OPTIMpAL SCAPE TRAJECTOY
FROM A
IGH EARTH OBIT
BY USE CF SOLAR ?RADIATION FPESSTE
by
Andrew Joseoh Green
A.S., Moh-;<.Valley Community College
(1970)
B.S., United States Military Academy
(1974)
SUBMITTED IN PARTIAL FULFILLM,1ENT
OF THE REqUIREMENTS
DEGREE
MASTER
FOR THE
OF
OF SCIENCE
at the
MAISSACHUSETTS INSTITUTE
OF TECHNOLOGY
September ]977
Sigrature of Author
Department of Aemnautics
and Astronautics,
24 August 1977
Ji
Thesis Supervisor
Certified by
Certified
by
-
1
,I
Accepted by
____
V
Chairman, Departmental Committee on
Graduate Studies
ARCHIVES
SEP 21
Thesis Supervisor
1977
OPTIMAL ESCAPE TRAJECTORY FROM A HIGH EARTH ORBIT
BY USE OF SOLAR RADIATION
PRESSURE
by
Andrew Joseph Green
Submitted to the Department of Aeronautics and Astronautics
on 24 August 1977, in partial fulfillment of the requirements
for the degree of Master of Science.
ABSTRACT
This thesis develops the optimal control law necessary for
a minimum time escape from a high Earth orbit utilizing solar
radiation pressure.
A force model is developed for an ideal
flat sail and a proposed Jet Propulsion Laboratory square sail.
A two-body inverse square force field model is assumed.
The
Earth's orbit about the Sun is assumed circular, and the solar
flux is assumed constant during the escape maneuver.
The initial
state is given, and the only condition on the final state is
that the energy of the spacecraft equals zero.
A modified
Newton-Raphson method is used to solve the two-point boundary
value problem.
It is found that the performance of any solar
sail escape trajectory tends to fall between the limiting cases
of similar polar and ecliptic escape trajectories.
A velocity-
dependent control law is also examined and found to be a good
approximation to the optimal control law.,
Thesis Supervisor:
Title:
_je~t#-
DEDICATED TO:
Theodore N. Edelbaum
ACKNOWLEDGMENTS
I wish to thank the United States Army and the Fannie
and John Hertz Foundation for making it possible for me to
continue my education.
I would also like to express my gratitude
to Lester L. Sackett and Louis A. D'Amario for their help,
encouragement and understanding during this research.
CONTENTS
Page
List of Symbols-------------.----------------------
1
-
Chapters
1. Introduction
1.1 Background ..----------------------------
1.2 ObJective----------------------____….
1.3 Synopsis
4
... 4
-----------------------------------
2. Force Model
2.1 General Force Model-----------------------2.2 JPL Force Model ---------------------------2.3 Effects of Solar Wind and Earth Shine------
6
7
12
15
3. Optimization
3.1
3.2
3.3
3.4
Coordinate System -------------------------18
Constraints ---.......------------------19
The Optimal Control Law -------------------- 20
Boundary Conditions ----------------------27
3.5 Summary
I-. .------.
mm-------------------.
29
4. Discussion
4.1
4.2
4.3
4.4
Introduction -----------------------------Optimal Control ---------------------------Open-Loop Control-------------------------Velocity-Dependent Control -----------------
5. Conclusions
...-------------------------------
Bibliography ----------............------------------
31
33
36
37
49
51
Appendices
A. Force Equation Parameters ---------------------
54
B. Optimal Control Theory---------- ---------- i
61
C. An Energy Dependent Newton-Raphson Iterator----
66
D. Computer Program ----------------------------...
71
LIST OF SYMBOLS
a - absorption coefficient
ao - characteristic acceleration
ac - acceleration on the sail due to solar radiation pressure
A - area of sail
BF
,B-
non-Lambertian coefficients of front and back surfaces
c - speed of light
c, ,c, ,C3 - sail force coefficients
C, ,C-
rotation matrix
E - energy
E-
sunlight power absorbed by Earth
E - re-radiated Earth power
RE
F. - total radiation force
F-
force due to absorption
F,-
force due to reflection
F - force due to re-radiation
F - maximum force at 1 A.U. from the Sun
g - gravity
H - Hamiltonian
HPP
-d/
h - Plank's constant
i - unit vector in the direction of the x-axis
J-
unit vector in the direction of the y-axis
J - performance index
k - unit vector in the direction of the z-axis
1
L - integrand of the performance index
m - unit vector in the direction of spectrally reflected sunlight
n - unit vector normal to sail's surface
m - mass of sail
p - solar pressure
P - solar intensity
Po,-
solar wind intensity
Pi - intensity of re-radiated energy from Earth
P
- Earth shine intensity
r - reflectivity of sail's surface or magnitude of radius vector
re- albedo of Earth
r - radius vector
R - distance of spacecraft from the Sun
R - distance of Earth from the Sun
Re- radius
of the Earth
s - specular reflection coefficient
S - terminal cost function of the performance index
t - time
T - temperature
V - velocity vector
V,-
circular orbital velocity
X - state of the spacecraft
(
6
- angle between the normal of the sail and -i
-
cone angle of
,
- limiting cone angle
'
angle of Sun with respect to the inertial coordinate frame
2
•,6emmissivity
f
of the front and back surface
- unit vector in the direction of the radiation force
~(rf- photons per unit area per unit time per unit frequency
-
G
unit vector tangent to the sail's surface
- cone angle of the radiation force vector
O, - limiting cone angle
- co-state vector
AAv
position co-state vector
- velocity co-state vector
- gravitational constant
- frequency of photon or angle or Lagrange multiplier
- Stefan-Boltzman constant or clockangle of A,
- transmissivity of surface
2
- centerline angle
- clockangle of the radiation force vector
w
- rotation rate of Earth about the Sun
Superscripts
i - with respect to the inertial coordinate frame
n - nth term or variable
T - transpose of the matrix
R - with respect to the rotating coordinate frame
Subscripts
o - initial
f - final
r - of or pertaining to the radius vector
v - of or pertaining to the velocity vector
3
1. INTRODUCTION
1.1 Background
A solar sail is a device which creates a propulsive force
from solar radiation pressure.
The solar radiation pressure
results from changes in the momentum of incident photons on the
sail's reflective surface.
This force can be used to increase
or decrease the energy of the solar sail, thereby allowing travel
in the vicinity of the radiation source.
One of the first examinations of a solar sail powered
vehicle supposedly occurred in a paper by the Russian author,
F.A. Tsander, in 1924(ref 3).
It was not until the late 1950's
that serious interest in the concept of solar sail propulsion
occurred.
The majority of the papers written on solar sail
propulsion have been concerned with the solar sail as a means
of interplanetary travel.
The use of a solar sail as a means
of planetary escape has been considered(ref 4 and 11), but the
analysis has been rather restricted.
Since the Space Shuttle program is envisioned to make low
Earth orbits readily attainable, the solar sail is at present
being considered for various high Earth orbit and interplanetary
missions.
The solar sail is rather interesting in regards to
interplanetary travel in that it has the potential for improved
performance over interplanetary ballistic and solar electric
propulsion systems.
Inherent to any interplanetary mission is
the successful escape from Earth.
1.2 Objective
Since the solar sail draws its energy from a limitless
4
source, the optimal escape trajectory
from Earth should be the
trajectory that reaches escape in the minimum time.
numerical integration of
A straight
the equations of motion, in order to
determine the optimal trajectory, usually results in numerical
problems.
These numerical problems are due to a large escape
time(in the order of months) and to the small integration time
step needed.
The method of averaging the orbital elements as
discussed in references 8,9,and 10, has proved successful in the
determination of optimal orbit raising trajectories.
As the ratio
of the propulsive fforce to the local gravitational force
becomes large, the method of averaging becomes inaccurate.
For
the solar sails discussed in Appendix A, it appears that the
method of averaging should be terminated when the trajectory
reaches the vicinity of 100000 kilometers from the center of the
Earth.
It
s therefore the purpose of this thesis to develop
an analytical model that will calculate the control law necessary
for a minimum time escape from a high Earth orbit with a solar
sail.
It is felt that a model similar to the one presented in
this thesis could be combined with an averaging method to determine the complete optimal escape trajectory of a solar sail
from a low Earth orbit.
Some basic assumptions are made in the solution of this
problems
1. The state is completely given at the time of launch.
2. The solar flux is constant during the escape maneuver.
3.
The distance between the Sun and the spacecraft is equal
to the distance between the Sun and Earth.
5
4. The Earth's orbit about the Sun is circular.
5. The Earth has an inverse square gravitational field.
6. The only gravitational field which affects the spacecraft is that of the Earth.
7. The effects of Earth shadowing is neglected.
1.3 Synopsis
This thesis is divided into five chapters.
the introduction.
Chapter one is
Chapter two developes an equation for the
magnitude and direction of the resultant force on a solar sail
due to solar radiation pressure.
Chapter three developes the
differential equations of motion for the state and co-state.
Optimal control theory is applied to determine expressions for
the time history of the control variables and the necessary
boundary conditions on the co-state.
Chapter four explains some
of the results of optimal control escape trajectories.
Two
other types of control laws, open-loop control and velocitydependent control, are also examined.
Chapter five lists some
basic conclusions about solar sail escape trajectories.
The
computer program, as described in Appendix D, was the main tool
for the analysis of solar sail escape trajectories.
6
2. FORCE MODEL
2.1 General Force Model
The intensity of sunlight can be expressed mathematically
as follows(ref 6):
o
a
where
P - solar intensity (watt/m2 )
h - Plank's constant (Joule-sec)
/-- frequency of a photon (1/sec)
7(f)-number of photons per unit area per unit time per
unit frequency (1/m2 )
Experimentally this can be expressed as(ref 5):
P
/3.5S3.7
where
R6
-
mean radius of Earth's orbit=1 a.u. =1.495*101 1 m
R - distance to the Sun (m)
The solar pressure, p, can then be determined ass
3.5A/3
o(1)
where
p - solar pressure(newton/m2 )
c - velocity
of light = 3*108 m/sec
The characteristics of a reflecting surface may be defined
in terms of the following parameters:
r - reflectivity of surface
a - absorption of surface
t
-
transmissivity of surface
7
s - specular reflection coefficient
'3(B-
emmissivity of the front and back surface
8B- non-Lambertian coefficients, front and back surface
P
where r+a+=l1.
Since T=O for the solar sail's reflecting
surface, a=l-r.
When light strikes a surface, three types of forces are
created due to absorption, reflection and re-radiation of
photons.
The total force can be expressed as(ref 6):
F -
F
where
F - total force (newtons)
F - force due to reflected photons (newtons)
F
-
force due to absorbed photons (newtons)
F - force due to re-radiated photons (newtons)
By defining the unit vectors i', n,
,
and m as in figure 1,
the following relations are obtained:
= a4v
1
-
o
where A is the area of the sail and p is the solar pressure.
Since it may be assumed that all incident photons are initially
absorbed by the reflector, F can be expressed as
C42
$6+
op94 , L0(2)
8
5
.
"I-
he
"I
Figure
1
__
The force due to reflected photons is composed of two parts,
due to specular(qF) and non-specular(F,) reflection.
reflected light causes a force in the -m direction.
The speculaly
This is
expressed as
The non-speculaly reflected light can be thought of as uniformly
distributed over a hemisphere on the reflected side(see figure 2).
The net flux will be in the -n direction, thereby causing a
force in the direction of n.
The total non-speculaly reflected
hemispheric power is given by c(1-s)r(o*A).
Lambert's Law applies and using Bto
Assuming that
indicate that the sail's
surface is not an ideal Lambertian surface, the force due to
non-speculaly reflected light can be expressed as(ref 6):
F.S =
B·(J -
f5
(7
Combining the above expressions, the following equation for FR
can be obtained:
- rcs
F'- (g. )[sfQ)r
9
c
I,
ulaly
id light
Figure
2
p1L9(r5ca
°< ,f(I-S)rr m) rn
FR-:
The power emitted
by 6-T
-r5s
nvzJd
ao
(3)
from a unit area of a surface is given
is the emmissivity of the surface, o-is the
where
Stefan-Boltzman constant, and T is the temperature of the surface.
Lambert's Law states that the force per unit area due to thermal
re-radiation on an ideal Lambertian. surface is - .3 e-r~L
c --- ~J where n
is the outward normal of
the surface.
Using Band
B to indicate
that the solar sail is not an ideal Lambertian surface, the
following expression for F can be obtained:
F
C
Figure 3 shows a small area of the sail that is in a steady
state condition.
Since the power in has to equal the power out,
the following expression can be obtained(ref 6)s
a 19Jaco = ( F 1
10
T0
I
I
Lure
iFI 3
Dividing each side of the equation by c and rearranging terms:
(4)
? .
-_(
Subsitution of this result into the expression for F givess
r
(
r
B:Ff -
3
8
(5)
Equation 5 gives the impression that it would be best to
.. Upon inspection of equation
build a solar sail with a low
4, one realizes that G6 is the only parameter a designer has for
regulation of the temperature of the sail, since the characteristics of the front side will probably be fixed such that maximum
reflection occurs.
Combining equations 2,3,and 5 gives the following general
force model:
r
yca4
t-5)c
*(l-rI
SB
*i,rs)< } (6)
c
11
-
For a perfect reflector r=l, s=l, and a=O; therefore
equation 6 reduces to
_.
~~
a=
~
2
(7)
Equation 6 was derived assuming a flat sail with area A.
A similar differential force model could be obtained.
It must
also be remembered, with reference to equation 6, that all of
the sail's surface characteristics are functions of the temperature of the sail,
face material used.
the frequency of the radiation, and the surIf the operating point is fairly constant,
these values can be approximated by constants.
2.2 JPL Force Model
The Jet Propulsion Laboratory, using a differential form
of equation 6, integrated the force over the surface of a sail
which may be used for future missions.
defined as shown in figure 4.
Angles 9 and 0 were
The incident angle ~ was varied
and a curve fit was performed on the data.
The force on the
sail can be expressed as:
E= Fe
where
F
i
force function magnitude(newtons)
-
unit vector in the direction of thrust vector
The force function magnitude can be expressed as(ref 16):
E -C (c
where
+
CCo29
C2
ee)
(8)
Fo =pA
8 - cone angle of the thrust vector with respect to
the Sun-spacecraft line.
12
Cl
c ,2
c3
- sail force equation constants
From figure 4 the following expression can be obtained:
< = /o.where
- cenerline angle,i.e. the angle between the surface
normal and the thrust vector, measured in the
direction from n to i'.
- incident angle as defined in figure 1
A relationship exists between
e
and
S, which is shown
graphically in Appendix A, along with a plot of equation 8.
If a certain
e
is desired, the relationship between
is used to obtain
,
thereby determining
0(.
direction of the normal of the sail is fixed.
and
Knowing o(,
z
the
The vectors
F, and n are all in the same plane.
I
I
I
_
/
~1
Figure 4
A perfect reflecting sail would be represented as:
F=/(o.r * o.trThis is equivalent to equation 7.
13
(9)
',
Equation 8 will be called the JPL reflector and equation 9
will be called the perfect reflector.
The direction of the force function can be represented as:
P-i~~aei t a~ne ~~a;~f~mrB L8BQ
(10)
where the cone angle(e) and clock angle(y) are defined in
figure 5.
'C
I
Figure
5
In order to compare the performance of different sails, a
term a,
the characteristic acceleration, is defined as
follows:
Therefore the acceleration of the sail is
as -
C-c(c/
Cdb s'C 3 Ceash)
where
a. - acceleration (m/sec2)
mS - total mass of solar sail vehicle
a, = Fo
(kg)
t
_X-
Therefore
¢C = 420 (C
tos
i c3 t40)f
The maximum variation of a
(11
due to variation of R(see
definition above) at the sphere of influence of Earth is approximately 1%.
Since this-paper is concerned with Earth escape
maneuvers that are well within the sphere of influence of
Earth, it is reasonable to assume a
constant over the maneuver.
2.3 Effects of Solar Wind and Earth Shine
The only force acting on the solar sail has been assumed
to be the force due to solar radiation pressure.
Other forces
that might be considered are the forces from solar wind and
Earth shine.
The solar wind intensity has been experimentally determined
as follows(ref 5):
The intensity of P
can increase by two or three orders of mag-
nitude during periods of solar activity.
Comparing P
and P5
it is evident that solar wind pressure is at least three to four
orders of magnitude less than photon pressure.
be neglected in this thesis.
Solar wind will
The major effect of solar wind is
to deteriorate the sails surface and thereby decrease the
reflectivity.
15
To determine the effects of Earth shine an argument similar
to the one developed in Villers (ref 15) is presented.
The ab-
sorption of sunlight energy by the Earth is given as:
where
E
- power absorbed by the Earth (watts)
Re - radius of the Earth (6378.16*103 m)
r,
- albedo/reflectivity of the Earth (.32 to
52)
Ps - solar intensity (watt/m2 )
Assuming that all Earth shine effects occur at the Earth's
surface and that the Earth re-radiates isotropically:
HAL=
Zi
-
{47
)
where
ELk- sunlight power re-radiated from Earth (watt)
PAp- Earth re-radiated intensity at the Earth's
surface (watt/m2 )
substitution gives
The maximum possible Earth shine intensity(P)
is then
At a distance D above the Earth's surface the energy recieved
from Earth shine has to be less than or equal to the maximum
power emmitted at the Earth's surface.
16
per
< a
i
t(12)
This paper assumes an operating point of at least 10 Earth
radii. The effects of Ps will be at least two orders of magnitude less than P
and therefore will be neglected.
17
3. OPTIMIZATION
3.1 Coordinate Systems
An inertial coordinate system(i) and a rotating coordinate
system(R) are defined as in figure 6 with the following assumptions
and definitions t
1. Launch occurs at t=O.
2. The inertial and rotating coordinate system are
centered at Earth with the x, y axes in the ecliptic
plane and the z axis perpendicular to the ecliptic.
3. At t=O, the x axis of the inertial coordinate frame
points at the Sun and remains fixed with respect to
inertial space throughout the mission.
4. The rotating coordinate system has its x axis pointing
at the Sun through out the mission.
5.
Earth's orbit about the Sun is circular.
6. The difference between the Sun-spacecraft distance
and the Sun-Earth distance is negligible.
18
The angle
gf
,-~
where
Y-
~~~
as defined in figure 6 can be expressed as:
t-(13)
angle of Sun with respect to the inertial
coordinate frame (rad)
r - rotation rate of Earth about the Sun (1.9913859*107rad)
see
t - time (see)
A rotation matrix that transforms the mathematical representation of a vector in the R frame to its equivalent vector
representation in the I frame can be defined as:
z = On CaY (14)
3.2 Constraints
The accelerations on the solar sail are caused by gravity
and solar pressure.
be written as:
The differential equations of motion can
(m)
y=
r-
Vc
(15a
(15b)
where
c - 'Earth'z::gravitational
constant(398601.2 km3 /sec2 )
V_ velocity vector (km/sec)
,-
radius vector (kin)
19
The solar sail has as control variables the cone and clock
angles(andy)
of the vector fR). The clock angle is unconstrained.
The cone angle is constrained to be between 0, and 180 degrees.
The upper constraint on G results from the mathematical definition of e and
f .
The lower constraint results from the fact
that.the force vector can never point towards the Sun.
The quantity
For the perfect reflector, it
8L is the limiting cone angle.
is 90 degrees and for the JPL reflector it is the solution of the
following equations
oe
C, tC,
redue
this reduces to the following:
fclo
C3
(16)
CooL = Therefore the constraints on the control variables are:
,4
6& '-
/go 0
-
(17
For the JPL square sail as given in Appendix A, 9L is equal to
118.851 degrees.
3.3 The Optimal Control Law
Since the solar sail extracts energy from a limitless source
for propulsion and attitude control, a performance index for
trajectory optimization should not involve the energy source.
There is an inherent assumption that the control law developed
does not require changes in the control variables in a period of
time that cannot be provided for by the attitude control system
of the solar sail.
The performance index J is given bys
*
20
20lt
=;
(18)
The final objective of the maneuver is to achieve escape.
Therefore the final energy (E ) has to equal zero.
·_
_
It is also assumed that the state (r and V) is given at
t=O.
This can be written as:
r (o) =
£do
(2 0a)
C,
(20b)
V
The problem is then to minimize J subject to the boundary
conditions in equations 19and20 and the .constraints.in
equations
15 and 17.
is to
The method used to achieve this optimization
define a Hamiltonian which must be maximized in order to
minimize J.
The Hamiltonian is defined ass
wher=-/
(21)
where
H
Hamiltonian
r
-
2)
(km/sec
position co-state vector with respect to inertial
space (sec/km)
-v()-
velocity co-state vector (also called primer vector)
with respect to inertial space(sec2 /km)
Applying the Euler-Lagrange equations:
therefore
therefore
21
V
combining and rearranging terms gives:
_
_
(22b)
whereas equations 15a and 15b are the state differential
The force vector and primer vector directions
equations.
can be
represented
as (see figugere 7):
(ii)3rt(,lfr( (
-v
(23a)
_~(r
Vt
(R
(23b)
/L2eBn4
J
C.~
8 c~~-
· L~~~u .~~C·77 ~~~~Pc~~
X,
~~ c~oc
~
'MAI~H~
_CLBrL6I
(
1
I
4
22
I
Since the control variable ? has no constraints, the
Hamiltonian can be maximized with respect to
partial of the Hamiltonian with respect to
The Hamiltonian of
f
o
by setting the
I equal to zero.
concern is:
-° I
Y ="--t
,f
o
)]
c
= o'-'
(24)
Fromequation 24 it is evident that the force vector, the
primer vector, and the x axis of the rotating coordinate frame
will all be in the same plane.
Therefore the force vector's
direction can be expressed as (ref 12)s
O[i]MuLAYJ
t
W
Au4'99)
|
23
.
IZk'A&
,is
Qun
f
!
$(t)_ <(,@-d)
*rM Ad
Because the control variable
5-i
(25)
is constrained as expressed
in equation 17, the Maximum Principle must be applied in the
determination of the control law for e.
Substitution of equation
24 into the expression for the Hamiltonian and rearranging terms
gives the following Hamiltonian of concern:
For the ideal reflector:
U'
therefore
The ambiguity of the sign in the above equation is removed by
useof the necessary condition thatthe second partial of H'
with respect to
imizes H'.
has to be less than zero at a point that max-
Substitution
of the above value of 6 into -
yields:
24
Since-
is in the second quadrant, the sign of the radical has
to be negative.
This same result would have been achieved if
physical reasoning were used to remove the ambiguity of the
above sign.
If the sign was positive,
A
would be greater than
e
zero and therefore p would violate the constraint equation.
Therefore the cone angle control law for a perfect reflector is:
f-oo
• *J
=1/3'
od
(26a)
At"~P
( > g00
(26b)
Proceeding in a similar manner for the JPL reflector, it
is found that the equations do not solve analytically in closed
form.
A second order Newton iteration method can be used to
solve for the optimal 9.
The Hamiltonian of concern can be
written as:
where
E(, a
so-c
c,
defining the following:
a/1"w = >VEpe&
L~~-/EJ)
2
Therefore the optimal e is given by:
25
JEa9
(m+/)
9
C-n)
=
/]o
e')
(27a)
/.//J(et
) is given by equation 26
o
e4 G !/oc/
(27b)
( a is given in equation 16)
(27c)
From equation 24 and 26, it is evident that expressions
for g and c-must be obtained.
expression for
,
From equation 23b the following
can be obtained:
therefore
(28)
111
where
£9.
andW are the x and y components of the primer vector
with respect to the inertial
coordinate frame.
In a similar manner it can be shown that:
0a-=/=A
(29)
From equations 26 and 27 it is clear thatA must be determined before the optimal thrust cone angle ()
can be determined.
Since in this thesis the optimal 9 will be determined and then
checked to see if it is within
the constraints,
considerable
work could be avoided if the constraints in equation 17 could
26
The optimal
be expressed in terms of ,.
partial of the Hamiltonian with respect to
(HP as defined
If it is assumed that the situation
above) is set equal to zero.
(and therefore/A=/
is such that
is obtained when the
) is the optimal
, the
following would result:
O6eL)
Z(9
9'04
'<
Since
-7=0
A!
is not equal to zero for all possible
L., it can be
concluded that:
Therefore
Upon inspection of equation 26 it is realized that as /t
approaches 180 degrees,
9approaches 180 degrees.
Therefore the
constraints imposed upon O in equation 17 can be replaced by the
following:
-<_
°
(30)
3.4 Boundary Conditions
The boundary conditions are now addressed.
This thesis
assumes that the state is completely described at launch as
given in equation 20.
Boundary conditions on the co-state must
27
now be determined based upon equation 19.
From Appendix B the following conditions on the co-state
must exist at the final time:
Li'ir ]
(31a)
@t')=p)=
© (31b)
(31c)
For the solar sail escape maneuver, the following terms can be
identified:
L=/
=
2
~U~
i"
l.I
I
' = Lv O
L'-'r
28
Substitution of these terms into equations 31a and 31b and
eliminating K from the equations results in the following
boundary conditions for the co-state:
(32a)
CLOZ;
tf
(32b)
O)
Ttt-t
where
-
3
r
t - tr
-
The final time (ti) can be obtained from equation 31c(equation
19).
3.5 Summary
The analysis of an optimal Earth escape maneuver to zero
energy results in a two-point boundary value problem of a 12t h
order system.
The implementation of the analysis in this thesis
is to guess an initial co-state and then by use of a Runge-Kutta
algorithm, integrate equations 15 and 22 until the energy equals
zero.
At each point of the integration the control law is
determined by equations 24 and 27 in conjunction with equations
28 and 29.
equation 32.
The final co-state is checked to see if it satisfies
If it does not (to a desired accuracy),-an iteration
scheme is developed that improves the initial guess of the
co-state. This iteration is continued until the desired accuracy
is achieved.
29
A Newton-Raphson iteration scheme was used to improve the
initial guess of the co-state.
A development of the Newton-
Raphson iterator is presented in Appendix C.
The computer
program developed to implement the above analysis is presented
in Appendix D.
30
4. DISCUSSION
4.1 Introduction
The purpose of this thesis is to develop the otilmal control
law necessary for a minimum time escape from a high Earth orbit
with a solar sail.
The end result is a computer program
(see Appendix D) that determines the time history of the optimal
control law.
With minor changes, this computer program could
be used to determine optimal escape trajectories from planets
other than Earth.
The computer program developed does not have active constraints to prevent the optimal trajectory from going below
a certain desired altitude.
This was not a serious Droblem for
escape trajectories whose inital orbit was above 100000 kilometers.
Some escape trajectories from initial ecliptic orbits with a
semi-major axis of 100000 kilometers or less did intersect the
surface of the Earth.
A recommended solution to this problem
is to add a penalty function to the performance index which
heavily penalizes trajectories close to the surface of the Earth.
A Newton-Raphson iteration scheme was used to solve the
two-point boundary value problem.
The computer program devel-
oped had problems converging for some initial orbits.
The con-
vergence problems were usually caused because the initial guess
of the co-state was too far from the optimal initial co-state
for the Newton-Raphson iterator to work.
One possible solution
to this problem is to develop a gradient method for solving the
two-point boundary value problem with a velocity-dependent
31
control law (discussed below) as the initial guess.
Another
possible solution to this problem is to use the velocitydependent control law to obtain an estimate of
escape trajectory.
the optimal
Using equation 32 and the estimate of the
optimal escape trajectory, an approximation to the final optimal
co-state can then be obtained.
The computer program in Appen-
dix D can then be used to integrate the state and co-state
backwards in time until the initial or near initial state is
reached.
The initial co-state as given by the backward integra-
tion will usually be sufficiently close to the optimal initial
co-state such that a converged escape trajectory can be obtained.
This latter method has proved successful in this thesis.
Although this thesis is concerned mainly with the optimal
control law, two other sub-optimal control laws were examined.
They are called open-loop control and velocity-dependent control.
An open-loop control trajectory results when equations 15
and 22 are integrated forward in time with the initial co-state
vectors as given below:
ro=ro
(33a)
(L)
(L)
The quantities ra' and
(33b)
oV
0 are the initial state vectors with
respect to the inital coordinate frame, and V
-C
is the circular
orbital velocity associated with ri.
At each point of the inte-
gration the control variables
are determined by equations
and
32
24 and 27 in conjunction with equations 28 and 29.
is made in open-loop control to satisfy equation 32.
No attempt
An open-
loop control trajectory would be the optimal control trajectory
if the initial guess of the co-state (equation 33) was the
optimal initial co-state.
Equation 33 results from the fact
that tangential thrust has been shown to be a good approximation
to the optimal thrust for initial circular orbits (ref 4).
Velocity-dependent control results when the acceleration
vector (equation 11) is required to lie in the plane of the x
axis of the rotating coordinate frame and the velocity vector
and is oriented in the direction that maximizes its projection
on the velocity vector.
A velocity-dependent control trajectory
results when equation 15 is integrated forward in time with
Y
and 9 being determined at each point of the integration by equations 24 and 27 in conjunction with ,- and
(coo,
,_
as defined below:
()V l(3a)
(34b)
te
=
C&O
Satisfaction of the constraints in equation 17 is also required..
Velocity-dependent control relies only upon the state, and therefore there is no need for the co-state equations and boundary
conditions.
4.2 Optimal Control
Figures 8 and 9 are the time histories of the optimal
33
control variables for a minimum time escape from an initial
150000 kilometer circular ecliptic Earth orbit.
rate of change of the control variable
degree per hour.
The maximum
is approximately one
The JPL square sail can sustain an attitude
rate of between 20 to 25 degrees per hour.
tude rates of the control variable
The infinite atti-
results from the mathemat-
ical singularities associated with cone and clock angle representation.
fore
At 1.7 and 4.5 days
is at 180 degrees and there-
can have any value for the same force vector.
the JPL square sail is not ideal, its
16, is not 90 degrees.
,
as given by equation
This is indicated as a straight line
between 5.9 and 9.1 days on figure 8.
During this period
be rotated through the necessary 180 degrees while
at
..
Because
can
is maintained
This results in an attitude rate of about 2.5 degrees
per hour.
Similar results were obtained for initial 100000
kilometer circular ecliptic orbits with resulting attitude rates
about twice as large.
The attitude rates of the solar sail
tend to be inversely proportional to the semi-major axis of the
initial orbit.
Figure 10 demonstrates some of the basic characteristics
of optimal escape trajectories from ecliptic orbits with a solar
sail.
The trajectory tends to become very eccentric quickly
with the instantaneous semi-major axis of the trajectory perpendicular to the Sun-spacecraft line in the orbital plane.
The
instantaneous pericenter of the trajectory is on the side of the
Earth on which the solar sail can achieve its maximum thrust in
34
the direction of the velocity vector.
Figure 11 shows the
increase in eccentricity for various initial ecliptic orbits.
The otimal
control law for an initial orbit inclined to
the ecliptic plane is shown in figure 12.
The initial orbit
considered is the initial circular ecliptic orbit in figure 10,
rotated out of the ecliptic plane by 30 degrees.
the x-z plane is maintained.
to figure 8.
Symmetry about
The control law for
is similar
The time to escape is 0.1 days longer than that
from a similar initial ecliptic orbit.
'
The control law for
is less demanding, and the trajectory tends to become eccentric
slower than it does for a similar initial ecliptic orbit.
The
instantaneous semi-major axis of the orbit is in the orbital
plane perpendicular to the Sun-spacecraft line.
As the inclina-
tion of the initial orbit is increased, the control law tends
to be less demanding, and the escape time tends to be longer..
The optimal control law for an initial 150000 kilometer
circular polar orbit is shown in figures 13 and 14.
variable
has a very low attitude rate.
The control
The attitude rate of
f' is about 2.2 degrees per hour at first and then falls off
considerably.
The optimal escape trajectory from circular polar
orbits maintains a low eccentricity throughout most of the escape
maneuver as shown in figure 15.
Similar results were obtained
for lower initial polar orbits.
As can be seen, the attitude
rates for initial polar orbits are less than those for similar
initial ecliptic orbits but the final time is longer.
It appears that the performance of a solar sail will tend
35
to be between the limiting cases of the resulting escape trajectories from similar ecliptic and polar initial orbits.
The
ecliptic initial orbit will yield the highest attitude rates
that can be expected and the smallest final escape time.
The
polar initial orbit will yield the lowest attitude rates and
the largest final time.
4.3 Open-Loop Control
Open-loop control is, as the name implies, an open loop
trajectory of the optimal control law with the initial guess of
the co-state vectors as given in equation 33.
For initial
circular ecliptic orbits, open-loop control is a good approximation to the optimal control.
For the initial 150000 kilometer circular ecliptic orbit
in figure 8, the final escape time predicted by open-loop control
varies from the optimal control final escape time by less than
1%.
For the initial 100000 kilometer circular ecliptic orbit
in figure 11, the error is around 4%.
As the eccentricity of
the initial orbit is increased the error between open-loop and
optimal control increases.
For the initial eccentric ecliptic
orbit in figure 11 (100000 kilometer pericenter, 0.25 eccentricity),
the error in the final escape time between these two methods is
about 8.5%.
As the inclination of the initial orbit with
respect to the ecliptic is increased, the error between optimal
and open-loop control also increases.
It can generally be
concluded that the open-loop control law is a good approximation
to the optimal control law for a minimum time escape from high
36
Earth orbits with low inclination and low eccentricity.
Figures 16 a and 16b were constructed using open-loon control.
Figure 16a show.ls
the final escape time for four various starting
positions from a 150000 km circular ecliptic orbit,
Figure 16b
shows how the energy increases during the escape maneuver of two
of the trajectories considered in figure 16a.
In figure 16b the
sharp increases in energy followed by little or no increases is
because in the ecliptic plane the solar sail increases its energy
from solar radiation pressure only when it is traveling away
from the Sun.
From figure 16b it can be seen that the initial en-
erqy of the solar sail determines how much energy must be added inorder to achieve 'escape. Given the initial energy, certain starting positions make it possible for the solar sail
to achieve es-
cape prior to making another revolution over that portion of the
trajectory where the vehicle is traveling
is the case for f
towards the Sun.
equal to 270 degrees in figure 16b.
For
This
'
equal
to 90 degrees in figure 16b, the effects of having to traverse the
'backside" of the trajectory an additional time in order to reach
escape condition is obvious.
It can also be seen that if the
eccentricity of the orbit is changed, the initial energy and starting position remaining the same, the final escape time will be
affected because the time in which the solar sail is in useftiul
sunlight is changed. It therefore appears that the final escape
time from ecliptic orbits is dependent not only upon the initial
energy of the vehicle but also upon its starting position within
the orbit and the eccentricity of the initial orbit.
37
4.4
Velocity-DePendent Control
The velocity-dependent control law orients the force vector
of the sail in a direction that maximizes its projection on the
velocity vector in the plane of the velocity vector and the Sunspacecraft line while still satisfying equation 17.
Velocity-
dependent control is rather interesting in that it is simplier to
implement and analyze and yet gives results that are close to
optimal.
For the initial 150000 km circular ecliptic orbit in figure
8, velocity-dependent control yields a final time of 20.33 days.
The optimal final escape time was 20.29 days, and the open-loop
control law yielded a final time of 20.29 days.
This gives an
error of about 1.2% between velocity-dependent and optimal control.
For the 100000 km circular ecliptic initial orbit in figure 11,
the error in final time between velocity-dependent and optimal
control was 1.7%.
As the initial orbit is inclined to the ecliptic
plane, the error between velocity-dependent and optimal control
increases.
For the initial 150000 km circular polar orbit in
figures 13 and 14, the optimal escape trajectory gave an escape
time of 29.45 days.
The velocity-dependent control law yielded
a final escape time of 30.42 days.
This is an error of about
3.3% between velocity-dependent and optimal control.
The open-
loop control law gave a final escape time of 32.7 days which
results in an error of 11% from the optimal control law.
The velocity-dependent control law was used to obtain
figurejsl7a and 17b.
Figure 17a shows that starting position
in the ecliptic plane has little effect upon the final escape
38
time.
Figure 17b shows that for a polar orbit the increase in
energy is nearly uniform through out the escape maneuver.
There-
fore the final escape time from polar orbits is almost totally
dependent upon the initial energy of the vehicle.
Starting
position in the orbit and the eccentricity of the initial orbit
have little effect upon the escape time from polar orbits
The velocity-dependent control law appears to be a good
approximation for the optimal control law of a minimum time
escape from high Earth orbits.
As the orbit is inclined the
velocity-dependent control law is a better approximation to the
optimal control law than open-loop control.
39
Optimal Control
-
Theta versus time
initial circular ecliptic orbit
r{= 150000 km i
V=
"0
JPL square sail
1.63014
km/sec
a, = 1 mm/sec 2
...
I
I .. ....
-
.
)
I
.....
I..
v
r
I
'.: .... ...
..
v-
.
.
-
.
-
.
.
.. ..
. .
.
.
.
.
..
·
.
..
........ ,..i. .......
.3.
.: ..
:lo
.
.
......________.__.... _.._.
... __. ._...._._.. ._.._........-... , ....._.. ._......._
....................... 1........1.- ..I
...
.. ...... ........ __.__ __... ,...-__.....
..... _......._. ..._..._..
.... .._....___.......
......
.... ,.,____,
... _.___......__ ..,......__ ..._ . ....._.___.....-,....-...
...... ..-.-. I-........... .I I...... ..--. 1..-.. .........-I..-.-.......
.
.
.._. .._..._..__. ..._.._.__ . ... _........_...._.. ............
_... ..
..1_... ._.._.
......... _................._ ....-..... I. ..........-- .....
~~~~~~~~~~~~~~~~~~~~..
IL.I_
.1-·II-.· -·DI-I..·-····.··i- - -----·II-· --.-.·-·--·I·---·.·- -I·_-· --·--·_···_·-··-·
··--
---
p.2
.-.
**~..-
.I ~~ t
-
6 __,1. Llh
_.::~~o. [..Z
Figure
.iI1I.XI~i
~8 ~ -~ 0till
iiii i
/d
/' I
-.
Z
-....-.-
t
4L_
I
. IIS
Optimal Control -
Psi versus Time
initial circular orbit
(60
= 150000 km
r
V) = 1.6301i4km/sec ,
a=
JPL square sail
=
1
mm/sec
2
q1o
t.o
80'
!
I.k
(Ms
'o
w
2
0
-
-
-
-'10
--
--60'
~.
-f
00
'"'
- -:-I--:
:-:- 111_........
. . -1-1.1
I , -_- - .__
. .- - ..: - . : I - . .. I . . -I .-I-I ._- . I - 1.
'- o
;1
..........-.
.....
...
·-i-1b
-WI-Jo-?·--*
................-
,P
r
IIA m
a I~WLA
Figure 9
-- -
'. "
_...
£ /c
ii/..
-
2' .....
.'.
.' 10
/
.
...
L
--
/8. ·
.
_Optimal Escape Trajectory
JPL square sail
a,, =
initial circular ecliptic orbit
mm/sec2
o = 150000 km
V
-0
=1.63014
km/sec
:
F
F~~
.
... .. .
4r
~~
1
:
sc-pe7
I)
0,1
N
0
0
II
II
II
0
~
0I=(
\~~~~~~~~~~~
'0
0i' '
'
o~~~~~~~~t
r4(
F~~~~~~
·-· ~-· ~ ~
H ~~
A~~~~~~~~~
4)
02
0 0
E
NB
5?~
0)r
~
4)
....
'0
0%
0
v.4
00
N
N
0
.4N
N
O
NC14
C\l
N
I
4)
.0
0
N
~% o
:
o
'0
N
0% 0
.0~~(%
,,I
414
0
C1
.4i
0
N.j
.4
.4 02C
or 0.1
1
,
^
c
0b
o 4)
.4
.4H0-
."4
N~
N'
.41 44
s~
·IM
t
9
...:
.
S~~~..
..
:
a.
02--
rzi
•i
0
C
.44
N
.1
.4
0
:s
..
W~ ~ .- ' ..,...
.
.
': ..
.
.
.
.
I
...... ,
'"
.
.
.
,
I
-. I
-
....
~..
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~OB
Figure
...
i ;
----
-
. ..i ....:..i.:i..;i.:;:';;''".''i;.-..
, .. . :..-. ..
~~~~9
h
.....
9
....
..
',,'
. Q..
. .
~~~
r
· · ·
B
..
'
,-
~
-
:.......
ry
.
t
· ·-
·
.
.
.
.
Optimal Control - Psi versus Time
Ot.d
initial circular inclined orbit
JPL square sail
IbO
ro
a,= 1 mm/sec 2
km
150000
V"=
1.4117
-o
.r
+ 0.81507 k km/sec
itoc
too
.
.
i
.
. .
.
.6
60
-
.0
-20
.1. .. . ..---.. . . - .. ... .
O .
-
-6..
.%tD
s
---o
Ii
i
...... -·· --.--. t0.0
!-20
i
I
.
..
.. ... ... :
L
-
. ...
-
-
I -
- .
. . .. -
. )
-1
- i SO
.. .. --..
-t. o
.. .
..
..
.
.
.
.40
............--;.'.
i u
¢._SO
1.-
Figure.l 2
-:-a -
-..
:
-
. we
S43
43
1.....
·.
1.
..
.
:
m
.'
_-'-
ODtimal Control - Theta versus Time
initial circular polar orbit
")=
,P
JPL square sail
a, = 1 mm/sec 2
150000km k
1.63014 km/sec
#"=
-O
0
1)
t
^J
U
Is
Itc
O
::::_-
0bt
:
--- -cFigure 13
Q
44
-
_Optmal
Control
Psi versus Time
(ik
Initial circular polar orbit - r,=
150000 km k'
JPL square sail
1.63014 km/sec
0
I"
ao= 1 mm/sec2
4J
'4
'i~~~~~~~~~~~~~~
--.
I
1
I
aS
:::':::-::-
II --_:_:I_:-::_-__-_
:
_-i
::i-
i
w
.CC)~~~~~~~~~~~~
,:, . ....-_.......
Fi~. ..e ... -.
~ ~~" . .
.............
.--
j rt'~"
-
0
,.o
-.o 'o
45
,lY-"
-_
-
0
.-
:- re
-ii~::r!:
: 14-:-4
:~-~~.~::
:~~'': ' ' ;i:''. . '...
.
10
!.
0
Control - Eccentricity versus Time
Otimal
JPL square
2
sail - a= 1 mm/sec
0
mn
e~
i
.iur
.5
_
~
~
.~-
~4
Io
Final Time versus Different Starting Positions
,en-Loop Control
circular ecliptic orbits
le
fY= angle in the orbital plane as
I, t
shown below:
id~o
J
L
I-X.
I');
-\
~.4
130
_
rIZI&
circular orbits with
ro
sai_.0
JPL square sail
ioo
a=
rs.ckm
1 mm/sec
tit
P-W
o70
70
.. 0
5.
::
...
-
I
:::--
_
.30
. 10
.0
-
.-. .. ar
Figure
16a
-l _
Foig
·
·-
° . .-'.
'
".
'
_
-- -.
---.'--..'
".4a
..
.
I
".
- ..
.. - ..
"-'- :
"''"`'~""
" '
r
UM
o
oO
4-J
.H
O
i
>I
tc
4J
U
.w,
.IJ
,rl
Ca
U
a
u
4-J
J
a
'rG
4
uQ)
Li
t4-4
(
J H
.. .
cI
C
Q
o
4-4
a)
r=(
11
Or
1'
rrz:
a
I
Figure
16b
47b
I
v
I
q
Li
14,
Final Time versus Different Starting Positions
JPL square sail a = 1 mm/sec 2
velocity-dependent control
O
circular polar orbits with. r = 15000U:- km
r=
angle in the orbital plane
c
(o)
lhn.mn
ma
vnrv., hr lnu?
I
I
3D-
to
. ..
A
'-.
A.
-M
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
lB .
II
i.
...
, Jr
-.... .
Figure17a.--
.
-.
0,
.
..
48a
.
. ..
. $so
......
~~~~
~
f·
..
,
.
.
.
,
e
.
1
..
a)
o
I1
Cd
n
Cd
0
-o
II
Cd
5-4
0
I
I
a)
0z
-r4
I
co
0
U)
h4
(1)
tc
i
4o
J
TO
p
(3
a)
a)
a)
1i
4,
.o
a)
t
Figure 17b
I
Z
3
-2
I
II
48b
O
-t
I
CONCLUSIONS
1. Escape from a high Earth orbit with solar sail propulsion
is possible.
2. Optimal escape trajectories in the ecliptic plane will give
the minimum escape time, but will demand the greatest attitude
rates.
Ecliptic escape trajectories involve very fast increases
in eccentricity, with semi-major axes perpendicular to the Sunspacecraft line.
Escape time from ecliptic orbits is dependent
upon the initial energy of the spacecraft, the starting position
of the vehicle in the orbit, and the eccentricity of the initial
orbit.
3. Optimal polar escape trajectories will have the least demanding
attitude rates, but the escape times will be longer than similar
inclined and ecliptic escape trajectories.
Polar escape trajectories
maintain a low eccentricity throughout most of the trajectory.
Escape time from polar orbits is dependent upon the initial
energy of the spacecraft.
Starting position in the orbit and
the eccentricity of the initial orbit have little effect upon
the final escape time for polar escape trajectories.
4. The performance of optimal escape trajectories inclined with
respect to the ecliptic plane will tend to be between the limiting cases of similar polar and ecliptic escape trajectories.
49
5. Open-loop control is a good approximation to the optimal
control law of a minimum time escape with a solar sail from
a high Earth orbit with low inclination and low eccentricity.
6. Open-loop control, with the proper scale factor, serves as
a good initial guess for the co-state vectors in the determination of the optimal control law.
The scale factor should be
chosen such that the final co-state is of the same order of
magnitude as the calculated optimal final co-state (equation 32).
7. Velocity-dependent control serves as a good approximation
for the optimal control law of a minimum time escape from a
high Earth orbit.
50
BIBLIOGRAPHY
1. Battin,R.H. Astronautical Guidance. New York, New York:
McGraw-Hill, 1964.
2. Bryson,A.E.,Hr. and Yu-Chi Ho. Applied Optimal Control.
Waltham, Massachusetts: Ginn and Company, 1969.
3. Cunningham,J.D. Optimal Control of a Solar Radiation Pressure
Powered Space Vehicle.(ad/a-004-792) Unpublished MS thesis.
Wright-Patterson Air Force Base, Ohios Air Force Institute
of Technology, December, 1974.
4. Fimple,W.R. "A Generalized 3 Dimensional Trajectory Analysis
of Planetary Escape by Solar Sail," ARS Journal, 32:883-887
(June,1962).
5. Grodzovshii,G.L., Yu.N.Ivanov and V.V. Tokarev. Mechanics of
Low Thrust Saceflight.
Translated from Russian by A. Baruch.
Jerusalem, Israel: IPST Press, 1969.
6. Hyde, Roderick A. Design and Behavior of Ribless Solar
Reflectors. Unpublished Ph.D thesis. Cambridge, Massachusetts:
Massachusetts Institute of Technology, 1976.
7. London,H.S. "Some Exact Solutions to the Equations of Motion
of a Solar Sail with Constant Sail Settings," ARS Journal,
30:198-200 (February,1960).
51
8. Sacket,L.L and T.N. Edelbaum. "Optimal Solar Sail Spiral to
Escape,"
to be presented at the AIAA/AAS Astrodynamics Con-
ference, September, 1977.
9. Sacket,L.L.,H.L.Malchow and T.N.Edelbaum. Solar Electric Geocentric Transfer with Attitude Constraints: Analysis. NASA
CR-134927(Draper Lab R-901), Cambridge, Massachusetts:
The Charles Stark Draper Laboratory, Inc., August, 1975.
10. Sacket,L.L., H.L. Malchow and T.N. Edelbaum. Solar Electric
Geocentric Transfer with Attitude Constraints: Program
Manual. Report R-902, Cambridge, Massachusetts: The Charles
Stark Draper Laboratory, Inc., August,1975.
11. Sands,N. "Escape from Planetary Gravitational Fields by
use of Solar Sails," ARS Journal, 31:527-531(April,1961).
12. Sauer,C.G. ,Jr."Optimum Solar-Sail Interplanetary Trajectories,"
Paper 76-792, AIAA/AAS Astrodynamics Conference, San Diego,
California, August,1976.
13. Schultz,D.G. and J.L. Melsa. State Functions and Linear
Control Systems. New York, New York: McGraw-Hill,1967.
14. Tsu,T.C. "Interplanetary Travel by Solar Sail," ARS Journal,
29:422- 4 27 (June,
1959).
52
15.
Villers,P. On the Application of Solar Radiation Momentum
Transfer to Sace
Vehicle Propulsion. Unpublished MS thesis.
Cambridge, Massachusetts: Massachusetts Institute of
Technology, 1960.
16. Wright,J.L. "Solar Sail Force Model," Jet Propulsion Laboratory Interoffice Memorandum (312/77.3-201). 28 March 1977.
17. Tlright,J.L. and J. Warmke, "Solar Sail Mission Applications,"
Paper 76-808, AIAA/AAS Astrodynamics Conference, San Diego,
California, August, 1976.
53
APPENDIX A
Force Equation Parameters
The values of all constants used in this thesis are:
Perfect Sail
JPL Square Sail
JPL Heliogyro Sail
r
1
0.88
0.88
s
1
0.94
0.94
E4
0
0.05
0.05
0.55
0.55
2/3
0.79
0.79
2/3
0.55
0o.55
0.5
0.349
0.367
0.5
0.662
0.643
-0.011
-0.010
B2
O
c
0
-3
The values for the constants in the JPL square and heliogyro sails were obtained from a JPL interoffice memorandum dated
28 March 1977 (ref 16).
These are the proposed specifications
on possible future solar sails being considered by the Jet
Propulsion Laboratory.
At high altitudes, where the attitude
controls are slowly varying, the heliogyro sail can be approximated by the model developed in this thesis.
Typical values of a
for the JPL square and heliogyro sails
vary from about 0.3 to 1.2 millimeters/sec2 .
an a
of 1 mm/sec 2 for all calculations.
54
This thesis assumes
Figure A-1 is a graph of the normalized force versus
the three types of solar sails listed above.
generating function.
The quantity
for
Equation 8 is the
is the cone angle of the
sail's force vector with respect to the x axis of the rotating
coordinate frame (see figure 5).
Figures A-2 and A-3 are polar plots of equation 11 for the
ideal and JPL square sail.
The polar plots are in the plane of
the primer vector and the Sun-spacecraft line.
The only quadrant
of concern is the upper right quadrant since this is were
satisfies equation 17.
0
The upper constraint on the cone angle
in equation 17 (180 degrees) results from the mathematical definition of a cone angle.
The lower constraint in equation 17 (9L)
results from the physical fact that the sail's force vector can
never point toward the Sun.
Ok is around 118 degrees.
From figure A-1 it can be seen that
Figures A-2 and A-3 also show the
basic principle behind the optimal
a primer vector, the optimal
projection of F onto
a.
.
Given the direction of
is the angle that maximizes the
When
is less than
, the maximum
projection of F onto a results when F equals zero or
Figure A-4 is a plot of the centerline angle ()
the cone angle ().
equals
e9.
versus
The centerline angle is the angle between
the sail's normal and the thrust vector (see figure 4).
analysis developed in this thesis determines the optimal
The
.
Using figure A-4 or an analytical representation, the corresponding optimal
can be determined. The following equation can be
used to find the incident angle
55
(see figure 4):
c>- =
/-X
f
S/80 -
(A-i)
The analysis developed in this thesis also determines the optimal
9 of the force vector (see figure 5).
Knowing
and o,
the
orientation of the solar sail is defined and the sail can then
be positioned in space to achieve the optimal thrust.
56
Normalized Force versus Theta
-
all
/
-.2.
/lo
IZo
-
Figure A-1
130
.
- 57
i
- : so
..
D
Polar Plot of Normalized Acceleration Vector for Ideal Sail
polar plot in the plane of the primer
vector and the Sun-spacecraft line
.75
normalized acoeleration vector =
CZ0
0
LC
-75-
Figure A-2
58
_
..
Polar
Plot of N~ormalizedAecereration
Vector iiir
ffor JPL
Suare
Sall
Saiii
ii iii
ii iiiiiiiiiii
iP iiiare
Veto
Aceeaio
Poa Plo o Nraie
polar plot in the plane of the primer
vector and the Sun-spaceoraft line
/
II
3/~
I
I0q
I
1_1.
c.0
normalizedacceleration vector
The arrow on the curve. indicates
.
the
direction of the normalized acceleration
vector as theta goes from 90 degrees to
Fig.re....A-3
Fgure
A-3
·
:
':..
'1'80degrees. 'When theta. is less than L
theacceleration vector is negative.
[l-59...
Centerline Angle versus Cone Angle
JPLLsquare sail with E80.35
IL
II
/O
?o
::
:::
s
.
- 3
I-:
t
-
-
..̀-·
-10.
~2:U&
· .c..... ....,
tI'- ~..-Y
.......... ...
,(Q,.
-.......
.:..
......
A
~~~~~~~~~~~~~~~~~..
....
Figure
~Fig~.
A W-.a--
...
:--·----
....
.....
.............
.
..........
6.
.. .
·--·-60--- -.- ; ; ;6:
.
...-
......
J30
... ,,,,.>: ~-/
~~~~~~~~~~~~..
.
......
9
.
.
.
..............
I:
.
.
.
...
..-....
.
..
.e .
20
.
.
APPENDIX B
Optimal Control Theory
This appendix develops the basic theory of optimal control
used in this thesis (ref 2 and 13).
Let the integral I be defined as:
_=F((xF_,,)
t
(B-i)
If the problem is to find the function x that maximizes I, the
integrand must satisfy the Euler-Lagrange equation (ref 13):
. dti
-A-
Also if the end point xi
=
(B-2)
is assumed to be fixed (xi and ti
specified), the following generalized boundary condition must
be satisfied (ref' 1.3).
L?
(~jI _
0(B-3)
The problem that this thesis is concerned with is the
minimization of the performance index J:
wh= 5 +ere
JL
where
61
~~(B-4)
S=S(i(tfr),tf)
L=L(x,u,t)
x=state
u=control
tf=final time (free)
ti=initial time (fixed)
It is also assumed that the following constraint equations are
imposed:
(state equations)
__zt
(functions
specified
at final time
)
Using the method of Lagrange multipliers, the constraint equations
can be adjoined to J, and therefore the redefined performance
index J' can be treated in an unconstrained problem.
Therefore:
'5'ii
~·(~--f~l
where
i
~-
Lagrange multiplier for
constraint at final time.
- Lagrange multiplier for state equations (co-state).
The following performance index can be defined:
PL = fL;
A4(ijAt S+fx if
62
t)-S(x~,ti
Since
(ti) and t
are assumed fixed the minimization of PI is
- .L
f
the same as the minimization of J'.
written as:
'5T
~
i>-
Therefore J' can be re-
g
d
By use of the chain rule this can be re-writtem as:
Defining the Hamiltonian HI such that maximizing it minimizes J:
i
(B-5)
Therefore J' can be re-written as
/J
)·1tta
(~-,X,
-, eL*(B-6)
Applying the Euler-Lagrange equation to J' and remembering that
x in equation B-2 corresponds to x and u in equation B-6
14_ d
4L!_
(B-7?)
__l-
63
(B-8)
-o
2.>'2
from inspection of the
amiltonian, the following expres-
sion can be obtained:
=
3L
(B-9)
Applying the generalized boundary condition (equation B-3),
the following expression can be written:
Substitution of L' gives the following:
Substitution of S'
_
and H and rearranging terms gives:
~_
__
_
therefore
(B-10a)
Lt
f]
Y(x(tt)
(B-lOb)
(B-lOc)
64
Therefore in order to minimize equation B-4, a Hamiltonian,
defined as in equation B-5, must be maximized.
Equations B-7,
B-3 and B-9 give the differential equations and control law that
must be satisfied, subject to the boundary conditions on the
state at the initial time and the co-state at the final time
(equation B-10), in order to determine the optimal escape
trajectory.
65
APPENDIX
C
An Energy Dependent Newton-Raphson Iterator
The solution to the problem of minimizing the final time
on an Earth escape maneuver to zero energy has been shown to be
the solution of the following set of coupled differential
equations:
r
(C-la)
= ,/
r3
- i)_
V
(C-lb)
_"(~)r~ -it)
,iL)
;'VV
.(0
ii
..
)(i)
r
(C-lc)
(C-1d)
The optimal control variables,
and (Y, are determined by equations
24 and 27 in conjunction with equation 28 and 29.
The boundary
conditions for the above set of differential equations ares
r
(o)
r0
V Co)
V
(C-2 a)
(C-2b)
Atw)
(C-2c)
a1 ()
*t.
66
(C
If'
(C-2d)
t tf
Eix
(#) ) j
where
-"
(C-3)
and * denotes the optimal end conditions as developed in Appendix B3. For an escape trajectory Ef is zero.
As can be seen, this is a two-point boundary value problem.
The method of solution will be to guess
A
o) and
AvOa)
and then
integrate equation C-1 forward until equation C-2e is satisfied.
The results of equations C-2c and C-2d are then compared to the
final co-state of the numerical integration.
If the two co-
states agree to within a desired accuracy, the problem is solved.
If they do not agree, an iteration scheme will be used to improve
the initial guess on the co-state.
This iteration scheme is
developed as follows.
Define the following function h:
=(i
-be
(C-4)
where
f(t4)
fLir
.V0T
~i~s·I-t
~__51,X
~
(fromnumericalintegration)
(fromequationC-2cand'C-2d)
67
If it is assumed that as E(x(t)) approaches Ef, energy is
a monotonic increasing function, a one-to-one mapping will exist
between tf and Ef.
In other words if a given Ef is specified
for a given initial state and co-state, the final time is
uniquely specified.
If Ef is changed, tf changes correspondingly.
It is also evident that if Ef is held fixed and the initial
co-state is varied, tf,x(tf),(tf)
and h will all vary.
The
quantity h is therefore a function of the initial co-state and
the specified terminating energy condition for a fixed initial
state.
This can be represented as:
=h ico) )
(c-5)
By use of a Taylor's series expansion, the following is
true to first orders
h (\IGI_
()
F
For the problem defined in this thesis, Ef is fixed; therefore
£F,equals zero.
Also the desired change in ~(o)
should
be such that to first order, h is driven to zero (final costate and optimal final co-state are equal). Therefore:
Define the matrix G as follows:s
68
Rearranging terms and substitution gives:
(o)cC t)(
) (
(C-61
The improved guess on the initial co-state can be written ass
t)(0) 3 t(Oii.,
d
t
)(0)
(C-7)
The implementation of equation C-6 requires the determination of the matrix G.
Analytically this is very difficult.
The method used in this thesis is to calculate neighboring traJectories.
The development of equation C-5 has shown that this
problem can be considered in an "energy space" instead of the
conventional "time space".
Each neighboring trajectory should
therefore be terminated when the energy equals Ef.
The result
of this technique is that G is a six-by-six matrix and six
neighboring trajectories need to be calculated on each iteration.
An obvious difficulty of working in a so-called "energy
space" is that equation C-1 has to be integrated in a "time
space".
Since the Runge-Kutta integration scheme takes finite
time steps, satisfaction of the energy stopping condition will
generally occur between integration steps.
A simple interpola-
tion is used to adjust the size of the last integration step.
This is shown graphically in figure C-1.
When the energy becomes greater than Ef on a particular
step, the average slope of the energy curve over the last step
69
is determined as
(c-8)
- £~ -~~~~~3~
The stepsize ssn,
which, if taken from to, will yield an energy
approximately equal to Ef at t=t0+ssn , can be expressed as:
' SOO-
(c-9)
The state and co-state are re-initalized to their values at
t=t0 , and the tile step ssn is taken so that condition E=Ef
is approximately achieved.
-
Eli
h-
$Sc
A S
A>
tf
J-scr-eTe t.me
&F
Figure C-1
70
APPENDIX
D
Computer Program
SOLCO (Solar Control) is the name of the computer program
developed to determine the optimal minimum time escape trajectory
of a spacecraft propelled by solar radiation pressure.
SOLCO
solves the two-point boundary value problem by use of a NewtonRaphson iterator (see Appendix C).
The sensitivity matrix needed
in the Newton-Raphson iterator is determined by comparing
neighboring trajectories to the nominal trajectory.
SOLCO is written in Fortran IV and uses double precision
arithmetic.
SOLCO requires a central memory core of about
60*103 bytes and a CPU time of between one to six minutes on
an Amdahl 470 computer.
The CPU time depends mainly on the initial
orbit of concern and the guess for the unknown initial co-state.
SOLCO consists of a MAIN PROGRAM and 9 subprograms.
The
MAIN PROGRAM and each of the subprograms are discussed seperately.
MAIN PROGRAM
This is the controlling program for SOLCO.
Subprogram INPNCL
This subprogram is called by the MAIN PROGRAM.
data is read and constants set in this subprogram.
data and constants are also printed out.
All initial
The initial
This initial data is
read in MKS units and converted to reference orbit canonical
here 1 DU=6373.16 km, 1 T'=806.3159 sec and/
units
=1 DU3/TU2 .
The data Is read. in with respect to the inertial coordinate frame
as discribed in Chapter 3.
The input variables are described
be1 ow.
AC = chacteristlc acceleration (mm/sec2 )
y(1) -
(3) = radius vector (km)
y(4) - y(6) = velocity vector (km/sec)
Y(7)-- Y(9) = position co-state vector (sec/km)
y(10) - y(12 ) = velocity co-state vector (sec2 /km)
prmt(1) = initial time (days), usually set equal to zero.
prmt(2) = terminating time (days) usually set large enough so
that it does not interfer with the energy terminating
condition.
prmt(3) = integration stepsize (days).
For an orbit with a
semi-major axis of 100000 km, a stepsize of 0.08 day
gave good results.
For a semi-major axis of 150000 km,
0.01 day was used.
EF = terminating energy condition (km 2 /sec2 ).
Usually set to
escape condition, EF=O.
IROT = Earth rotation factor.
If the effects of the Earth's
rotation about the Sun is desired, IROT should not
equal zero.
IPR = Print step.
The value of IPR causes every IPRth output
of the Runge-Kutta subprogram to be printed; that is,
IPR=1 causes every output to be printed.
INP = Plotting factor.
This program has the option of storing
the data from each print step on tape/file so that it can
72
be used later.
INP equal to a non-zero number will cause
storage.
KSTEP = Stepsize factor.
KSTEP determines whether the neighbboring
trajectories in subprogram NEWIT are perturbed by a fractional amount of the nominal values (KSTEP=0) or by a
fixed amount (KSTEP=1).
KSTEP=0 was usually used.
NIMAX = Maximum number of iterations allowed in NEWIT.
FLIM = desired accuracy of the Newton-Raphson iterator.
FLIM =
0.1 gave good results.
xs(l) - xs(6) = stepsize of the neighboring trajectories of the
co-state vectors.
Usually taken as 10-5 with
KSTEP=0.
Subprogram NEWIT
This subprogram is called by the MAIN PROGRAM.
Newton-Raphson
terator as discused in Appendix C.
trajectory is calculated.
are determined.
This is the
A nominal
Errors in the final co-state conditions
The initial co-state vectors are varied and
neighboring trajectories are calculated in order to get a sixby-six sensitivity matrix.
This matrix is inverted and post-
multiplied by the final co-state error in subprogram DCROUT.
New values of the initial co-state vectors are obtained, and a
new nominal trajectory is calculated.
This is continued until
the magnitude of the error in the final co-state is less than
some desired accuracy (FLIM) or until convergence fails.
73
Subprogram DCROUT
This subprogram is called by NEWIT.
Inputs to DCROUT are
the six-by-six sensitivity matrix and the final co-state error
vector.
The sensitivity matrix is inverted and the error vector
is premultiplied by this matrix.
The output of this subprogram
is the vector quantity by which the initial co-state should be
changed.
Subprogram PRNT
This subprogram is called by NEWIT.
Value
of the inde-
pendent and dependent variables of the iterator are printed on
each iteration.
Subprogram TRAJE
This subprogram is called by NEWIT. A trajectory is calculated with the initial conditions as determined by NEWIT.
This
subprogram also calculates the error in the final co-state
conditions.
Subprogram RK4
This subprogram is called by NEWIT and the MAIN PROGRAM.
This subprogram is basically the
4 th
order Runge-Kutta integra-
tor of the IBM Scientific Subrountine Package without the
accuracy checks.
The SSP integrator was changed slightly to
allow the terminating energy condition to be satisfied.
74
SubDrogram FIGNC
This subprogram is called by RK4.
This subprogram eval-
uates the right hand side of the system of first order differential equations of the state and co-state.
The effects of the
Earth's orbit about the Sun is considered in this subprogram.
The control variables ( and
). are also calculated.
Subprogram OUTNC
This subprogram is called by RK4.
condition is checked in this subprogram.
The limiting energy
If the terminating
energy condition is exceeded, a new time step for the RK4 subprogram is determined such that the terminating energy condition
is satisfied.
All output is changed from reference orbit
canonical units to MKS units.
Subprogram OPCOV
This subprogram is called by the MAIN PROGRAM. A summary
of the converged trajectory is printed.
A flow diagram of the subprograms of SOLCO is shown below
in figure
D-1.
SOLCO
I1
__
T'iJ.
5L%
-.
L
.. 4
A program listing of SOLCO follows.
76
D
f
CI
'L
ucn
D U)
Ut
7')
U)
L
L,t
W
U
LL,
l4
J
a
IU
I-U
i-Z
LU
a
L.
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I-
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c'"
-
U
U
aa<<
W.
Q
-11
L.!
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L
U
C.T
U
-
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a
It
Du. J
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3U.
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LL
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LU
L
-
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r.U
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L
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(1)oa
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a
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