OPTIMUM TRANSFERS FROM CIRCULAR ORBITS
by
GIM
ER
JEW
B.Sc.
Queen Mary College, London University,
1974
S. M.
Massachusetts Institute of Technology,
1975
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June
1978
Massachusetts Institute of Technology
Signature of Author
Signature Redacted
Department
Certified by%.
1978
(Aeronautics
and Astronautics
Signature Redacted
Thesis Supervisor
Certified by
Signature Redacted
Thesis Supervisor
Certified by
Signature Redacted
Thesis Supervisor
Signature Redacted
Accepted by
Chairman, Yepartmental Graduate Committee
Archives
MfTTs
JUL 6
TITUTE
1978
LIBRARIES
Optimum Transfers From Circular Orbits
by
Der
Gim Jew
Submitted to the Department of Aeronautics and Astronautics
on
in partial fulfillment of the requirements
February 17, 1978
for the Degree of
Doctor of philosophy
ABSTRACT
a
The
problem
circular
orbit
orientation.
the minimum impulse transfer
is to determine
to an
elliptic
The optimum number
shape and
of arbitrary size,
orbit
from
4 3, location, magnitude and direct-
ion of the impulses are to be determined.
The basic
H.W.
approach
combines
technique
a numerical
developed
by
in his Ph.D. thesis at Stanford University with a number of
Small
analytic
approximations.
estimate
of
An analytic
optimal
the
trajectory
provides
approximation
and
Smallos
formulation
a first
allows
rapid convergence to an exact numerical solution.
The analytic
ing
of
this
approximations
problem
guidance
technique.
circular
orbit
to
which could
an
improve
the theoretical
provide
the basis
Knowledge
of
elliptic
orbit
the
optimum
could
for
understanda
subsequent
transfers
improve
the
envelope and fuel economy of the space tug.
Thesis Supervisor: Wallace E. Vander Velde
Title: Professor of Aeronautics and Astronautics
Thesis Supervisor: Richard H. Battin
Title: Lecturer of Aeronautics and Astronautics
Thesis Supervisor: Walter M. Hollister
Title: Associate Professor of Aeronautics and Astronautics
from
a
operaional
iii
In Memory Of My Teacher
Therodore
Neil
Edelbaum
iv
Acknowledgments
I
T.N.
wish
to
express
deepest
my
gratitude
and
respect
to
Edelbaum, who suggested and supported the work of this thesis.
and
It is a distinct pleasure to thank Professors
W.M. Hollister for many helpful criticisms.
L.L.
Sackett
R.H.
Battin
for his valuable
for
his
contributions.
teaching
and
support
W.E. Vander Velde
I am grateful to
I am also indebted to
through
out my graduate
studies.
This
research
under contract
was
supported
I.R.F. 18705.
by
the
C.S.
Draper
Laboratory
v
List Of Symbols
semi-major axis
d.
J
defined by eq. 49
e
eccentricity
e
unit vector
h
angular momentum per unit mass
*
a
h
reference angular momentum per unit mass
H
variational Hamiltonian
H
DH/Du
u
i
inclination
p
adjoint vector
P
parameter (semi-latus rectum)
Q
defined by eqs.52c and 60
R
radius vector
t
tan(Au/2)
u
central angle
v
true anomaly
V
velocity vector
AV
characteristic velocity
x
state vector
x.
state variable defined by eq. 49
O.
coefficients defined by eq. 54
1
thrust unit vector
X.
J
adjoint variable defined by eq. 49
X
primer vector (Lawden)
A.
functions of state and adjoint variables
defined in eq. 50
vi
Y
thrust angle w.r.t. eL axis (Fig. 1)
thrust angle w.r.t. e1 axis (Fig. 1)
w
argument of pericenter
argument of the ascending node
T
nondimensional velocity
y
gravitational constant
subsfripts:
i
initial orbit
s
final orbit
vii
Table of Contents
Abstract
Acknowledgments
List Of Symbols
Page
Chapter
1.
Introduction
1
2.
General Analysis
4
3.
Small Inclination Approximations
8
4.
Small Eccentricity Approximations
17
5.
Large Inclination And Eccentricity Approximations
24
6.
Computation Technique
32
7.
Conclusions
41
Appendix
A:
Optimum Single Impulse Transfers From
43
Circular Orbits
Appendix
B:
Computer Listings
46
Appendix
C:
Examples Of Exact Transfers
72
Figures
78
References
92
Biography
94
ONE
CHAPTER
1
Introduction
bolic
to
orbit
circular
only in
Edelbaum',
to the excellent paper of
50's
been
the
made
in
3-dimensions,
out
a
with
1976
efficient
very
and
were
not
(Lawden)
spiral
another
survey
no
had
3-
symmetric
solutions
almost
Small4
the
in
in
solutions
from
that
t.
Gobetz and Doll 2
had
In
refer
should
active
coaxial,
nodal,
that
except
For further
very
written by
arising
pointed
solutions
2-dimensional
was
speculated
are
readers
the
transfers.
limited
Marchal3
by
written
impulse
free.
, 2-dimensional
singularities
the
symmetric
and
limited
paper
to
due
three
and
two
transfers
circle,
The
transfers.
quasi-circular
complete
to
circle
i.e.
dimensions;
paper
special
some
only
that
out
pointed
survey
1969 a
In
601s.
and
involve
How many impulses ?
'
transfers
impulse
optimum
in
The para-
these will
rendezvous,
impulses
of
on the optimum number
Research
since
to be non-optimal when the time is
by many experts
discussion
free.
and
transfers
orbit
for
time
fixed
the time
a
from
Finite four impulse solutions which exist
considered.
are
solutions
transfers
Finite
infinity.
at
impulses
impulse
excluded
states are
end
with
orbit
elliptic
an
and hyperbolic
infinitesimal
optimum
investigates
thesis
This
had
progress
all
generalized
computer
program
optimum
impulse
in 1972.
5
transfers
the
orbit
in
problem
Marec
elements;
i.
become
7
a
of
complicated
first
due
orbit
circular
semi-major
The
pioneered
Breakwell
and
vicinity
the
inclination,
6
Nae
,
-
Eebu
Edelbaum
axis,
and
to
a,
second
the
t
with
small
changes
e,
eccentricity,
order
boundary
solutions
of
in
and
this
effects
of
the pericenter
of
layert
linearization:
1.
When
the
changes
in
a and
e
are
small,
of the transfer orbit is not well defined.
2
small,
i are
e and
in
When the changes
2.
the line of nodes
of the transfer orbit is not well defined.
Marchal
of
in
only
changes
thus
8 found
p
Marchal
as A
will
0, the first order solution does not agree with
a -w
is not well defined).
w
Ai (because
the matching of other linearized solutions.
then the
defined,
and
a,
If one calculates the limiting solution
allowed.
changes
If the variation of
in
W
arbitrary
i become
e and
in
We
impose difficulties
also find that the boundary layer effects
is well
small
of
i such that arbitrary variations in
Aa, Ae and
that of small
the case
for
solution
order
first
e and
, were
in r
a
and
this is the general problem we wish to solve.
2
Chapter
solution
of
circular
orbit
the
describes
allow
some
and
simplifications
order
first
from an
Transfers
transfers.
impulse
optimum
the
of
analysis
basic
assumptions
initial
with
no
penalty on generality.
The
contribution
theoretical
of
this
to extend
is
thesis
the
basic analysis to obtain the following analytic approximations:
Small inclination approximations in chapter 3.
Small eccentricity approximations in Chapter 4.
Large inclination and eccentricity approximations in chapter 5.
6
Chapter
approximations
to
an
combines
to compute
elliptic
of
estimate
orbit.
optimal
the
the
formulation
of
and
Small
the analytic
from a circular orbit
the general
transfers
An analytic
approximation
trajectory,
and
Small's
provides
a
first
formulation
allows
of
single
rapid convergence to an exact numerical solution.
Appendix
impulse
transfers
A
describes
the
basic
analysis
from a circular orbit in state space
space.
This is an extension of Der's
describes
the logic of the computer program.
M.S.
thesis
9.
optimum
and parametric
Appendix
B
The listing is included
3
for
reference.
transfers.
Appendix
C
consists
of
ten
examples
of
exact
4
2
CHAPTER
TWO
the
time
free
0
essentially
General Analysis
An
of
mination
14
the
S
transfer
requires the deterstate
adjoint
and
the true anomalies of depart-
of the transfer orbit,
v2
and
vi
optimum impulse
dimensional
five
p of the transfer orbit,
and arrival
ure
and
1
parameters:
and
x
vectors
0.
orbits
two elliptic
between
of
solution
analytic
and the respective
true
The primer
of the initial and final orbits.
analies
v. and v
s
1
10
vector of Lawden can be defined as a vector formed of components which
are
Lagrange
the
optimum
referred
to
rocket
as
constitute
duced
in
the
,
normal
which
is
a
components
arbitrary point
adjoint
primer
with
vector,
the
conic
of
in the variational
problem.
velocity
association
0
and
trajectory
the
which
orbit,
introduced
multipliers
The
vector
are
velocity
primer
because
vector.
the
vector
the
Along
is
and
of
often
multipliers,
intro-
the
transfer
circumferential
radial,
are given by the classical
T (Fig.1)
vector
adjoint variables
the
trajectory,
the
primer
formulation
its
derivative
at an
expressions
p1esin v + p cos v
p1p + p 2 / *P
3 sin v (1+1/$
+ p 5 cos
p4 sin v /*
p V2 + p
1
/3
-
where
*
=
e
=
2
-p
1+ecos v
and
3
the
sin
v
esin v + p 3 (e+cos v)
v) + p 5 sin v
p (e+cos
eccentricity,
v/i
-pi
2
)
=
P
=
parameter,
primer
vector
1 =
gravitational
constants
p
1,...
constant,
,
p5
are
5
can
which
constants
integration
be
identified
the
as
Lagrange
multipliers associated with orbit element state variables.
points ( v
X.
=
by
developed
conditions
optimum
The
Lawden
at
the
junction
v 1, v 2 ) are
= 1
X
(1)
,
0
The
impulsive
terms
the
of
velocities
at
elements of the
points
may
be expressed
transfer
and
final
junction
the
initial,
follows:
e sin v
esin v-
AV
-
(1+ecosv)
-
JI (+e
'Ti
cos v) cos i
(1+e cos v) sin i
L
e sin v
-
esin v
S
-
AV 2
(l+e cos v) cos(i -i) -
I
(1+ecos
S
-
where
E(1+e cos v) sin(i -i)
= inclination of the transfer orbit plane,
i
i
S
= inclination of the final orbit plane,
e
= eccentricity of the initial orbit,
e
=
S
eccentricity of the final orbit,
v)
in
orbits as
6
P. = parameter of the initial orbit,
P
and
= parameter of the final orbit,
S
i >
is- i
0,
The dynamic relations at the junction
0
points are
AV
k=
k
(3)
k AV k
where
The
IAVkI
=
V
,
k=1,2
geometric relations
obtained from the equation of orbit
and
2
2
a.(1-e.)/(1+e.cos v.) = a(l-e )/(l+ecos v
)
the latitudes at the junction points are
(4)
s
s
+ v
(
s
s
2
)
a (1-e 2 )/(1+e cos v ) = a(1-e 2 )/(1+ecos v
=0,
sin i sin( w + v 2 ) = sin i
where
w + v
sin(
)
(5)
a
= semi-major axis of the transfer orbit,
a.
=
a
= semi-major axis of
w
=
argument of the perigee of the transfer orbit,
Ws
=
argument of the perigee of the final orbit.
Although
transfer,
semi-major axis of the initial orbit,
Long11 lists
70
Eckel 12r,13reduced
the final orbit,
equations
to
define
a
the system of equations
2
impulse optimum
to 4n-5 equations
7
to be solved
However,
sufficient
the
14
(1)
equations
optimum
linearized
to determine
by
described
equations
transfer.
n impulse optimum
simultaneously to define a
to
from
transfers
(5)
are
a circular
orbit.
We will assume for the rest of this thesis that:
1.
9
the argument of the node,
the initial circular orbit, v.
The
plane
, reducing the state space to 4
In addition the true anomaly of departure on
dimensions.
2.
In this case we may neglect
The initial orbit is circular.
of
the
initial
,
becomes arbitrary.
circular
reference
is the
orbit
plane.
3.
orbit
is elliptic.
The
final
the
initial circular
incides
with
The line of intersection of
orbit co-
orbit and the final elliptic
a reference
line
such that
S.
1
=
S
= 0.
The
reference line is the line of the nodes as in Fig.l.
The transfer
itial
is in the outward direction.
lies
circular orbit
orbit, with
a,
1
N
a
S
.
4.
That is, the in-
entirely inside the final elliptic
8
THREE
CHAPTER
Small Inclination Approximations
Zero order solution ( i = 0
S
To investigate optimum transfers
)
3.1
consider
only
is
the
that
S
transfer orbit is
the
final
orbit
The zero order solution (
is the
i = 0
S
(Fig.2).
transfer
The
coplanar
)a +a.)/2
S 1
=
((1+e
e
=
((1+e
W
is arbitrary.
S
<1.
of
we first
orbit,
specified by
a
S
inclination
Hohmann
elliptic
the
of
i
element,
orbit
small
the
that
problem
from a circular
)
3
)a -a.)/((1+e )a +a.)
I
S
S
S
1
p1
= 1/2
,
The primer vector constants can be easily obtained as
2
=
p
=
3
(1-e
p
The characteristic
AVER
3.2
4
)/2
=
p
5
=
0.
velocity of the elliptic
Hohmann transfer is
(fl
-
- 1 +
7(
) )
.
p2
(6)
Study of the primer vector
When
the
inclination of
the
final orbit is small but not zero,
9
the first order term in
the
of
axis
semi-major
the
of
(Fig.3),
the
and
vector
primer
plane.
is impossible for the
i = 90 .
for
except
as in Ref.4,
derivative
its
orbit
initial
the
transfer)
of impulses
optimum number
study the
to
orbit
(1800
problem we are considering
2-body
depends only on the inclination of
AV
final
transfer
The elliptic Hohmann
of
i
To
the dominent terms
is
(when
is
small
but
are
odd
nonzero) can be expressed as
p esin v
+ p2
1$
p sin v /
4
iv
2
2sn
1
-
The
radial
functions
and
normal
v and
of
the
p4 (e+cos v)
components
derivation
may be three impulse transfers.
of
the
primer
vector
from the elliptic Hohmann transfer
The first
impulse can be at v1 = 0
and the other two close to the apocenter of the transfer orbit.
optimum conditions
v2 = v
(1)
and
(2)
give a family of
The
transfer orbits with
p
1/2
2
= (2+e 2 (1-2cos v-cos v)+2(1+e) JF* (1-2e-ecos v))
p2 = p (1+e)
lI
(1-2e-ecos
,
as an independent variable,
v)
(7)
pp4 = e 2 p 2
1
cos v
(
(3+2e+ecos v)
(1-2e)/e
10
The limiting solution of
= 1/2
p
= (1-e2 )/2
gives
,
p
v -w
(8)
p 2 = e 2 (3+e)(1-e)/4
4
plot
the
orbit
circular
the
initial
are
symmetrical
three equal maxima on the
transfer
impulse
exists
plane
two
types
a (1-e
S
S
orbit
three
impulses
Actually
the three
distance
of the final orbit
a (1-e
impulse
<<l
transfers
(sections
S
3.3,
range in which the pericenter distance
is neither very close to the radius of the initial
a
nor too large where the bi-parabolic transfer dominates,
impulse first order solution requires to determine
, vi
i,
W
Since
v
e 0
1
derivative
,
v
2
,
pl
S
3.4)
for larger pericenter distance.
)
e,
a,
third
solution.
transfer
In the practical
is coplanar to
of the theoretical
First order solution ( i
3.3
and
v
2 against
Then the primer vector has
axis.
transfer optimal
with the bi-parabolic
second
physical
at the vicinity
three
of
the
and
transfer orbit.
first
The variation of the pericenter
indicates
the
by
(8)
transfer orbit
first
the
the semi-major
to
and
the primer vector magnitude
of
Theoretically
(Fig.4,5).
(7)
)
and
transfer
of
solution
the
illustrate
can
)
We
....
p5
circular
the
11 unknowns:
of the first transfer orbit.
M 0 , the primer vector and its
r
, p 2, p
~ v
3
5
2
S
v 2 can
and the nondimensional velocity vector at v 1 and
, v
be expressed approximately as
11
p esin v + p
1
3
( p (1+e)2+ p )/(1+e)
I
( p sin v + p )/(1+e)
4
1
5
p esin
v 2- p
( p 1 (1-e) 2+
P2 /le
( p4 sin v 2 - p5 )(-e)
I
p (1+e) 2 + p (1+e)
2
1
1
1=
-
3
Fp3
-
2
L
-
2
esin v 1 + p3 (1+e)
p 4 (1+e)
p1 (1-e) 2 + p2 (1-e)
-
p3-e)
p2 esin v2
P4(1-e)
esin v/b
AV
1
1
AV
b 1- cos i
s in i
Oe sin v /b3 -
esin
AV 2
1
AV
2
b3 cos(is-i) - b 2
b sin(i -i)
S
3
v2/b2
b1
1
b2
F
where
b3
+
1
es'
b1- 1,
C=
c
,
12
.
= b 3- b
23
2
The optimum conditions (2)
X sin v + Y p3
p
4
give
= 0
(9)
=
X
e
=
0
X2
Y
=
(4+5e-e3 )/2
Y
=
(-4+5e-e )/2
The dynamic relations (3)
(p1(+e)2+ p )/(l+e)
give
=1
(10)
2
(p (1-e)
(p
4
+ p2
sin v + p)/(1+e)
1 p5 )1e
=1
= sin i
/c1
(11)
(p sin v -
4
2
p5 )/(1-e)
= - b sin(i -i)/c
s
2'
3
,
where
4
,
x 2sin v 2+ Y2 3
13
W sin v + p = 0
1
1
3
(12)
v- p
W sin
2
3
= R
2
= e(p - 1/(c b
1
1 1
))
W
= e(p + 1/(c b
))
R
=
W
1
2
2
1
2 2
e sin v /(c b
S
S
2 3
)
where
2
The geometric relations (4) and (5) give
a(1-e) = a.
(13)
A
=
+v
a (1+e
S
S
)
a(1+e)
= 0,
+ v2 )
sin i sin(
We notice that
anomaly
of
impulse case.
the
(9)
sin(
sin i
=
the
second
Once
w
transfer
(9)
-
11 unknowns of the first transfer orbit.
to justify our estimate
formulations
of
v
.
12 equations for
v
of the 'final*
S
orbit at arrival
is designated,
vS
(15)
+ v)
(15) actually consist of
-
The additional one is
12 unknowns.
true
(14)
orbit or the
in the
three
(14) uniquely determine
Then (15)
can be used
Remember that we will have similar
S
in chapter 4 and 5.
Incidentally the optimum conditions
(1) are the same as the dynamic relations (10).
Now the approximate
solution of the three
impulse transfer
a
= (a
S
(1+e
S
)+a.)/2
1
,
a circular orbit to an elliptic orbit is specified by
from
14
e
(a (1+e )-a.)/(a (1+e )+a.)
S
S
1
S
S
1
=
p1
= 1/2
,
V1
2
p2 = e (3+e)(1-e)/4
,
= (1-e 2 )/2
4
sin i
=
p
)
0,
sin v
sin v
c b 2 b sin i /(c b 2 b + c b 2
1 2 3
s
1 2 3
2 1
=
=
0
R /W
2
2
2
p5 = 0.
velocity of the three impulse transfer
AV
=
AV
+
EH
1
a(1-e 2 )
AVEH is given by (6)
where
4 s
=
eI(3+e)(1-e) /2
(16)
p 4 is, at this time, chosen such that
and
= -
eI(3+e)(1-e) /2
0
,
,r
given by
s
order effect of inclination in
the first
p
p i sin
is
(
The characteristic
AV. is positive, i.e.
1
7r
<
w
2
15
S
(l-e
S
)
)
of
Numerical
(3.3)
section
the
where
region
shaded
is
transfer
the
,
1
solution
the
in
a.
=
line in Fig.6.
order
a.i
a (1-e S ) :
S
E and
denoted by
i
S
S
first
pericenter distance
a
a (1-e ) :
a
valid
be
cannot
31)
by
the
that
shown
have
(denoted
<< 1,
by the solid
and is indicated
impulse
studies
S
pericenter distance
the
If
one
( i
First order solution
3.4
Incidentally
(denoted by 311).
are respectively the regions of
8
5
The
validity of the first order solutions of Edelbaum and Marchal .
the
regions
layer
locate these dotted lines is impractical because it
To
effects.
since these
are not exact,
cannot be uniformly valid due to boundary
order solutions
first
regions
these
separate
which
lines
dotted
M
will
require extremely long computation.
as
considered
the
total
first
transfer
that
to
similar
approximation
solution of
orbit
is
of
still
S
coplanar
three
impulses are
third
impulses
can
small compared
be
located
to
away
AV1
from
family
of
three
impulse transfers
anomaly, v, as an independent variable.
S
(Fig.7b)
the
elliptic
at a three impulse
Theoretically
initial
the
the
elliptic
6V 3 increase the
AV
.
All the
, therefore the second and
the
contribute to the inclination term in the total
a
orbit and
and
3.3.
to
and
shall be dominant in
we arrive
section
31.
initial
1
final orbit 0
The
0
the
S
6V and
second and third impulses
2
1 *
apocenter distance, therefore 6V must be opposite to
orbit
to
coplanar
to the
transfer is actually impossible,
Hohmann
inner
transfer (Fig.7c),
actual
the
of
AV
the
single impulse approximation
the
of
AV
Since
the
became the initial
Suppose that this nominal orbit 0
we wish to find the optimum transfers
study
a,= (a (1+e )+a. )/2
that
such
order
first
to
0
orbit
nominal
0.11 (Fig.7a),
orbit
circular
interested
are
be
a
choose
may
we
First
can
311 which
solution
we
section
this
In
specified
apocenter,
AV
by
.
(7)
and
they
Then we have
with
the true
16
The coplanar transfer orbit is specified by
= a.(ta cos v -a.)/(ta (1+cos v) -2a.)
a
1
= -
e
where
=
t
5
S
1
(ta -a.)/(ta cos v -a.)
s
1
S
1
2
)/(l+e cos v )
S
S
S
closely approximated
(1-e
transfers
is
study
perturbed
the
solution.
Since this family of three impulse
by
single
impulses,
The characteristic
we
will
velocity
not
of
the
three impulse transfer family is given by
where
1.< f
p
-
T.
(
1 ) +
-1
1
T
1- 2
a(1-e
p i sin w
S
P4 S
)
Av.
1p
=
e p
(3+2e+ecos v)
,
0
4
e p
+2e+ecos v)
,
r
<
WS
w
(17)
17
FOUR
CHAPTER
Small eccentricity approximations
Zero order solution (
4.1
If
the
zero
the
then
e =
S
inclination of the
(Fig.8).
chapter
2,
=
S
Hohmann
I tilted
or
transfer
Hohmann
final
( e
solution
order
0
)
4
The
transfer
must
the
transfer
is
be
0
)
is
nodal.
as
defined
that
transfer
coaxial
also
is
orbit
and
0
of
the
between
with
The
0
<
i
<rr,
generalized
two
circles
assumption
transfer
3
orbit
of
is
specified by
a
=
(a +a.)/2
S
1
e
=
(a -a.)/(a +a.)
S
1
S
1
0
i
.
Wi=
can be determined
b22
2
b1
sin i
from
b -2b cos i +1)/(1-2b cos(i -i)+b ))
2
S
2
1
sin(i -i)
S
(18)
=
b2
and
constants
vector
obtained
The
primer
and
(2) and dynamic relations (3) are
P1
=
p2
p
5
where
the optimum
2
2
(c2b (b -cos i)-c b (cos(i -i)-b 2))/(4ec1 c
conditions
(1)
b (b1-cos i)/c1- p1b,
= b 2 sin i /c
1
c =
1
p
=
0
.
=
1
b 2 -2blcos
1
1
have
p
from
'
b
where
i
+1
and
2
2=
1-2b cos(i -i)+b
We
also
18
The characteristic
velocity of the generalized Hohmann transfer
AV
4.2
2
Fb
- =
b cos i +1 +
b
-
2
csi
)i+ (19)
Study of the primer vector
When the eccentricity of the final
the
12
is
derivation
from
the
generalized
three impulse transfers.
orbit is small but not zero,
Hohmann
transfer
may
be
two or
The dominant terms of the primer vector and
its derivative can be expressed as
an
p 1 esin v
p1 * + p 2 /*
J1
p cos v /*
,,5
2
1
-
+2
p esin v
p sin v
If
we
assume
that
the
first
impulse
for a three impulse transfer, we obtain
is
p
1 /(
L + 2
=FI
2 ,(
=
M)
121
p2
5 -1 k1+e) -p
1 (1+eJ
2+2
+p2 )2
at
v =
0
as
in
section
3.2
19
L = 2+e 2 (1-2cos v -
where
cos 2 v)
M =
and
However, when we consider the limiting solution of
imaginary (since
=
We
of
type
that
conclude
discussed
transfers
are
only
transfers
perturbed
in
of
p 5 becomes
with the value
e p1 -(3+e)(1-e)
therefore
the
1)
v--.Pw,
.
p
e 4
v-2e Cos V).
V(-ecos
chapter
the
from
there
is
no three
3.
The
impulse
other
three
coaxial
type.
coaxial
and nodal type are
the
transfer
Therefore
two
impulse
impulse
dominant and
will be discussed in the next section.
If
nor
the
too
impulse
large
a,
change
order
),
and
the
the final orbit is neither
bi-parabolic
e,
i,
first impulse
we may put
cos(i -i) t:;
,
p
p^
0.
is
cos
expressed as
p esin v + p
=
1
3
( D (1+e) 2 + p )/(l+e)
p /(1+e)
dominates,
the
two
pP 5 of the transfer orbit.
W
S
0.
at
Since the
( less than 6* in zero
small
i
:
.
Also we have
The primer vector and its
non-dimensional velocity vector
1
close to zero
) requires determination of 11
to the case of
the
v
the
at
transfer
, v1 , v 2 ' p1 . ') .
W
solution is restricted
plane
v2
where
of
order solution ( es<<l
first
unknowns:
This
inclination
1
)
First order solution ( e s<
4.3
v1
and
v "" 0,
derivative
v2
can be
20
2
1
2
n
v - p
p esin
3
Lo
L
1
p (1-e2+ p )/(
~
-e)
p /(1-e)
rp (1+e)
2
+ p2 (1e
-
p esin v + p (1+e)
-
p (1+e) + p sin v
2
1
3
1
p (1-e) 2 + p(1-e)
P73T
- p esin v
2
Lp (e)
O
-
2
p (1-e)
3
+ p5 sin
esin v /b
A
1
~c1
AV1
Cos
i
1
sin i
Fe
2
AV
2
2
2
b 3 cos i - b
3
s
2
-
where
b
=
1+e
b2
b3
es'
sin v s/b3- esin v2/b2
b sin(i -i)
3 S
21
C=
c2
b-1
/b - 2b2 b3 cos i+ b.
The optimum conditions (2) give
0
X sin v + Y p - p p
1
2
13
45
(20)
2
2 3
2
where
= 0
p p
,
X sin v + Y p -
4 5
X = p e( p (1+e)
1
1
1
2
+ p (1+e))
2
- ep + p2 /(1+e)
2
5
X = p e( p (1-e)
2
1
1
2
+ p (1-e))
2
- Bep - p2 /(I-e)
2
5
Y = (1+e)( p(l+e) + p + 1)
1
1
(1-e)( p1 (l-e) + p
B
= (b3
B)
,
= -
+
Y
osi- b )/c
The dynamic relations (3)
give
( p,(1+e) 2 + p2 )/(1+e)
= 1
(21)
( p,(1-e)2 + p2 )/(-e)
p5/(l+e)
=
B
= sin i /c,
(22)
p /(1-e)
53
W sin
1
= b sin(i -i)
s
v + p
1
3
=
/c
2
0
(23)
W sin v -
2
2
p
3
R
2
22
where
W = e
1
( p - 1/(c b ))
W = e
2
( p + 1/(c b ))
1 1
1
1
22
)
R = e sin v /(c b
2
S
S
2 3
The geometric relations (4) and (5) give
a(l-e) = a.,
(24)
)
a(l+e) = a (1+e
S
S
sin i
The
=
(25)
0
sin( w + v2 )
of
solution
= sin iS
-
(20)
(25)
sin( w + v )
(26)
,
+ V
,
W
gives the two impulse transfer specified
by
=
(a
e
=
(a
sin i
(i
=
S
S
=
(1+e ) + a.)/2
I
S
(1+e
S
)
-
a.)/(a (1+e ) + a.)
I
S
S
I
c b2 b sin i
S
1 2 3
- V,
v1
p =b 2 sin i /c
1
1
5
=
(1+e-B(1-e))/(4e)
p
=
1+e- p1(1+e)
,
p
/ (c b2 b cos i + c b 2
1 2 3
S
2 1
)
a
23
p
3
=WX R / (W W (Y1 2 2
1 2 1
sin v=
sin v
p4
2
Y )-X W - X W)
2
1 2
2 1
-p3 /W1
=
( R + p ) / W
3
2
= ( X sin
41
v + Y p
1
1 3
2
p
5
The characteristic velocity of the two impulse transfer is given by
A
b2 -2b cos i + 1
1
J
+
b2b b cos(i -i)+b 2
s
2
23
(27)
24
CHAPTER
5
FIVE
Large Inclination And Eccentricity Approximations
from
transfers
impulse
is valid
for
to
is
objective
Our
large
a circular
inclination
approximation
an
determine
orbit
to
an
elliptic
and eccentricity.
to obtain uniformly valid
range
of inclination and eccentricity.
make
orbit,
which
and eccentricity
approximations
over a larger
These improved approximations
insight into the nature of the transfers which is needed
yield further
to
optimum
To achieve this,
we will extend the analyses of the small inclination
approximations
to
for
assumptions
reasonable
arbitrary
inclination
large
and
eccentricity.
small
The
from
order
at
the
coplanar
may be extended to second
.
The two impulse
Horner14
approximation
solution
H to a fixed point
from
transfer
inclination
of
K (Fig.9),
with plane change only
K gives the characteristic velocity
AV
=
I
AV
7(x
AV 1 =
where
+
1
-1)
(f
2+res -F
AV 2
x
=
r
=
s=-l
(28)
AV,
2
(1+i2 /2)
s
-r)/(1-r
((
))
(1-r2)/(-r/
(29)
,
a.(1+e cos v )/P
S
S
1
S
= a.
e sin v
S
/ (
a.
(1+e cos v) +J
S
= x2 - 1
2
2
a
= a x /(1-e
,
e
)
The transfer orbit is specified by
)j
-)
i
= 0
,
25
= 0
V
,
is arbitrary
W
v2= cos-'((rx2 -1)/e)
The primer vector constants can be easily obtained as
p1
=
2
1/ (x (1+r))
P2= r x / (1+r)
In Horner1 1, ' 0 and
.
p 5=0
5=
p 3=p 4
4
p3
AV 1 + A V2 is minimum in (29),
be applied in the regions
and
A V.
Equating
AV
1
of the inclination
2
1 =
a(l-e
p4 =
= -
The
regions
obtained
WS
and (28),
we have
<
w
P4
-
X.sin
i
e/(3+e)(1-e) /2
,
the limiting value
s
-
2( A V + AV
2
1
validity
of
p4
0
s
AV
EH
)AV)
2
WS
have
7r/2
:.e:
31r/2
been reduced
*
line of nodes and the in-plane firing angle
The
stated half is associated with the short transfers
the
to half
of
that
The remaining half are transfers symmetrical to
the
while
3w/2
and
,
in chapter 3.
transfers)
w
i
s
=
e(3+e)(1-e) /2
of
and
w/2
)
where
<
of (16)
I
1
( .sino
s
1
AV2
=
L
i
L
0
AV can only
remaining
half
is
has opposite sign.
associated
(less than 1800
with
the
long
1
b323Sb3
26
Local minimum solutions of the
transfers (more than 1800 transfers).
the
remaining
half
this
illustrate
AV
>
AV.
1
is
AV
for
i
order
solution;
the
w
>
S
We will
of
examples
i
L
approximation
final
orbit, wS
important when
is
eccentricity becomes effective.
where
numerical
and
half
L
eccentricity
of
stated
AV.
larger
four
first
the
we now have
for
argument
they have
the
AV
I
in
respectively
but
in
symmetry
small
pericenter
first
regions,
As in Fig.10,
Appendix C.
The
exist,
transfers
With
A
be
may
extended
the
if
in the
, can be included
the second order term in
2
5
= a (l-e )/(l+e cos v
S
S
S
S
)
short
long and
v is defined by
S
v
+
S
v
5
W.^:
,
if
sin w
,
if
sin
S
+
W'
0
S
0,
W
S
0.
.<
The geometric relations require
=
a
( AS + a.)
/ 2
1
= ( A
e
5
- a.)
/ ( A S + a.)
1
1
Except that the transfer
we
obtain
using
the
.
(30)
orbit elements
a and
e are defined by (30),
a further extension of the small
eccentricity
first
4.3.
order
solution
of
section
approximation
The characteristic
velocity, similar to (27), is given by
AV
=
b2 -2b cos
+
i
+
2
b
-2b b cos(i -i) +b 2
_
1
(31)
27
where
bi , b2
and b3 are calculated
from the new
AV
and
AV can be compared to determine
E
e
becomes significant.
are valid
The extensions
cannot
be sufficient
orbit.
However,
the limiting
only within second order,
provide
e of (30).
e
s
when
a
two
impulse
w
s
and therefore
transfers to a general elliptic
to describe
they
a and
transfer mode
final
(Fig.11)
with the following properties:
1.
The first impulse is close to the pericenter of the transfer
orbit, such that
0, and
(32)
i , such that
, and so
<i
cos(i -i)
S
.-
cos is
S
The second impulse is close to the line of nodes,
v
S
+
W VOW
S
7w
4.
p =
The first plane change is small compared with
i
3.
0, and so
1
1
2.
v 13
such that
as in Fig.lla,
(33)
as in Fig.llb.
(34)
Horner's solution can be used to estimate
v2 of the trans-
fer orbit.
a
give
= a (A cos v -a )/(A (1+cos
2 i
s
i s
v ) -2a )
i
2
,
The geometric relations (4)
e
= (a -A )/(A cos v 2-a )
i s
s
2i
.
(35)
and
vector
primer
The
velocity vector at
its
and
derivative
v2 can be expressed as
v1 and
p esin v + p
3
1
1
p (1+e) + p2/(1+e)
=
(p sin v + p )/(1+e)
5
1
4
p esin v + p cos v2
3
2
1
P 4 + p /$
2 2
1 2
=
-
p sin v (1+1/
2
3
2
*
(p sin v + p cos v )/
2
5
2
4
4
)
2
2
p (1+e) 2 + p2 (1+e)
P
- p esin v + p (1+e)
3
1
2
3
- p (1+e) + p sin v
1
5
4
3
=-
P2
-
p esin v + p (e+cos v 2
3
2
-
p (e+cos v
4
=
1
J7
Aici
sin i
2
e
esin v
--A 1
2 sinv
)
2
3
2 2
p1 2
-
cos i
) + p sin v
5
2.
the
28
nondimensional
29
e sin v
* Cos i S
s
C2
-*
where
S
1
2
s-
=2
2
S
-
=
c
2
sin(i -i)
2
$2=1+eosv 2
s = 1+ecos v
$)
is +
2cos
$
-1
AV2
'
2
v2
,
AV
2 esin
are known quantities.
The optimum conditions (2) give
5
sin v 1 )/(1+e)
X sin v + Y p + (p sin v +p cos v )(-p (e+cos
4
2 5
2
2 3
4
2
2
= 0
(36)
,
X1sin v1+ Y 1 p3 + (p4 sin v 1 +p5 )(-p 4 (1+e)+p
v )+p sin v )/
5
2
2
= 0,
,
2
(37)
2
- p 2e
X, = p 1 e(p 1(1+e) + p 2 (1+e))
x =p
(p
1
X2
*
2
V ) -p e(p
+p
2
1
2
2
Y, = p1 (1+e) 2 + p 2 (1+e)
y
=
2
The dynamic relations (3)
p1 (1+e)
+
p2 /(1+e)
+ 1+e
)
p
Cos v (p* 2 +
2 1 2
2
1p
1
+p
2
~J
2
2
,
where
'
+ (e+cos v )(p
2
2
1
+ P /i)
2
2
2
give
=
1
(38)
p1
2
+ p2
p3 sin v (1+1/
=
C) is (fscos
)/c,
p esin v + p = esin v /(c fl7
1
31
1
1
)
,
30
(39)
= (essin vs-
(p sin v + p )/(I+e)
4
1
5
=
esin
sin i / c
v2)/(c2
,
p esin v + p cos v
(40)
(p sin v + p cos v )/ $'
Since
v2 is estimated by
= -
I
sin(i -i)
Hornerts solution,
/ c.
p1
,
p2
' p3 and
v1
can be solved by (38) and (39), giving
p1etan v 2 +(e ssin vs
s esin v2)/(c 2 cos v2 r
(42)
)
p1 = (tf+g-(1+e))/( *2 +tm-(1+e)2
2
=
sin v1
where
g
(43)
(1+e)(1-p (1+e))
=
=
p3 cr~iT
/ (e-p 1 ec1
'2 (PsCos is -
= (e ssin vs -
t
=
m
= e tan
(44)
/
C2
t
2Sesin v2 Mc 2 cos v 2 Ws
)
f
)i
)
p2
(41)
,
= -
p
v2
*
2sin v 2(1+1/ i2
p
= n 1sin i + n
p
=
n sin i + n
,
We can now solve (40) for
p 4 and
p5 in terms of sin i, as
(45)
(46)
31
2= -
2 sin i / (c cos
2
is'
'2
2
n3=
cos i -(1+e)c cos v )/(c c cos v (tan v -sin
s
2
2
1 2
2
2
(1+e-cln sin v )c,
1
n4= -
1
v (tan v -sin v
1
v1
n2 sin
We can now solve sin i from (36),
A (sin i)
A =
where
(n2
1
B =
+ B sin
sin v
3
and (46);
(n n + n n /(1+e))
-
n n
1 3
-
sin v
3 4
1
sin v
-
n n
+ Y p -
n n
n n
2 3
1 4
1
2 4
1 3
p
7 unknowns
solution of the
i.e.
(47)
1
C = (X + n2 + n2 /(l+e))
1
2
4
The
(45)
= 0 ,
+ C
+ n2 /(1+e))
1 2
2
2
2
v
2
)
n = (c 1
1
12
)
where
p
,
i obtained from the 7 equations (41) ,sin
.5
guess
arily satisfy (37) because of our initial
D
,
, sin v
p
,
p
(47) does not necessv
.
Theoretically
the
However,
is satisfied.
such that (37)
2
solution obtained from (41) - (47) is accurate enough to initialize a
we
v
can recompute
good
set
velocity
of
of
input
the
two
for
the
computer
impulse
transfer
program.
for
a
The
general
characteristic
elliptic
final
s
2
(48)
orbit is given by
2
)
2 V-2 Iwcos(i -i)+
AV
=E
(t2+e-2q7
cos i
+
2
1+e
s
]s(2
32
SIX
CHAPTER
Computation Technique
6
Optimal control theory
6.1
This
He
formulated
only a
to
section
set
a
condition
of
necessary
necessary
He
further
found
positive
definite.
If
conditions
characteristic
explicit
the
th
Small's thesis
primer vector
a
quartic
to
relationships
when
this
independent
in
the
concept,
variable
rate
4
to have
and reduced these conditions
for
to another
velocity,
for
orbits,
Using
the
analysis
basic
conditions
from one impulse
developed.
the
the
single maximum on elliptic
definite.
be
the
explores
quartic
a
is
for
be
the
just
computer
chosen
of change
of
positive
switching
ceases
program
was
proportional
state and
to
to
adjoint
x
d.(x., u).
=
J
,
variables are respectively
J i
(49)
=,
. . .
,5)
aj.
J
S
where
_ 3H , (ij =1,
the
,
unit
vector
in
the
direction
of
thrust,
and
u,
the
angular position of the radius to the vehicle measured in the instantorbit
plane,
are controls.
A.d.
=
x. is an arbitrary set
The Hamiltonian defined by
of orbit elements.
H
The state
.
aneous
JJ
is
constant
X.d.
the
e
h
If
is
summed
optimal
as the
A.
(j
along
J
over
J
u is
1,
j = 1,
...
and may be normalized
, 5.
that which maximizes
radial
=
an extremal,
...
and
,
jJJ
adjoint variables
X.
J
angular momentum
5)
are
functions
The optimal
-
)
2
.
8
to unity.
is
XA.d
R
R
unit vectors and
e=
eh
L
h
x.
of the state variables
(2. d.
Jj
We define
And
,
and
and
J
eR.
R*
and
X.d.
A
=
A ) e R33
sin (u +
1 si
2
[ A 3 + (1+$)
+
A 1 cos (u+A 2
eL
(50)
-1
+4i
[ A4 sin (u+A5
eh
H requires that
The optimal u which maximizes
2
-
[X.d.(u)]
J J
-2
Au other
for all
0
[.d.(u+Au)]
JJ
-
0,
than
21
(51)
,...... This maximizing test can be
written in the form
2
a
0
for
+ a t + a2 t
1
all
t
functions
tan(
=
3
4
+ a 3 t + a 40t
Au/2
)
and
of the eight parameters
0
ai(i
A.
=
,
0,
...
e, v, and
,
4)
u.
are
explicit
#
a0
If
0 and
let
Z, = 4a
a4 - a1 a3
z 2 CZ +a
2
Z2
2 = ay
1 a4 + a9o a3
3
then,
the
test
for
the
a
a1 a2
a 2 a33
positive
quartic
definite
can
be
expressed
completely by
a
>
0
(52a)
3 2
2
Q(a.)
The
left
=
Z1 [(a 2
member
of
Z1 )2 +9a 2 Z2 ] -a2 Z2
(52c),
Q(a.
),
switching condition which represents
2
27 Z
equals
zero
/4
>
(52c)
0.
corresponds
a double maximum of
to
the
H, and will
be violated before (52a) or (52b).
The
case
a0 >0
corresponds
case corresponds to impulsive extremals.
to either
impulsive
firing
included
in
The
the
a
= 0
testing
34
conditions
(52)
or
non-optimal by Marchal
condition
switching
The
firing
continuous
15
be
can
was
which
reduced
a
to
It is desired to determine
independent parameters.
From (51), we have
222
2 =4cos 2t (ao+a1t+a2
proved
function
be
to
of
five
a. from a set of
five independent parameters.
=
4
3
+a3t +a4t
)
-Xd(u)] 2 X]
[X.d.(u)] -I[X.d.(u+Au)]
02
(1+t 2)[*-2te sin v + (2-,)t2 2
(53)
and the particular set of
a
t
a s is
2
2
2
[1-4k -(2+ *k )tan *
=
a, = 44
tan
]
-
.2
j
(2k-cos y)-2esin v (1-2 tan 2
a3= a1 +8(-1)tan
The
= [(3-4))
2
cos y-2k][2k-cos y] - (*-1) sin y
firing angles having the
limits of
0 4 y.4
, 0
accord with the geometry and cos 4
]
5
.
a
2
2
(cos y-k)+4esinv [(cos 1-k) -tan 2
]
,
a 2= a +a +8esin v tan *(cos y-k)-4($-1)[(cos y-k) 2-tan 2
and
2w
W/2
4lr/2
The thrust vector
in (53).
is defined by
and the parameters
k and
cosy
j
eL + cos4
sin
y eh
(55)
are defined by
cos (u + A2)]/
k
5
[A 3 + A
j
S
{A 4 sin (u+A5 ) sin $ + A
* cos
*
)
sin 4 eR + cos4
=
cos (u+A5 ) cos
*
cos Y
+ [A1 cos (u+A 2 ) esin v - A1 sin (u+A 2 ) ecos v]
cos 4 sin y} / cos 2
[A4 cos (u+ A 5 )/cos *+
esin v sin i ] cos y
+ [tan 4 - k e sin v] sin y
.
=
(56)
35
On using
H
= 0, we have the functional relationship
U
j sin y + k e sin v
tan
*
(57)
1 + k $cos y
can be chosen as
A desirable set of independent parameters
( e, v,
=
1/2
set
an initial
Having
.
,
y, k
1/2
)
q
q1
now define
we
,
nondimensional
the
*
independent variable
h
T=
-
AV
which is desired to yield
after
impulse
an
The radius
( e
axes
an impulse,
during
for
determined
,
therefore
e
h /*
given
T
h*
and
h
is a
R
XV.
to the
Figure 12 remain constant
are constants.
terms
in
=
firing with respect
f
and
2
set
.
per unit mass
) in
2
,
expressed
increment
velocity
a
R and the direction of
vector
orthogonal
fixed
be
value of the angular momentum
reference
the
can
T(r
) so that the independent
=
of
the
Equating
instantaneous
elements and that of the impulse, three integrals result
[f$'
cos
y
~
sin
h AT
]
cos
(58)
*
A
=
*
e sin v
[F
sin
These determine
in terms of
during
AT
y ]
=
0
the three independent
.
A
.
h
an impulse.
The remaining two
Thus,
we
have
parameters
1/2
k*1
and
q
=
( e, v, Y ) of q
.- 1/2
j41 are constants
q( AT ) which leads
to the
36
important functional relationship
(59)
a.i(AT)
=
(
a.
F
Finally, (52c) and (59) give the switching condition in the form
>
Q(Ar)
(60)
0
where the left hand
in
h
6.2
side can be expressed as a
12th degree polynomial
1/2 AT /h*
Two point boundary value problem
An optimum orbit transfer
which
may
the initial adjoint variables
be solved if
the
general
i
relates
=
( p
an input
variables
initial
A.
adjoint
,
1
adjoint
the
vector
are known.
p
are
In
arbitrary
A linearized solution establishes an estimated
independent variables.
set of
of
elements
five
is a two point boundary value problem
...
),
p
,
_5
set of
q
defined
in
( e, v, Y
=
(49).
A.
variables
the optimal control
while
Therefore
are
formulation
1/2
. -1/2
) to the adjoint
,,k$1 , jiP
known
if
an
the
estimated
set
of
input
can
be
q
J_
calculated from the estimated
p
V,
A
First
we
define
elliptic orbit (Fig.13).
R,
,
AR , K, 4, 3, of the first transfer
37
=
P/) eR
esin v
0
+
p cos
.3
p
P1
+
/$
+ p 2 si
- p sin v (1+1/
V/
+ p5 cos v /
*
Lsi
2
p 4, + p2
p3, sin v
-
p esin v + p (e+cos v)
-
p (e+cos v) + p5sin
p sin v + p cos v
p cos v 4
AKF
-
p e
3
,.
4=
e
0
3=
4,
'
1
$
V
v
)
p esin V
-
sin i
L
cos i
0
p sin
5
v
v
38
i
is the same as the velocity adjoint
x
If we define the state vector
= -
1.
~R . I
- *
=
V
.V
cos v /*
V'(sin
h1
v
eR+ cos v
eL
'
Xe
.R-
R
,
=2
+
S
o
) where
S
,
then by contact transform, we have
h,
L
W
(61)
h'
,
L = log
( L, e, i,
=
.
where the primer vector
=
K
4
=
K. 3
and
N
I sin v e
V
cos v /e
(62)
)cos v )+ x(1+ * )sin v /e
~ (L+ X (e+(+
/
X~4.cos u +( ) -(x +x)cos i)sin u
A
i
It
is
=
d.
0
-
X
sin
ij
that
Small
proved in
U
)
h
X V / h*
V
and so (50) and (62) give
Q
=
1
-A
2
0=
O
'3
cos A
A
1
=
112
A
3
= h ( X
2
sin A
h ( X
cos
sin
03/
e)
/ h*
e
=
L
h ( X sin
e
+ e
e
)/
03+
h*,
X
cos oi/ e) / h*,
(63)
39
Q
5
=h( X
A 4 cos A 5
Q4
. / h*
=h
= A sin A
5
4
cos i )/ (h* sin i)
-
1
k give
The definition of the firing angles and
cos$
A 4 sin( u+ A 5
siny
cos
A 4 sin(u+A 5
=
k
=(
A3 +
A
cos(
sin $
=
cos $
1/2sin y
Q1 sin u - Q 2 cos u
u + A2
= ( Q 4 sin u + Q 5 cos u ) /
e
Now with an initial circular orbit,
constants
-1/2
(65)
= ( Q3 + Q 1cos U + Q 2 sin u ) / *-1/2
* 1/2k
q= (
is reduced to
(64)
)
Finally (63) and (64) give
on using (50), (55) and (56).
cos $
)'
/
=
,
sin
an
during
, k.
impulse
, j. ).
and
u =
v.
0 ,4)=1
$
Since
A
+
V,
,p /2s
=
0 ,
,
the input
are
,1/2k
(65)
can
rewritten as
1.
Q2
= -
'
sin
cos *. sin Y.
1
1
cos *. k.
11
i3
=
=
/ *-1/2
Q
5
( 0 + Q
1
) / *-1/2
Equations (57) and (66) enable us to compute the input
(66)
be
40
q
which
in
=
( 0 , 0
turn
,
gives
.
the
k.
,
j. )
initial
point boundary value problem.
(67)
adjoint
variables
to
solve
the
two
41
CHAPTER
7
SEVEN
Conclusions
The
ion
to
general
linearized
The
transfers.
of
this
solutions
or
analysis
of
analysis
general
simplified,
but
in
solution.
However,
the
provides
thesis
we
of
approximations
transfers
still
cannot
approximations
a
from
impulse
optimum
circular
obtain
are
formulat-
a simple
a
analytic
unique
estimates
excellent
is
orbit
since
the numerical solutions have rapid convergence.
An
only
important
exist
at
the
transfers
cannot
transfers
from
long
theoretical
transfers
is
vicinity of Hohmann
provide
a
result
minimum
circular
as
are
orbit
that
3
transfers
and when
The
solutions.
are
not
illustrated
simple
in
the
of
Small
impulse
due
first
transfers
the Hohmann
symmetries
to the
four
of
short
and
examples
in
Appendix C.
The extremal
condition
quartic
using
defined
equation
double
computed
generation
by
contribution
can generally
some
of
approximations
this
and
There
are
often
not
compute
mations.
every
approximations
of
The
is
solution
8 digits
are
The practical
of
scheme of
this
solutions
space.
combination
of
of accuracy
numerical
state
generation
the
changes
matching
the
of
this
between
characteristic
(for
exact
in
the
analytic
however,
trajectory
The
The
the
analytical
Small
to provide
solution.
combinations
possible,
the
errors
extremal
different
inclination.
equation.
therefore
thesis
the
consists of a switching
be computed up to
acceptable
the exact numerical
and
quartic
precision,
with
different
a
scheme
problem
any
approximations
in semi-major
these
is
not
axis,
approximations
necessary
obtained
achieving
the
with
eccentricity
since
two families of analytic
velocities
of
analytic
associated
from
same
we
is
can
approxi-
two
analytic
final
orbit),
42
are
local
minimum
but
approximations,
some
near
are
trajectories
In
solutions
two
families
analytic
approximations
case,
this
transfers
may
two
to
lead
different
of
families
if the
analytic
solutions are identical.
the
from
farther
solutions.
numerical
different
boundary
different
the
approximations,
analytic
of
between
In
for
of
slightly
numerical
respective
especially
cases,
boundary
the
the
may be
which
one is
and the better
both are local minimum solutions
obvious.
limits
The
the
validity
of
define,
although
to
difficult
in
the changes
of
analytic
different
the
close
of
Results
compute.
are
approximations
trajectories
some
inclination
and
eccentricity
extensive
not
computations
are
limits
obtained.
When the eccentricity of the final orbit is less than
and/or
inclination
the
of
the
final
orbit
is
to
are
the
to
difficult
for
less
than
.9,
60*,
the
of
the
numerical solutions usually converge rapidly.
This
parking
orbit
guidance
to
an
thesis
the
improves
problem
technique.
which
could
Knowledge
elliptic orbit
in
theoretical
understanding
provide the basis
of
transfers
3-dimensions
could
from
for a subsequent
a circular
improve
the
orbit
operational
envelope and fuel economy of the space tug.
A good
transfers in
4,
a
the
project
3 dimensions.
parametric
simple
problem
research
figure.
will
be the general
ellipse to ellipse
Since the number of input parameters is
space will no longer be visualized or constructed in
A
similar
approach
as that of the circular orbit
should give us some theoretical understanding,
solution will not be forthcoming in the near future.
but the general
43
Appendix
Optimum single impulse transfers from circular orbits
A
This appendix continues
basic
of
analysis
the study of Der t s M.S.
space
parametric
and
space.
considered
in
intersect,
the firing angles and the normalized
can
be
state
determined
precisely,
unique.
necessarily
but
Figure
14
between two intersecting orbits.
then the change in
h '/h =
=
D
and
I
=
T
),
adjoints
are
not
physical
transfer
T
h cos
h is
+ RX oA
(1+
D cos y /fr)
cos E eh
cos i'
,
~
-
= 3
D sin y
/fi)
eL
But
sin E eL
eh = cos
.
and
a
,
orbits
If we define
(values after a firing are primed).
eL
(y
AV,
initial
illustrates
two
When
is
*
C = h'/h
the
The
.
from a circular orbit
transfers
optimum impulse
9
thesis
Therefore we have
C cos i' = 1 +
D cos y /F
C sin i' = D sin
Since
C and
D
are given, we can determine
C
I/F =
cosy
With a fixed
have
i
y/F
f
T
- 2 C cos i'
( C cos i'
-
+ 1
,
(Al)
1 )/ D .
(A2)
and the given initial
and final
states
(x
,
x'), we
44
*
tan
'sin v'/
=
- e sin v/
)I D
(A3)
on using the first equation of (58).
From (57)
we also have
*
tan
= (j sin y + k e sin v)/(l + k * cos y
(A4)
.
above, we have
D
*
Finally from the definition of
)
T
$ D /(h cos # ).
h
=
the
When
=
and
orbit
initial
(h/h)
IC2 - 2C cos i'
D
= (C cos i' -
tan *
=
e'sin v' /(DC)
tan
=
j
=h
a
for
plot of
single
values of
-1/2j
of (*-1/2
,
(i)
(ii)
(iii)
1/2k
(A9)
,
(Alo)
,
,
t
T
)
) uniquely from
from
a circular
in Figures
(A7), (A8) and
orbit,
15a,b reveals
are not unique in general.
but
the
that the
The solutions
can be:
*
max
(A8)
.
,
and *1/2k
1,
=
i
(A7)
,
impulse transfer
1/2
1/2
(*
0,
(A6)
+k cos y )
( y
We can determine
(AlO)
)/D
)
D/( cos
1, e =
,
cos
T
h =
(Al) to (A5) can be simplified to
+
sin y /(1
then
circular,
is
h'.
=
(A5)
>
02
, no single impulse solution,
, a unique solution for (-1/2 j1
=
0
, a family of solution for (*-1/2
2k),
1/2k).
shows
are the unknowns
the non-uniqueness
in this equation.
Figures
15
a,b also
(initial adjoints)
multiple
impulse
within the boundaries
For
to
But
solutions since
j
and
j
k are adjoints,
k
thus
ki
transfers
from
of
of the input variables
show the boundaries
j* 1/2 and
discussion on
9
M.S. thesis.
further
Der's
the
adjoints are not unique in general.
the initial
All
of
j- 1/2and
these
1/2at an out of plane angle
a circular
k$1/2
boundaries,
orbit must
originate
associated with the
the
reader
y
.
also
should
y
.
(A9)
45
and
refer
46
subroutines
3
Appendix
PEST,
XKUJ,
MAINE,
subroutines
Small
4
and OUTPUT
MATX
circular
orbits.
SWITCH,
TIME
START,
are
program
applicable
only
are
DTDU
2 and
to
extremal-generation
The
and
four
the
and
RUN
developed
by
The program logic is illutrated in Table Bl.
.
The
extremals
at
with the nominal
derivative
and
are
trajectories
will
matrix which
variables
adjoint
the beginning
of
to
run
the neighboring's
at
i=2,3,4.
required
are
for
the
computing
partial
used in a 3-dimensional Newton method.
be
for
of
each
are stored at
I extremals
the
the J impulses in the array S(I,J,K).
of
each
and
I=1
(I)
arranged
subroutines
extremal-generation
neighoring
State
main
The
C.
from
transfers
impulse
The
of
results
generate
to
used
program
computer
the
is
listing
following
The
the
Computer listings
B
Appendix
The K
variables are identified in Table B2.
of computation is
The procedure
specify in
We first
1.
I =1,
J1
summarized
as follows:
RUN
,
IM = 4
JM = 2 or 3,
,
= an initial
state (a circular orbit),
= a final state (an elliptic orbit).
S
RUN calls PEST to estimate the primer vector
x
j
=
1,...,5,
and
PEST calls
XKUJ to compute
the
constants
input vector
p.,
J
q
of (67).
2.
RUN
calls
MAINE
extremal-generation
initial
calls
(52).
values
SWITCH
of
to
which
is
scheme.
the S(I,J,K)
check
the
the
driving
MAINE
calls
array
input
subroutine
START
to
for K = 1,......,20.
satisfies
the
of
the
set
the
START
inequalities
of
47
3.
order
first
which
update
the
not
next
4.
the
step 3 is
JM,
impulse J+l.
calculate
I=1,
at
the given final state and nominal
x
xz=
to
Au,
find
T, by a
check
the
and
calls DTDU
the
exact
If
T
to
J+l
of the
MAINE returns to RUN.
the
the
to
impulse J+1.
next
repeated
to
trajectory
nominal
the
J+l equals JM,
If
OUTPUT
RUN calls
for
array
AV,
found, MAINE calls
is
T
angle
the coast
calculates
S(I,J,K)
equals
When the exact
(52).
inequalities of
SWITCH
calling
and
method
Newton
SWITCH
to find the exact normalized
TIME
calls
MAINE
final
norm
of
state
x(I).
For
the difference
of
final state
-X(i)
is computed.
5.
If xz
step
2
not
is
4
for
each
increment
of
the
input
method.
With
may
ical
to
be repeated
than
less
the
Then
I.
vector
new input
until
10
xz
solution is acquired.
is
,
we put
MATX to compute
RUN calls
6q
vector
less than
by
( q
10
I=2,3,4 and repeat
a
3-dimensional
+
6q
the
Newton
), step 2 to 5
and the exact numer-
48
Table Bi
Diagram
Flow
I
RUN
MATX
PE ST
I
-
XKUJ
MAINE
I
S TART
f
,F
I:
TIME
SWITCH
DTDU
OUTPUT
49
Table B2
The variables stored in the S(I,J,K)
s(I,J,K)
K
S(I,J,K)
1
T-
11
* 1/2 (cos y - k)
2*
AT
12
cos(u+ 1
13
sin(u+ A
)
1/2 tan(Au/2)
)
K
3*
*
array
(h/h*) *-1/2 cos
14*
cos Au
5
*- 1/2 esin v
15*
sin Au
6
k *1/2
16
cos i
7
tan
17
sin i cos u
8
A4 cos(u+ A 5)/cos
18
sin i sin u
9
*1/2 sin y
19
sin i cos
10
'p
20
sin i sin
Information
for the
'
*
4
J th impulse is stored in
S(I,J+1,K)
00010 C RUN
00020 C
00030
IMPLICIT REAL*8(A-H,O-Z)
00040
00050
DIMENSION S(4,7,20) ,DP(5) ,X(4,4) ,DY(3) ,P(5)
COMMON/KLN/JM
00060
00070
00080
00090
00100
00110
00120 51
00130 81
00140 82
COMMON/STR/S
COMMON/BIN/BPOBPTE1,V1,PI,DTFTAUl
COMMON/FCOMP/X
COMMON/FIN/H2,ECES,SNCV1
COMMON/TIN/ETVTWTPTSTSIT,OM
COMMON/ONM/TPHI ,CUPSUP
FORMAT(4G10.7,I2,GI0.7)
FORMAT(4D10.5,I2,D10.5)
FORMAT(/' CASE',13,//' INPUT',/,' H2
00150
00160 83
JM
SIN I
1ESINW
FORMAT(4(D12.5,3X),15,8X,D10.5)
00170 91
00180 92
00190
FORMAT(' OUTPUT')
FORMAT(/' H**2
UP
PHI
2
ECOSW
DTAU')
00200
IM =4
00210
DP(2)
00220
DP(3) =1.D-6
00230
DP(4)
00240
Ox =100
00250
00260
00270
ZZ =0.DO
ONE =1.DO
PI =DARCOS(-ONE)
00280
00290
KN =100
El =0.DO
00300
VI
00310
00320
00330
00340
DO 1 I =1,4
DO 2 J =1,7
DO 3 K =1,20
S(I,J,K) =I*4+J*7+K*20
=1.D-6
=1.D-6
=0.DO
ESINW
ECOSW
CLIM')
SIN I
TAU
V
00350
00360
00370
00380
00390
00400
00410
00420
00430
00440
00450
00460
00470
00480
00490
00500
00510
00520
00530
00540
00550
00560
00570
00580
00590
00600
00610
00620
00630
00640
00650
00660
00670
00680
3
2
1
CONTINUE
CONTINUE
CONTINUE
C
DO 79 NN =1,10
READ (5,51) H2,ECESSNJMCLIM
C
11
12
WPITE (6,82) NN
WRITE (6,83) H2,ECES,SN,JMCLIM
IF (JM-1) 12,12,11
CALL PEST(P,PS,PUPK,JM)
WRITE (6,91)
WRITE (6,92)
C
C
17
MULTIPLE IMPULSE TRANSFERS
IF(PS.EQ.0.DO) GO TO 18
PHI =DARSIN(PS)
PC =DCOS(PHI)
UP =DARSIN(PU/PC)
GO TO 19
18
19
PHI =DARSIN(PS)-PI*.001D0/180.D0
PC =DCOS(PHI)
UP =DAPSIN(PU/PC)-PI*.lD0/180.D0
SIUP =DSIN(UP)
XK =PK/PC
IF
21
(UP) 22,21,22
XJ =0.DO
GO TO 23
22
XJ =DTAN(PHI)*(l.DO+XK*DCOS(UP))/SIUP
23
DO 76 KL =2,KN
73
DP)
CALL MAINE (ElVlXKUPXJONEIM,JM,
CALL OUTPUT (1,JM,0,XXH2,0,OX)
IF (DABS(X(3,1))-.D-5) 73,73,74
X(3,1) =ES
H
00690
00700
74
XZ =DSQRT((EC-X(2,1))**2+(ES-X(3,1))**2+(SN-X(4,1))**2)
GO
TO 77
IF (OX.GE..179995D3.AND.XZ.LE..5D-4)
75
CALL OUTPUT(2,JM,0,XXH2,0,OX)
CALL OUTPUT(3,JM,0,XXH2,0,OX)
CALL OUTPUT(4,JM,0,XXH2,0,OX)
CALL MATX(ECES,SN,DY)
XK =XK+DP(2)*DY(1)*DMIN1(ONECLIM/XZ)
UP =UP+DP(3)*DY(2)*DMIN1(ONE,CLIM/XZ)
XJ =XJ+DP(4)*DY(3)*DMIN1(ONE,CLIM/XZ)
CONTINUE
S(1,JM+1,2) =DTF
IF(XZ-1.D-7) 77,77,75
00710
00720
00730
00740
00750
00760
00770
00780
00790
00800
76
77
00810
DO 78 J =1,JM
00820 78
00830
00840 79
00850
CALL OUTPUT(1,J,1,ZZXX,1,OX)
CALL OUTPUT(1,JM,0,XXH2,2,OX)
CONTINUE
END
U,
00010
00020
SUBROUTINE PEST(P,PS,PU,PK,JM)
IMPLICIT REAL*8(A-H,O-Z)
00030
00040
DIMENSION P(5),V(50),X(50),Y(50),Z(50),DF(50),F(50)
COMMON/FIN/H2,ECESSN,CV1
COMMON/TIN/ETVTWT,PT,STSITOM
00050
ONE =1.DO
00060
PI =DARCOS(-l.DO)
00070
P12 =.5D0*PI
00080
FIRST ORDER ESTIMATE OF
FORMAT (/'
00090 10
INITIAL STATE
00100 C
AI =1.DO
00110
EI =0.DO
00120
00130
00140
00150
00160 C
00170
00180
00190
00200
TAU
SI =0.DO
WI =0.DO
XMI =0.DO
FINAL STATE
E =DSQRT(EC*EC+ES*ES)
W =DATAN2(ES,EC)
S =DARSIN(SN)
A =H2/(1.DO-E*E)
00210 C
TRANSFERS, GENERAL
00220 C
TWO IMPULSE
00230
00240
00250
00260 11
00270
00280
00290 12
00300
00310 13
CIS =DCOS(S)
SIS =SN
IF (W) 11,11,12
ECVS =EC
ESVS =-ES
GO TO 13
ECVS =-EC
ESVS =ES
OM =W
00320
00330
00340
IF (OM.LE.PI2.AND.OM.GE.-PI2) ONE =-l.DO
IF (JM.EQ.3) ONE =1.DO
ES =ONE*ES
=',D15.10,/)
00350
00360
00370
00380
00390
00400
00410
00420
00430
00440
00450
00460
00470
00480
00490
00500
00510
00520
00530
00540
00550
00560
00570
00580
00590
00600
00610
00620
00630
00640
00650
00660 14
00670
00680
EC =ONE*EC
BU =1.DO-E
BV =1.DO+E
AT =.5DO*(A*BV+1.DO)
ET =(A*BV-1.D0)/(A*BV+1.D0)
BS =1.DO+ET
BT =1.DO-ET
Bl =DSQPT(BS)
B2 =DSQRT(BT)
B3 =DSQRT(BU)
Cl =Bl-1.D0
C2 =DSQRT(BT-2.DO*B2*B3*CIS+BU)
BB =(B3*CIS-B2)/C2
P(1) =.25DO*(BS-BB*BT)/ET
P(2) =BS-P(1)*BS*BS
SIT =CI*BT*B3*SIS/(C1*BT*B3*CIS+C2*BS)
Gi =Cl/(B4-P(1)*ET*Cl)
G2 =C2/(P(1)*ET*C2-B5)
RHO =(1.DO+ECVS)/H2
R2 =DSQRT(2.DO/(1.DO+RHO))
S =ESVS/(1.DO+ECVS+P2*DSQRT(H2))
DS2 =DSQRT(1.DO+S*S)
Xl =R2*(DS2-RHO)/(1.DO-RHO*DS2)
ETI =X1*X1-1.D0
CV2 =(RHO*X1*X-1l.DO)/ET1
SV2 =0.DO
IF (ES.NE.0.DO) SV2 =DSQRT(1.DO-CV2*CV2)
Wl =ET*(P(l)-l.D0/Cl/Bl)
W2 =ET*(P(1)+1.D0/C2/B2)
R22 =ESVS/C2/B3
IF (JM-2) 14,14,15
P(5) =-Bl*B1*SIT/Cl
Xl =P(1)*ET*BS*(P(1)*BS+P(2))-P(2)*ET+P(5)*P(5)/BS
X2 =P(1)*ET*BT*(P(1)*BT+P(2))-BB*ET*P(2)-P(5)*P(5)/BT
00690
Yl =BS*(P(1)*BS+P(2)+1.DO)
00700
Y2
00710
00720
00730
00740
00750 15
P(3) =Wl*X2*R22/(W1*W2*(YI-Y2)-Xl*W2-W1*X2)
SV1 =-P(3)/W1
P(4) =(X1*SV1+Y1*P(3))/P(5)
GO TO 16
P(5) =0.D0
00760
P(3)
00770
SV1 =0.DO
00780
00790
00800
P(4)
P(1)
P(2)
00810 16
SV2 =(R22+P(3))/W2
=-BT*(P(1)*BT+P(2)+BB)
=0.DO
=-.5D0*ONE*ET*B2*DSQRT(3.DO+ET)
=.5DO
=.5DO*BS*BT
00820
CV2 =-DSQRT(1.DO-SV2*SV2)
00830
VT =DARSIN(SV1)
00840
00850
00860
WT =-VT
ST =1.DO+ET*DCOS(VT)
PT =AT*(1.D0-ET*ET)
00870
00880
00890 17
00900
S2 =1.DO+ET*CV2
IF(JM-2) 17,17,18
CC =Cl+C2*DSQRT(S2/BS)
GO TO 19
00910 18
CC =Cl+B2*(B3-B2)/Bl+DABS(P(4)*DSIN(W))*S/Bl/B2
00920 19
WRITE (6,10) CC
00930
00940
00950
CALL XKUJ
RETURN
END
(PPSPUPKJM)
1-l
U,1
00010 C XKUJ
00020 C
SUBROUTINE XKUJ (PPSPUPKJM)
00030
IMPLICIT REAL*8(A-H,0-Z)
00040
DIMENSION P(5)
00050
00060
00070
00080
00090
00100
00110
00120
00130
00140
00150
00160
00170
00180
00190
00200
00210
00220
00230
00240
00250
00260
00270
00280
00290
00300
00310
00320
00330
00340
COMMON/FIN/H2,ECESSNCV1
COMMON/TIN/ETVT,WT,PTSTSIT
QP =DSQRT(PT)
QS =DSQRT(ST)
SW =DSIN(WT)
CW =DCOS(WT)
SV =DSIN(VT)
CV =DCOS(VT)
UT =WT+VT
SU =DSIN(UT)
CU =DCOS(UT)
R =PT/ST
VR =ET*SV/QP
VL =ST/QP
YR =P(1)*ET*SV+P(3)*CV
YL =P(1)*ST+P(2)/ST-P(3)*SV*(1.DO+1.DO/ST)
YH =P (4) *SV/ST+P (5) *CV/ST
ZR =(P(1)*ST*ST+P(2)*ST-P(3)*ST*SV)/QP/PT
ZL =(-P(2)*ET*SV+P(3)*(ET+CV))/QP/PT
ZH =(-P(4)*(ET+CV)+P(5)*SV)/QP/PT
XKR =(P(4)*SV+P(5)*CV)/QP
XKL =(P(4)*CV-P(5)*SV)/QP
XKH =-P(3)*ET/QP
SX =SIT
CX =DSQRT(1.DO-SX*SX)
XL =2.DO*R*ZR-(YR*VP+YL*VL)
XE =-R*ZR*CV/ST+(SV*YR+CV*YL)/QP
XW =XKH
XI =XKR
k-n
00350
XM =XKL*SX+XKH*CX
00360 C
00370
Ql =(XE*CW-XW*SW/ET)*QP
00380
Q2 =(XE*SW+XW*CW/ET)*QP
00390
00400
00410 11
00420
00430
00440 12
00450
00460 13
00470
00480
00490
00500
Q3 =(XL+ET*XE)*QP
IF (SIT) 11,11,12
Q4 =0.DO
Q5 =0.DO
GO TO 13
Q4 =(XM-XW*CX)/SIT*QP
05 =XI*OP
PS =Ql*SU-Q2*CU
PU =(Q4*SU+Q5*CU)/QS
PK =(Q3+Ql*CU+Q2*SU)/QS
RETURN
END
00010 C MAINE
00020 C
00030
00040
00050
00060
SUBROUTINE MAINE (EV,XKUPXJHRIMJMDP)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION S(4,7,20),T(30) ,DF(30),F(30),ZK(30),ZJ(30)
DIMENSION D(4),DP(4)
00070
COMMON/STR/S
00080
00090
00100 91
00110
00120
COMMON/BIN/BPOBPT,E1,V1,PIDTFTAUl
COMMON/ONM/TPHICUPSUP
FORMAT (6D15.10,' FAILED GENERAL TEST')
ZZ =0.DO
X =0.DO
00130 C
00140
CALL START
00150
00160 110
00170 111
00180 112
00190
IF (NF) 220,220,110
IF (IM-1) 210,210,111
DO 112 K =2,4
D(K) =0.DO
DO 113 IA =2,IM
00200
00210
D(IA) =DP(IA)
CALL START (E,V,XK+D(2),UP+D(3),XJ+D(4),HRIA,1,NF)
00220
00230
00240
00250
00260
00270
00280
00290
00300
00310
00320
00330
00340
113
210
211
212
213
214
(E,VXKUPXJ,HR,1,1,NF)
D(IA) =0.DO
J =0
BPO =(DSQRT(2.DO*(1.DO+E))-l.DO-E)/HR
J =J+1
DO 218 I =1,IM
IF (I-1) 213,213,212
CALL TIME (I,JJMS(1,J+1,2),S(1,J+1,2)/100.DOS(IJ+1,2))
GO TO 218
IF (JM-1) 214,214,217
DXK =1.D-5
N =1
ZK(1)
ZJ(1)
=XK
=XJ
00350
CALL TIME
(IJJM,0.DO,SPANT(l))
00360
F(l)
00370
ZK(2)
00380
00390 215
ZJ(2) =TPHI*(1.DO+ZK(2)*CUP)/SUP
CALL START (E,VZK(N+1),UPZJ(N+1),HR,1,1,NF)
00400
00410
00420
00430 216
00440
CALL TIME (I,JJMT(N),T(N)/100.DOT(N+1))
F(N+1) =T(N+1)-TAU1
GO TO 221
IF (DABS(F(N+1)).LE.1.D-5)
DF(N+1) =(T(N+1)-T(N))/(ZK(N+1)-ZK(N))
DZK =F(N+1)/DF(N+1)
00450
N =N+1
00460
00470
ZK(N+1) =ZK(N)-DZK
ZJ(N+1) =TPHI*(1.DO+ZK(N+1)*CUP)/SUP
00480
GO TO 215
00490 217
00500 218
CALL TIME (I,JJMZZSPANS(1,J+l,2))
CALL SWITCH (I,J,JMS(IJ+1,2),X,NX,0)
00510
00520 219
IF (J+1-JM) 211,219,219
RETURN
00530
00540
00550
220
221
=T(1)-TAUl
=XK+DXK
WRITE (6,91) E,VXKUP,XJHR
RETURN
END
U,
00010 C START
00020 C
00030
00040
00050
SUBROUTINE START (EFZKUAZJHR,I,J,NF)
IMPLICIT REAL*8 (A-HO-Z)
DIMENSION S(4,7,20)
00060
COMMON/STR/S
00070
COMMON/TIN/ETVTWT,PTSTSITOM
00080 C
00090
00100
00110
00120
00130
00140
00150
00160
00170
S(I,J,12)= DCOS(F)
S(I,J,13)=
DSIN(F)
S(IJ,10)= 1.DO+E*S(I,J,12)
PAD= DSQRT(S(I,J,10))
S(I,J,5)=
E*S(I,J,13)/RAD
S(I,J,9)=
RAD*DSIN(UA)
RAD*DCOS(UA)-ZK
S(I,J,11)=
S(I,J,7)= (ZJ*S(I,J,9)+ZK*S(I,J,5))/(1.DO+ZK*(ZK+S(I,J,11)))
S(I,J,4)= HR/DSQRT((1.DO+S(I,J,7)**2)*S(I,J,10))
00180
S(I,J,6)= ZK
00190
00200
00210
S(I,J,8)= ZJ*(ZK+S(I,J,11))+(ZK*S(I,J,7)-S(I,J,5))*S(I,J,9)
CALL SWITCH(I,1,JM,O.DO,QX,NF,1)
IF (NF) 202,202,201
00220 201
00230
S(I,J,1)= 0.DO
S(I,J,2)= 0.DO
00240
S(I,J,3)= 0.DO
00250
00260
00270
00280
00290
00300
S(I,J,14)=
S(I,J,15)=
S(I,J,16)=
S(I,J,17)=
S(I,J,18)=
S(I,J,19)=
00310
S(I,J,20)= 0.DO
00320 202
RETURN
00330
END
1.DO
0.DO
1.DO
0.DO
0.DO
0.DO
00010 C TIME
00020 C
(I,J,JMDT1,SPAN,DTS)
(A-HO-Z)
00030
00040
SUBROUTINE TIME
IMPLICIT REAL*8
00050
00060 C
DIMENSION DT(50) ,Q(51)
00070
00080
00090
601
602
603
FORMAT(8D16.8)
FORMAT(413,6D16.8)
FORMAT(' FAILED KSP')
K= 2
00100
00110
IKS= 70
00120
00130
K75= 0
M4= 20
00140
DTNEG= 5.D-10
IF (DT1) 305,500,506
00150
00160
00170
00180
00190
00200
00210
00220
00230
500
503
IF (NF) 306,306,501
DT(2)= 1.D-3
CALL SWITCH(I,J,JMDT(K) ,Q(K) ,NF,2)
IF (NF) 505,505,503
IF (K-7) 504,306,306
504
DT(K+1)= DT(K)*5.DO
501
502
K= K+1
GO TO 502
00240
00250
00260
505
(DT(K)-DT(K-1))/5.D0
506
CALL SWITCH(IJ,JMDT1,Q(1),NF,1)
507
DT(1)= DT1
DT(2)= DT(1)+SPAN
CALL SWITCH(I,J,JM,DT(2),Q(2),NF,2)
IF (NF) 50,50,305
DT(2)= DT1
IF
00290
00300
00310
00320
00330
00340
SPAN=
GO TO 79
00270
00280
DT(1)= 1.D-6
CALL SWITCH(I,J,JMDT(1),Q(l),NF,1)
508
(NF) 508,508,507
00350
00360
00370
00380
00390
50
00400 C NOW
00410
00420
51
00430
00440
00450
53
00460
54
55
00470
00480
00490
56
00500
00510
00520
00530 75
76
00540
00550
00560
78
00570
79
00580
00590
80
00600
100
00610
00620
304
00630
305
00640
00650
00660
00670
306
00680
00690
Q(2)= Q(1)
DT(1)= DMAX1(1.D-6,DT1-SPAN)
CALL SWITCH(I,J,JMDT(1) ,Q(1) ,NF,2)
IF (NF) 305,305,50
KS= K-1
GOOD AT DT(KS),BAD AT DT(KS)+SPAN
KSP= KS+IKS
DO= Q(K)-Q(K-1)
DDT= (DT(K-1)-DT(K))/DQ*Q(K)
IF (DDT*(DTNEG-DDT)) 54,53,53
IF (NF) 75,75,100
IF (KSP-K) 304,304,55
DT(K+1)= DT(K)+DDT-DTNEG/2.DO
IF ((DT(K+1)-DT(KS)+DTNEG)*(DT(KS)+SPAN-DT(K+1)))
K= K+1
CALL SWITCH (I,J,JMDT(K),Q(K),NF,2)
DQ= Q(K)-Q(K-1)
51,100,51
IF (DQ)
IF (K75-M4) 76,76,305
K75= K75+1
K= KS
SPAN= SPAN/5.D0
K= K+1
DT(K)= DT(K-1)+SPAN
CALL SWITCH(I,J,JM,DT(K) ,Q(K) ,NF,2)
IF (NF) 50,50,78
DTS= DT(K)+DTNEG
RETURN
WRITE (6,603)
WRITE (6,602) I,J,K,K75,DTlSPANDT(1),Q(1),ZKZJ
WRITE (6,601) (DT(IK) ,Q(IK) , IK=1,K)
KK= 51
Q(KK)= 1.DO
RETURN
END
75,75,56
00010 C SWITCH
00020 C
00030
SUBROUTINE SWITCH(I,J,JM,WW,QlNFKDU)
IMPLICIT REAL*8(A-H,O-Z)
00040
00050
DIMENSION S(4,7,20)
00060
COMMON/KLN/KL
00070
COMMON/STR/S
00080
COMMON/FIN/H2,ECESSN,CV1
00090 C
00100
20
21
IF (KDU-1) 27,20,21
ASK= 1.DO-S(I,J,9)*S(I ,J,9)-S(I ,J,6)*S(I,J,6)
TF= 1.DO-2.DO*S(I,J,7)*S(I,J,7)
QS= S(I,J,5)+(S(I,J,6)-S(IJ,11))*S(I,J,7)
QS8= 8.DO*S(I,J,7)*QS
ASKO= ASK*(2.DO*TF-l.DO)
THO= -3.DO*(2.DO*TF-l.DO)
D22= -(2.DO+TF)*S(I,J,7)
D21= -D22*S(I,J,6)-QS*TF
XE= TF-S(IJ,6)**2-((S(IJ,8)+S(IJ,5)*S(IJ, 9)) **2+
1
(S(I,J,5)*S(I,J,6)-S(I,J,7))**2)/S(I,J,10)
F= S(I,J,11)+WW*S(I,J,4)
DFF =WW*S(I,J,4)
F2= F*F
P= (F+S(I,J,6))*(F+S(IJ,6))+S(I,J,9)*S(I,J,9
XA= P*(ASK+F2)-4.DO*F2
XB= S(I,J,7)*(2.DO*P*(F+S(I,J,6))-4.DO*F)+(S( I,J,5)+S(I,J,7)
)
00110
00120
00130
00140
00150
00160
00170
00180
00190
00200
00210
00220
00230
00240
00250
00260
*WW
00270
00280
00290
00300
00310
00320
00330
00340
00350
1*S(IJ,4))*(2.DO*F2-P)
XC= ASKO+QS8*F+THO*F2+P*XE
XD= D21+D22*F
X4= XA*XE-XB*XD
Z4= XE*XB*XB+XA*(XD*XD-XE*XC)
01= (XC*XC+16.DO*X4)*X4*X4-Z4*(XC*(XC*XC+18.DO*X4)+27.DO*Z4)
IF (XE) 25,25,22
22
23
IF(Q1)25,23,23
PP= 2.DO*XC-3.DO*XD*XD/XE
a'
00360
00370
00380
24
25
IF (PP) 24,26,26
IF (XC*XC+12.DO*X4-PP*PP) 25,25,26
NF= 0
26
NF= 1
00390
00400
00410
00420
00430
00440
00450
00460
00470
RETURN
27
28
00480
PETURN
DER= XD*XD/XE/XE
TR= DER/2.DO*(XC/XE-3.D0/4.DO*DER)+2.DO*X4/XE/XE
OP= XC/XE-3.D0/2.DO*DER
00= (XD*(DER-XC/XE)+2.DO*XB)/XE
IF (00) 29,28,29
S(I,J+1,3)= 1.D20
S(IJ+1,14)= -l.DO
00490
S(IJ+1,15)= 0.DO
00500
00510
00520
00530
GO TO 30
PPR= OP+DSORT(XC*XC+12.DO*X4)/XE
YS= -DABS(OQ)/OQ*DSQRT(TR/PPR)
ZS =XD/2.DO/XE
29
00540
00550
S3 =YS-ZS
S(I,J+1,3)= 1.DO/S3
00560
DEN= 1.DO+P*S(IJ+1,3)*S(I,J+1,3)
00570
00580
00590
S(IJ+1,14)= 2.DO/DEN-1.DO
S(IJ+1,15) =2.DO*S(IJ+1,3)*DSQRT(P)/DEN
CALL DTDU(I,J,JMWWJ+1,1)
00600
00610
30
RETURN
END
a'
4S
00010 C DTDU
00020 C
SUBROUTINE DTDU(I,LJMDTM,ISW)
00030
IMPLICIT REAL*8 (A-HO-Z)
00040
00050
DIMENSION S(4,7,20)
00060
COMMON/STR/S
00070 C
00080
DF= S(I,L,4)*DT
VL= 1.DO+DF*(S(I,L,6)+S(I ,L,1l))/S(IL,10)
00090
00100
00110
00120
00130
00140
00150
00160
00170
00180
00190
00200
00210
00220
00230
00240
00250
00260
00270
00280
00290
00300
00310
00320
00330
00340
VH= DF*S(I,L,9)/S(IL,10)
RAD2= VL*VL+VH*VH
S(I ,M,1)= S(I ,L,1)+DT
S(I,M,2)= DT
S(I,M,5)= S(I,L,5)+DF*S(I ,L,7)
S(I,M,8)= S(I,L,8)*VL-VH* ((S(I ,L,10)-1.DO) *S(I ,L,7)-
I
S(I,L,11)*S(I,L,5))
S(IM,10)= S(I,L,10)*RAD2
S(IM,11)= S(IL,11)+DF
RAD= DSQRT(RAD2)
CX= VL/RAD
SX= VH/RAD
S(IM,16)= CX*S(I ,L,16)-SX*S(I,L,17)
S(IM,17)= CX*S(IL,17)+SX*S(IL,16)
S(I,M,18)= S(IL,18)
TX= (CX+S(I,M,16))/(1.DO+S(I,L,16))
S(I ,M,19)= TX*S(I,L,19)+SX*S(IL,12)
11
S(I,M,20)= TX*S(I,L,20)+SX*S(IL,13)
IF (ISW) 12,12,11
CDU= S(IL+1,14)
SDU= S(I,L+1,15)
SOR= SDU/DSQPT(S(I,M,10))
SPT=
(1.DO-CDU)/S(I,M,10)
TP= CDU+SRT-SOR*S(I,M,5)
TTO= CDU*S(I,L,7)+SOP*S(IM,11)
U,
00350
00360
00370
00380
00390
00400
00410
00420
00430
00440
00450
00460
00470
00480
00490
00500
00510
00520
00530
00540
00550
00560
CSOC= DSQRT(1.DO+S(I,L,7)*S(I,L,7)-TTO*TTO)
FPR= CSOC/DSQRT(TP)
S(I,M,4)= FPR*S(I,L,4)
S(I,M,5)= (CDU*S(I,M,5)+(S(I,M,10)-1.DO)*SOR)*FPR/CSOC
S(I,M,6)=
(S(I ,L,6)-SOR*S(IL,7)-SRT*S(I ,M,11))/FPR/TP
S(I,M,7)= TTO/CSOC
12
S(I,M,9)= (CDU*S(I,L,9)+SOR*S(I ,M,8))/FPR/TP
S(I,M,8)= (CDU*S(I,M,8)-SOR*S(I,L,9)*S(I,M,10))/CSOC
S(IM,11)= (CDU*S(I ,M,11)-SOR*S(I,L,7)*S(I,M,10))/FPP
S(IM,10)= TP*S(I,M,10)
Tl= S(I,M,17)
S(I,M,17)= CDU*S (IM,17) -SDU*S (I ,M,18)
S(I,M,18)= CDU*S (I ,M,18) +SDU*T1
SSI= S(I,M,19)*S (I,M, 19) +S(I,M,20)*S(I,M,20)
13
S(I,M,12)=
14
S(I,M,12)=
S(I,M,13)=
RETURN
END
IF
15
(SSI) 13,13,1 4
CDU*S (I ,L,12) -SDU*S(I ,L,13)
S(I,M,13)= CDU*S (I ,L,13) +SDU*S (IrL,12)
GO TO 15
(S(I, M,17)*S(IM,19)-S(IM,18)*S(IM,20))/SSI
(S(I, M,18) *S(IM,19)+S (IM,17) *S(IM,20) )/SSI
a'
00010 C MATX
00020 C
00030
00040
00050
SUBROUTINE MATX(ECESSNDY)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION S(4,7,20),DY(3),X(4,4)
00060
00070
COMMON/STR/S
COMMON/FCOMP/X
00080
00090
00100
All =X(2,2)-X(2,1)
A12 =X(2,3)-X(2,1)
A13 =X(2,4)-X(2,1)
00110
00120
A21 =X(3,2)-X(3,1)
A22 =X(3,3)-X(3,1)
00130
A23 =X(3,4)-X(3,1)
00140
00150
A31 =X(4,2)-X(4,1)
A32 =X(4,3)-X(4,1)
00160
A33 =X(4,4)-X(4,1)
00170
00180
00190
00200
00210
00220
00230
00240
00250
00260
00270
00280
00290
00300
00310 21
00320
00330
B11 =A22*A33-A23*A32
B12 =A21*A33-A23*A31
B13 =A21*A32-A22*A31
B21 =A12*A33-A13*A32
B22 =All*A33-A13*A31
B23 =A11*A32-Al2*A31
B31 =A12*A23-A13*A22
B32 =A1*A23-A13*A21
B33 =All*A22-Al2*A21
Yl =EC-X(2,1)
Y2 =ES-X(3,1)
Y3 =SN-X(4,1)
S15 =S(1,2,15)
IF (S15) 21,22,21
DA =All*Bl1-A21*B21+A31*B31
DY(1) =(Yl*Bll-Y2*B21+Y3*B31)/DA
DY(2) =(-Y*B2+Y2*B22-Y3*B23)/DA
C'
00340
00350
00360 22
00370
00380
00390
00400 23
00410
DY(3) =(Yl*Bl3-Y2*B23+Y3*B33)/DA
GO TO 23
DA =B23
DY(1) =(Yl*A32-Y3*A12)/DA
DY(2) =(-Yl*A31+Y3*All)/DA
DY(3) =0.DO
RETURN
END
00010 C OUTPUT
00020 C
00030
00040
00050
00060
00070
00080
00090
00100
00110 61
00120 62
OMEGA
00130
SUBROUTINE OUTPUT(I,J,KKDT,Z,IOUT,OX)
IMPLICIT REAL*8(A-H,O-Z)
DIMENSION S(4,7,20) ,X(4,7)
COMMON/KLN/JM
COMMON/STR/S
COMMON/BIN/BPOBPTElV1,PIDTFTAUl
COMMON/FCOMP/X
COMMON/TIN/ETVT,WTPTSTSITOM
FORMAT(4(D9.3,1X),D8.3,IX,D1O.4,1X,D9.3,1X,D9.3,1XD8.3)
FORMAT(/' TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
1 =',D14.5//)
00140
00150
OME =1.DO
ONE =1.DO
00160
P12 =.5DO*PI
00170
00180
IF(OM.LE.PI2.AND.OM.GE.-PI2)
IF (JM.EQ.3) ONE =1.DO
00190
00200 11
IF(KK)11,11,12
P =Z/S(I,J,4)**2/(1.DO+S(I,J,7)**2)
00210
PP =P-S(I,J,9)**2
00220
00230
00240
00250
00260 12
DF
DF
DT
GO
DF
00270
PP =(S(I,J,6)+S(I,J,11)+DF)**2
00280
P =PP+S(I,J,9)**2
00290 13
00300
00310
RAD =DSQRT(P)
ESF =RAD*(S(I,J,5)+DF*S(I ,J,7))
VL =1.DO+DF*(S(IJ,6)+S(I,J,11))/S(I,J,10)
00320
VH =DF*S(I,J,9)/S(I,J,10)
00330
00340
RAD2 =VL*VL+VH*VH
CX =VL/DSQRT(RAD2)
ONE =--1.DO
=DSQRT(PP)-S (IJ,6)-S (IJ,11)
=DF*S (I,J,6)/DABS (S(I ,J,6))
=DF/S(I,J,4)
TO 13
=DT*S(I,J,4)
00350
SX=VH/DSQRT(RAD2)
00360
00370
00380
00390
00400 51
00410
S16 =CX*S(I,J,16)-SX*S(IJ,17)
S17 =CX*S(I,J,17)+SX*S(I,J,16)
S18 =S(I,J,18)
IF (S18) 52,51,52
UX =0.DO
GO TO 53
00420
00430
00440
00450
UX =DATAN2(S18,S17)
TX =(CX+S16)/(1.DO+S(I,J,16))
S19 =TX*S(I,J,19)+SX*S(I,J,12)
S20 =TX*S(I,J,20)+SX*S(I,J,13)
00460
00470
00480
00490
00500
00510
00520
00530
00540
52
53
41
42
43
44
45
IF (S20) 44,41,44
IF (S19) 42,43,43
OMS =PI
GO TO 45
OMS =0.DO
GO TO 45
OMS =DATAN2(S20,S19)
SSI =S19*S19+S20*S20
OX =180.DO/PI*OMS
00550
SI
00560
00570 14
IF(SI) 14,14,15
CU =S(I,J,12)
=DSQRT(SSI)
00580
SU =S(I,J,13)
00590
00600
00610 15
00620
00630 16
00640
00650
OMS =0.DO
GO TO 16
CU =DCOS(UX)
SU =DSIN(UX)
H2 =P*S(I,J,4)**2*(1.DO+S(I,J,7)**2)
X(1,I) =H2
X(2,I) =(P-l.D0)*CU+ESF*SU
00660
X(3,I)
00670
X(4,I) =SI
=(P-1.DO)*SU-ESF*CU
00680
TAU =S(I,J,1)+DT
00690
00700
00710
00720
00730
00740
00750
00760
00770
00780
00790
00800
00810
00820
1,2)
00830
00840
00850
00860
00870
00880
00890
00900
00910
DTF =DT
X21 =X(2,I)
X31 =X(3,I)
IF(IOUT-1) 23,21,22
21
PHI =DATAN2(S(I,J,7),1.DO)*180.DO/PI
SUP =S(I,J,9)/DSQRT(S(I,J,10))
UP =ONE*(DARSIN(SUP)*180.DO/PI)
X(2,I) =ONE*X(2,I)
X(3,I) =ONE*X(3,I)
31
VT =0.DO
IF
(ESF) 32,31,32
GO TO 33
32
33
22
23
VT =DATAN2(ESF,P-1.DO)*180.D0/PI
WRITE(6,61) X(1,I),X(2,I),X(3,I),X(4,I),TAU,VTPHI,UPS(IJ+
RETURN
X(2,I) =ONE*X(2,I)
X(3,I) =ONE*X(3,I)
WRITE(6,61) X(1,I),X(2,I),X(3,I),X(4,I),TAU
IF (OM.LE.0.DO) OME =-1.DO
OMEGA =-90.DO*OME*(ONE-1.DO)-OMS*180.D0/PI
WRITE (6,62) OMEGA
RETURN
END
-_
72
Appendix
Examples of exact transfers
C
7
the state vector
total normalized
=
TE=
AV
( h , ecos w
,
esin w , sin i
)
An exact numerical solution may be described by:
TAU
location of the impulse
defined by the true anomaly
=
v
PHI
the firing angles
UP
AC=
magnitude of each impulse
DTAU
location of the first impulse
w.r.t. the reference frame
The
following 10 examples
from a circular
orbit
TE
=
=
A
OMEGA
illustrate
2 and 3 impulse transfers
( 1, 0, 0, 0 ). The final states
7
are
chosen with no restriction.
Case
1 - 4
The
transfers.
describe
final
the
elliptic
symmetrical
properties of
orbits have
large
2 impulse
inclinations
and
eccentricities.
Case
5
6
-
are
3
impulse
transfers.
The
final
elliptic
transfers.
The
final
elliptic
orbits have small inclinations.
Case
7
8
-
are
2 impulse
orbits have small eccentricities.
Case 9 -
10
are elliptic Hohmann transfers.
CASE
1
INPUT
H2
ECOSW
0.42000D+01
FIRST ORDER ESTIMATE OF
I
JM
CLIM
0.42262D+00
SIN
2
.15000D+00
ESINW
0.34641D+00
TAU
0.20000D+00
=.4708666384D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
V
PHI
UP
DTAU
0.100D+01 0.0
0.0
0.0
.0
0.0
-.513D+00 0.880D+01 .321D+00
0.174D+01 0.738D+00 0.379D-02 0.373D-01 .321D+00 0.1708D+03 0.399D+01 -. 544D+02 .153D+00
0.420D+01 0.346D+00 0.200D+00 0.423D+00 .475D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
CASE
:
OMEGA
0.81642D+01
2
INPUT
H2
0.42000D+01
FIRST ORDER
ECOSW
-0.34641D+00
ESTIMATE OF
TAU
ES INW
SIN I
0.20000D+00
CLIM
.15000D+00
JM
2
0.42262D+00
=.4708666384D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
PHI
V
UP
DTAU
0.513D+00 -. 880D+01 .321D+00
0.0
.0
0.100D+01 0.0
0.0
0.0
0.174D+01 -. 738D+00 0.379D-02 0.373D-01 .321D+00 -. 1708D+03 -. 399D+01 0.544D+02 .153D+00
0.420D+01 -. 346D+00 0.200D+00 0.423D+00 .475D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS :
OMEGA =
0.17184D+03
CASE
3
INPUT
H2
-0.34641D+00
FIRST ORDER ESTIMATE OF
SIN I
FSINW
ECOSW
0.42000D+01
TAU
-0.20000D+00
0.42262D+00
JM
CLIM
2
.15000D+00
=.4708666384D+00
OUTPUT
H**2
ECOSW
SIN I
ESINW
TAU
V
PHI
DTAU
UP
0.0
-. 513D+00 -. 880D+01
.0
0.0
0.0
0.100D+01 0.0
0.174D+01 -. 738D+00 -. 379D-02 0.373D-01 .321D+00 0.1708D+03 0.399D+01 0.544D+02
0.420D+01 -. 346D+00 -.200D+00 0.423D+00 .475D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
CASE
: OMEGA
=
.321D+00
.153D+00
-0.17134D+03
4
INPUT
F2
0.42000D+01
ECOSW
FIRST ORDER ESTIMATE OF
TAU
CLIM
SIN I
ESINW
0.34641D+00
-0.20000D+00
2
0.42262D+00
.15000D+00
=.4708666384D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
PHI
V
UP
DTAU
0.513D+00 0.880D+01 .321D+00
0.0
.0
0.0
0.0
0.100D+01 0.0
0.174D+01 0.738D+00 -. 379D-02 0.373D-01 .321D+00 -. 1708D+03 -. 399D+01 -. 544D+02 .153D+00
0.420D+01 0.346D+00 -.200D+00 0.423D+00 .475D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
:
OMEGA =
-0.81642D+01
CASE
5
INPUT
H2
0.39600D+01
ECOSW
FIRST ORDER ESTIMATE OF
SIN I
ES INW
0.87155D-02
TAU
0.99619D-01
0.87156D-01
JM
CLIM
3
.50000D-01
=.4519980884D+00
OUTPUT
H**2
0.100D+01
0.159D+01
0.259D+01
0. 396D+01
ECOSW
SIN I
ESINW
0.0
-. 586D+00
-. 245D+00
0.872D-02
0.0
0.723D-03
0.325D+00
0.996D-01
0.0
0.299D-02
0.846D-01
0.872D-01
V
TAU
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
CASE
PHI
UP
DTAU
.0
0.0
0.127D+00 -. 831D+00 .259D+00
.259D+00 -. 1558D+03 -.703D+01 0.209D+02 .109D+00
.369D+00 0.1730D+03 -. 432D+01 -. 256D+02 .953D-01
.464D+00
:
OMEGA =
0.93696D+02
6
INPUT
H2
0.39600D+01
ECOSW
0.0
FIRST ORDER ESTIMATE OF
ESINW
0.10000D+00
TAU
SIN I
0.82808D-01
JM
CLIM
3
.5000OD-01
=.4520750429D+00
OUTPUT
H**2
0.100D+01
0.159D+01
0. 254D+01
0. 396D+01
ECOSW
-.139D-16
-. 589D+00
-. 266D+00
0.228D-07
ESINW
0.0
0.637D-03
0.318D+00
0.100D+00
SIN I
0.0
0.261D-02
0.805D-01
0.828D-01
TAU
PHI
V
UP
DTAU
0.111D+00 -. 724D+00 .261D+00
.0
0.0
.261D+00 -.1561D+03 -. 697D+01 0.207D+02 .103D+00
.364D+00 0.1722D+03 -. 390D+01 -. 248D+02 .990D-01
.463D+00
TO MATCH ACTUAL FINAL OPBITFIPST IMPULSE SHIFTS
:
OMEGA =
0 .90170D+02
CASE
7
INPUT
H2
ECOSW
0.49875D+01
ES INW
0.45853D-01
FIRST ORDER ESTIMATE OF
TAU
SIN I
0.19937D-01
CLIM
.10000D+00
JM
2
0.64279D+00
=.5807860994D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
V
PHI
UP
DTAU
0.100D+01 0.0
0.0
0.0
.0
0.0
-.675D-01 0.135D+02 .302D+00
0.168D+01 0.679D+00 0.461D-03 0.543D-01 .302D+00 0.1792D+03 0.358D+00 -. 699D+02 .273D+00
0.499D+01 0.459D-01 0.199D-01 0.643D+00 .575D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
CASE
:
OMEGA
=
0.70359D+00
8
INPUT
H2
0.79928D+01
ECOSW
0.15000D-01
FIRST ORDER ESTIMATE OF
TAU
SIN I
ESINW
0.25981D-01
JM
2
0.81915D+00
CLIM
.10000D+00
=.6187259196D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
V
PHI
UP
DTAU
0.0
-. 467D-01 0.933D+01 .338D+00
0.0
.0
0.100D+01 0.0
0.0
0.178D+01 0.781D+00 0.368D-03 0.410D-01 .338D+00 0.1793D+03 0.352D+00 -. 804D+02 .281D+00
0.799D+01 0.150D-01 0.260D-01 0.819D+00 .619D+00
TO MATCH ACTUAL FINAL ORBITFIPST IMPULSE SHIFTS
: OMEGA =
0.68374D+00
0
CASE
9
INPUT
H2
0.19200D+01
ECOSW
0.20000D+00
FIRST ORDER ESTIMATE OF
TAU
ES INW
SIN I
0.0
0.17365D+00
JM
CLIM
2
.20000D+00
=.3124921504D+00
OUTPUT
H**2
ECOSW
ESINW
SIN
I
TAU
V
PHI
UP
DTAU
0.100D+01 0.0
0.0
0.0
.0
0.0
-. 642D-04 0.192D+02 .197D+00
0.141D+01 0.412D+00 0.263D-06 0.545D-01 .197D+00 0.1800D+03 0.845D-03 -.415D+02 .104D+00
0.192D+01 0.200D+00 0.273D-04 0.174D+00 .302D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE
SHIFTS
:
OMEGA
=
-0.36000D+03
CASE 10
INPUT
H2
0.19200D+01
ECOSW
-0.20000D+00
FIRST ORDER ESTIMATE OF
TAU
SIN I
ESINW
0.0
0.17365D+00
JM
CLIM
2
.20000D+00
=.3124921504D+00
OUTPUT
H**2
ECOSW
ESINW
SIN I
TAU
V
PHI
UP
DTAU
0.100D+01 0.0
0.0
0.0
.0
0.0
-.642D-04 -.192D+02 .197D+00
0.141D+01 -.412D+00 -.263D-06 0.545D-01 .197D+00 0.1800D+03 0.845D-03 0.415D+02 .104D+00
0.192D+01 -.200D+00 -. 273D-04 0.174D+00 .302D+00
TO MATCH ACTUAL FINAL ORBITFIRST IMPULSE SHIFTS
:
OMEGA =
-0.18000D+03
--j
Figures
78
A
SO
2
T
AA
Circular orbit
: Final orbit
: Transfer orbit
: Initial
orbit plane
: Final orbit plane
Fig.1
The coordinate system of the non-coplanar transfer from a
circular orbit
(Fixed reference axes
2 3
rx
1, A2,3)
79
0
Fig.
2
An elliptic Hohmann transfer,
is
=
0.
1
-
s s
S
N
Fig. 3
.
80
axis inclination relative
to tli,. initialorbit-plane
1
First order influence of inclination (with impossible elliptic
Hohmann transfer orbit
0 ).
81
09:
First transfer orbit
: Second transfer orbi
7
0
a
400
A
I
AV
I
Fig.4
Illustration of a three impulse transfer.
3 iMgulse transfer family
hi;
000
0
4-J
C-)
1.
*Tr
Limiting solution
Zt
0
cc
VS
Fig.5
vs
\V of the first
(transfer angle)
transfer
orbit
0
,
(Cos
1i 2
)
True anomoly
(
e
)
Single impulse
01.0
4J
toP
LH
0I
121
I
Ratio of the semi-major ases
rig.
Fi rsL order so-LUL1
6
UL
and 3I1,
nl i .3IaLLUL,
(i
/K00
).
mo
(d5
4e41
I
Va
-
0
0,s
08 O
Fig. 7a
Single impulse
approximation
Fig - 7b
Three impulse
approximation
Fig- 7c
Actual three impulse
approximation
84
-
Fig.8
0
Generalised Hohmann transfer between circles, (k= 0 ).
85
o,
060
Fig. 9
A two impulse transfer with plane change at further impulse.
I*
86
-Pd
-I-)
0
U
U)
-P0
fd
Cd
In>nto
ofj
th
fialori
U
04'
Inclination of the final orbit
Determination of the limit of the first order solution of
small inclination.
(
4V/g=
first
order,
V
=
second order
)
Fig. 10
(
)
--. 0
18Q)0
1 .
')70
0
87
05
0
/
,
A
(b)
0
-c:::
90
20
Fig.1l a,b
2 impulse transfers visualization with an elliptic final orbit.
88
and
6
in
4
e
the plane of
)
Fixed orthogonal set (
e
4 4-
Fig. 12a.
A supporting impulse
-qo
-< 1 4 o0'
e
.2
Fig . 12 b.
An opposing impulse
4, 2702
v
and
89
A
S3
(a~)
In the plane of
e
and
W.*\
e,4
.A
I
Omo
(b)
Iv
4"
1
K
ei
A
.2
Fig.
13
(a, b)
Definitions of the transfer
parameters
e
90
3
I.
1--
AA
Fig. 14
A single impulse for two intersecting orbits.
91
0J
=constant plane
> 0
% constant
,
I
. x 4
0
/5a
A supporting impulse from a circular orbit with
I
= constant and
i
0
1
I
- constant
,
Fig-
'O.
~'
constant
A
-~.--
4I
Fig./
/5b
Nonuniqueness
of
at
~J = constant.
I
92
References
1.
T.N.
How Many Impulses?
Edelbaum
Astronautics and Aeronautics,
Vol. 5, no.11, Nov.1967, pp.64-69.
2.
F.W.
Gobetz
J.R.
and
Doll
Survey
Of
Impulsive
Trajectories.
Of
Trajectories.
AIAA J. Vol. 7, no.5, May 1969, pp. 801-834.
3.
C.
Marchal
Survey
-
Paper
Optimization
I.A.F. - 27th Congress - Anaheim
4.
H.W.
Minimum
Small
Ph.D.
Orbits.
nautics
5.
T.N.
Thesis,
Marec
Congress,
7.
J.V.
and
a
Madrid,
Nearby
Methods
for
Editor.
8.
C.
Non-coplanar
Elliptic
Flight
Thesis,
in The Near
of Aero-
Vicinity of
Impulsionnels
Orbit.
Economiques
Proc.
M.
I.
T.
Advanced
Optimization.
17th
IAF
Problems
and
B.
Fraeijs
de
de
Hohmann
as Cas des
Veubeke
1969.
des
Transfers
Faiblement
Optimum Single
M.S.
Department
Minimum Impulse Transfer Between a Circular Orbit
Excentriques
Publication ONERA No. 124,
Der
Elliptic
1966.
Planetaires
G.J.
between
non-coplanaires.
Generalisation
Inclines.
University,
Infinitesimaux
Pergamon Press,
Orbites
9.
oct.
Space
Marchal
Transfer
s.8.
J. Astronaut. Sci. 14, 2, 1967.
Transferts
Breakwell
no.76010
1972.
Quasi-Circulaires
orbites
Time-free
Minimum Impulse Transfers
Edelbaum
J.P.
10-16 Oct. 1976
Stanford
and Astronautics,
a Circular Orbit.
6.
Fuel
Space
Faiblement
1968.
Impulse Transfers
Department
et
from Circular
of Aeronautics
Orbits.
and Astronautics,
1975.
10. D.F.
Lawden
Butterworth,
11. R.S.
Long
Optimal
Eckel
Orbits.
13. K.G.
Transfer
Optimum
between
N-Impulses.
Space
Non-coplanar
Elliptical
Navigation.
Optimum
Orbits.
6, 1960.
Transfers
Astronautica Acta,
Eckel
for
London, 1963.
Astronautica Acta, Vol.
12. K.G.
Trajectories
Vol.
between
8,
Transfer
Astronautica Acta, Vol.
Fasc.
in
Non-coplanar
4,
1962.
Central
9, Fasc.
Elliptical
Force
5-6,
1963.
Field
with
14. J.M.
Optimum
Horner
coplanar terminals.
15. C.
Marchal
ltOptimalite
1968.
Impulse
des
Transfer
between
93
Arbitrary
ARS Journal, Jan. 1962.
Tridimensionel
Generalisation
Champ Newtonien),
pp.3-13,
Two
Arcs
a Pousee
Intermediaire
La Recherche Aerospatiale,
et
de Lawden
No.
Etude
(Dans
de
un
123, Mars-Avril,
94
Biography
Raised
1949.
London
in
University,
a
with
graduated
September
from
B.Sc.
degree
1974,
he
studied
He
Hong Kong.
England,
was born in Canton,
Der
Jew
Gim'
at
1971
Aeronautical
in
China,
Queen Mary College,
to
June
and
1974,
First
Engineering,
Class Honours.
In
September,
Technology
as
a graduate
He
Astronautics.
Astronautics
Laboratory
as
in
a
June,
student
in
obtained
a
1975.
research
to
came
Massachusetts
the Department
S.M.
He has
assistant of
orbit transfers from 1975 to 1977.
the
degree
of
in
N.
Aeronautics
and
Aeronautics
and
in
S.
Draper
Edelbaum on high
thrust
been working
T.
of
Institute
C.