
-Again,
-C
\1
{'L
may replace r
in
~
'(M)
S
~
(A-10)
because only the gradients
of this function are important.
From Eq.
-Ct
(3.3.13)
1JLU)
(A-ll)
m
-72-
or-
+ (r
3
0
As with
V,
lm =Consfdni
(v)tai
-
ve.ctor
one attempts to write the
R
generating
function relation to correspond with the
integrals",
(A-12).
"ordinary
However, in the case of
b
R
one must
_ (Y) to make the com-
first perform the gradient of
parison.
(A-12)
T__
)
-A k.=C
-
-A
(A-13)
1-_1R
one may write
-(R
m+J
AriiL)
m
-
With the above results,
in the form
(A-14)
the relation
(A-10) may be put
(A-12).
C
IAT
--
-A
V),
_:A
-
(A - 1 5
)
Also,
-----
-I1
-73-
Therefore, one expects that
I
-4-
' (vm
To prove that
for
A
AI
(iA)
Jivd4 IAVIIIJp~JstiV i(A-16)
-I
0
)
)
one proceeds just as one did
The result is
.
-Ac x AA
=L
I-At
P) P ItJill-VA +j A-(R)
-A
The question of augmenting
S
by
V
(A-17)
Fe)
,
where
augmenting S
by (- O
.
If one can find
0
and
&
IT
. , arises along with the question of
such that they are equal then this is the ideal situation.
The author could not quite obtainI9
= &,
but the following
and
A
.
development explains what was accomplished to rid
Note that
I.At~)IJ_ --T
-'/'
A
-74-
Also
A
3
T
+ T
=
(A-19)
= (~i71
However
J
-j- n -A
-IAjJ
'Y
UMAv + Lkrill.A(v)
(A-20)
y-i.A(V)1(I
A
(t)R 1)
(
(A-21)
.' =
&fc Y
Am-lI
Ih (
AI))
Define an augmented S
(A-22)
by
S =S-G
As a result,
(A-23)
the term
transformation.
3 b- 0=
__
disappears from the
But what of _A
?
Note that
CrcIfC[
4-
IJ)
II
J
II'
]
j
(A-24)
-75-
So,
9
is
'
plus a remainder that in the V
leaves the second term of Eq.
From Eqs.
3
(A-l),
transformation
(A-24) 's final form.
(A-3) and
(A-24) one has that
gives the following V transformation:
V
={-N
r
-0iYn)
.A
And S
gives the followingR
+
transformation:
roJ-L
T.
ILA,.I
I.,IJ
From Eq.
(3.3.7)
term of Eq.
h t
for V'
(A-26)
one has that the additional
(A-25) may be combined with L
would become
/P
,
+I[4vj4aI
.ALoltlecnIJ
If one imagines scaling all of the
terms then one has from Eqs.
*LI
(A-25) and
that many terms of the transformation,
and
(A-26)
like V and R,
are not invarient under scaling, although _r
are invariant.
where
and
If one could form slightly different
arguements of the logs, where the arguements are in-
variant under scaling, then V and R would be invariant
too.
Namely,
'
and 3
of Sect. 3.1 are so normalized.
It is stated without further motivation that if one
desires to change to doubly starred variables such that
-76-
R
-AV)
R
(A-27)
-AR
(A-2 8)
V)
=R
+(w
,xe,(A-2
OLroA
4-LLC
-AIN
L401fAcRu
_
)
9)
(A-30)
J-kALVJI
then one requires the generating function
S
=
()
R + -t
V
(A-31)
Using
So"on the transformation of S
of Sect.
3. 1.
3S
gives the results
was defined after some experimentation.
-72-
Schmieder's Sweeping Parameters
APPENDIX B
-MAay b(ts) s=
DIAGRAM F-\
It will be shown that
o(
and Q( ME ~y be used as
the two degrees of freedom necessary to solve the
problem of Chapter 4.
Define
=
J
(B-1)
and notice from Diag. B-1 that
fixed att,.
,
=d
(B-2)
(B-3)
_3 = fd n-
As already mentioned in Chapter 4,
at
and
Eqs.
(B_4)
and
(B-4)
A
V0
one must define
to define a trajectory.
From
(B-I):
sI
COs
= s(t +N
(1-
= co5( , + o)
(B-5)
(B-6)
---
-78-
Hence,
CE,
condition
defines
_(Jtff,j.
Using the transversality
(4.3.13) one can show that
or
The result
-
=0
(t (,
>-(nC
To define
(B-7)
rt,
,rlti)
(B-9) depends on
Therefore, writing Eq.
(B-8)
(B-9)
r (t) =0,
i\r.)(one needs
2 'irh+L
also determines
0
Sri
. J(til Sd
0(
b(
(t,).
From Eq.
O9fSC)
(B-10) at
txr
(B-4)
(B-10)
t,
one has that
(B-11)
But
(B-12)
and from Newton's Laws
=
u(
0-S
In C( -- 3COW
(B-13)
-
-I
-79-
Using the above three equations one has
((B-14)
or
\ (-
(B-15)
0.E.D4
-80-
(/0om)
y(/3 rn
6.5/
8.0
Ii
7. S-
'/
.
r(t..
6.YSO
7.0
,
rEt 2 a)
6.47
6.46
6.0
78/
(t.2)A
/
Lr
/
I
0
/
6.41
3,~0
A
I
I
6.43
rGt 6, -.10
I kmz.62 miles
A
I
4/
I
a4-
a-
6.4!
I4..
3.0
J(i )42.74
'
6.401
2.5
f, .z
SJ5
I
-
60
/20
/80
300
240
-RO
.360
FIG.
/
o
EXAMPLE
r(t)
M.K.S. UNITS
-1
180
540
600
-L
(sec.)
7-e (de.)
3, (def)
i
24
60
22
I
'I
I
so
/
it
/
I
0
A
it
/190
/a
40
NA
I
30
/
/6
/
N
I
/
S
/4
I
/
XL
/
S
/0
,J\.
*1
/0
S
I
0
I,
I
I
V
Co
a
6
11
*
I
-&0
oll
J-
9
4-
tLz 3w55
'p.
0
0
60
/20
/80
2-f0
300
-360
420
FIG. 2
E XA MPLE
A2T
6(t)~
r
2 e
%..
480
5-0 600
(sec .)
COMM
trou
it
Val
.r) (t=0)
tr
ArI (t,)
1trae
A
f- I II-
.7
-Attr u e
S
.5
~r~traAe
0
60
/20
180
2-0
JOO
360
-20
FIG. 3
TRUE
A
AND FITTED
LINEAR FIELD A,
480
50
600
($c.
83-
(/o
6
)
J,; A,
i
.3
.2
.1
..-.
J
---
~- -w
60
-.1
120
/80
2A0
300 ,36&<'2O
480 510
.- ~Z...
A
M=r
-4
FIG. 4
I&A,
-
.3
\
I
600
(s ec.)
A&
j
( /0
y
4.0
3.5
-
3.0
A
|1
A
I.-
I1
.100"
'A
Aa
7000
.000Aa
.7
.A0
1.5,
A-o
/.0
A-01
0.5
I0 t2 ---- 55'5-9
60
120
/80
2-40
300
J60
420
A~
FIG. 5
y &A 2
-80
540
600
1
sec.
(norm dliZE-D)
/1
C,
/6
/.4
/
2*1
/
N
N
//
N./.-*/
-.--
018
0.6
~-.
---
------
.
.-
0.-1
0. 2
t z
0.0
60
/20
/80
240
300
360
420
-0.2
-
0.4
-0.6
FIG.6
(a, ,a), a3 , a4 ,a,)
490
540
655
600
W-
c 5 e C.)
A (normaliZED)
3.0
7
/
b,
'.5-
/
6N
N
0.0
N
-
N
--...........
N.
I
NI
*1
za'
0.0
60
/20
/80
240
300
360
-0.5
-
120
-180
Jr0
I1
/.0
---
-..-
-2.0
- --.-...,..........B
4
Io'
.000
-3.0
FIG. 7
)
(b,,b b3 b
3
600
(sec)
-87-
References
1)
Bellman, Richard E. and Dreyfus, Stuart E.,
Applied Dynamic Programming, Princeton University
Press, 1962, Chapters V, VI, XII.
2)
Bliss,
Gilbert Ames,
Lectures on the Calculus of
Variations, University of Chicago Press, Phoenix
Science Series,
3)
1961,
Chapters III, VII, VIII,IX.
Bliss, Gilbert Ames, The Problem of Mayer With
Variable End Points, Transactions of the American
Mathematical Society, XIX, 1918, pp 305-314.
4)
Boyce, M. G., An Application of Calculus of
Variations to the Optimization of Multistage
Trajectories, Progress Report No. 3 on Studies
in the Fields of Space Flight and Guidance Theory,
NASA MTP-AERO-63-12, 1963.
5)
Brouwer, Dirk and Clemence, Gerald M., Methods of
Celestial Mechanics, Academic Press, New York, 1961,
Chapters XI, XII, XVII.
6)
Bryson, Arthur E. and Ho, Yu-Chi, Optimal Programming,
Estimation and Control, Harvard University,
(unpublished 1966)
7)
Goldstein, Herbert,
Wesley Press, Inc.,
8)
Miner, William E., Low Thrust Optimization through
Hamilton-Jacobi Theory, NASA-Electronics Research
Center, (unpublished).
9)
Rosenberg, R. M., An Optimization Problem in
Dynamics, 4th U. S. National Congress of Applied
Mechanics, Vol. 1,
10)
Classical Mechanics, Addison1950, Chapters 2,7,8,9.
1962.
Ross, S., Optimal Ascent into Orbit-A New Look at an
Old Problem, NASA, Washington, D.C., (XV International
Astronautical Congress, Warsaw, 1964).
- -
.1910111
1. 1
111
....
..
.------
1
-88-
NASA,
11)
Schmieder, David, Sweeping Parameters,
Huntsville, (internal note).
12)
Polovitch, R. S. and Morgan, W. B., Path Adaptive
Guidance for Saturn V Three Dimensional Ascent to
Orbit, NASA, Huntsville, MTP-Aero-63-70, 1963.
13)
Baker, Clyde D., Saturn Guidance Concepts, NASA
research Achievements Review Series #15-16,
NASA TMX-53373, 1965.
14)
Perkins, F. M., Explicit Tangent-Steering Guidance
Equations for Multi Stage Boosters, Aerospace
Corporation,
#TDR-469(5540-lo) -3,
1965.