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AN APPROXIMATE GUIDANCE LAW FOR ATMOSPHERIC PLANE CHANGE
MANEUVER OF AN AEROCRUISE SPACE VEHICLE
S. Hougui* and D. Mishne**
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Abstract
I n t h i s work a sub-optimal feedback guidance
law i s developed i n o r d e r t o minimize t h e f u e l
comsumption d u r i n g t h e atmospheric p o r t i o n of an
a e r o a s s i s t e d o r b i t a l plane change.
The guidance
law is developed through t h e use of an expansion of
t h e Hamilton-Jacobi-Bellman equation.
During t h e
atmospheric p o r t i o n of t h e t r a j e c t o r y , t h r u s t is
applied t o the vehicle (aerocruise) i n order t o
counterbalance t h e aerodynamic drag. The performance of t h i s maneuver is compared t o a z e r o t h r u s t
plane change maneuver i n t h e atmosphere (aeroglide).
The c r i t e r i a f o r t h e comparison a r e t h e
f u e l consumption, h e a t i n g r a t e , d u r a t i o n of t h e
maneuver and t h e s e n s i t i v i t y t o atmospheric d e n s i t y
variations.
Numerical c a l c u l a t i o n s a r e performed
f o r a t y p i c a l space v e h i c l e . The complete maneuver
is simulated u s i n g t h e e x a c t equations of motion
and atmospheric d e n s i t y p r o f i l e .
The c o n t r o l i s
determined from t h e guidance laws f o r t h e two types
of maneuver - a e r o c r u i s e and a e r o g l i d e . The t o t a l
f u e l consumption i s then c a l c u l a t e d i n c l u d i n g t h e
impulses belonging t o t h e p o r t i o n s of t h e
t r a j e c t o r y o u t s i d e t h e atmosphere.
The g e n e r a l scheme of f i n d i n g t h e guidance law
u s i n g t h e p e r t u r b e d Hamilton-Jacobi-Bellman
and 7 . I n Appendix A, a
equation can be found i n
s h o r t summary of t h i s technique i s given.
The Aerocruise Plane Change Maneuver
The a e r o c r u i s e p l a n e change maneuver c o n s i s t s
of two o r t h r e e t h r u s t impulses o u t s i d e t h e
atmosphere, and a l i f t i n g maneuver t o g e t h e r with a
continuous t h r u s t i n s i d e t h e atmosphere (see Fig.
1 ) . During t h e atmospheric p a s s , t h e t r a j e c t o r y i s
c o n t r o l l e d by changing t h e l i f t magnitude (by
c o n t r o l l i n g t h e angle of a t t a c k ) , t h e l i f t
d i r e c t i o n (by c o n t r o l l i n g t h e r o l l a n g l e ) , and t h e
t h r u s t ( i n a way t h a t i s explained i n t h e s e q u e l ) .
F i r s t , we d e r i v e t h e equations of motion f o r
t h e atmospheric p o r t i o n of t h e maneuver.
The Dynamical Equations
The equations of motion f o r t h e atmospheric
p o r t i o n of t h e maneuver are8:
Introduction
Aeroassisted t r a n s f e r has been recognized a s a
p o t e n t i a l way t o reduce t h e f u e l consumption of
o r b i t a l plane change maneuverlv2. Optimal guidance
laws f o r t h e a e r o a s s i s t e d plane change m
have been d i s c u s s e d i n t h e l i t e r a t u r e
Recently, approximate guidanc ,$aws, u s i n g feedback
These laws a r e
e x p a n s i o n s , were d e v e l o p e d
based on a n a l y t i c s o l u t i o n s which reduce t h e
computational requirements of onboard c a l c u l a t i o n s .
The g u i d a n c e law i s e x p r e s s e d a s a s e r i e s
expansion, where t h e z e r o t h o r d e r term is t h e
a n a l y t i c s o l u t i o n of t h e problem i n which t h e
i n e r t i a l f o r c e s a r e neglected with r e s p e c t t o t h e
aerodynamic f o r c e s . The e x i s t i n g solution^^'^ were
developed f o r t h e a e r o g l i d e maneuver, i.e. a
maneuver without any t h r u s t along t h e atmospheric
pass.
An a e r o c r u i s e maneuver, i . e . a maneuver i n
which t h r u s t i s a p p l i e d along t h e atmospher;~ pas$.
i s a l s o of i n t e r e s t .
The purpose of t h i s paper i s
t o develop an approximate feedback guidance law f o r
t h e a e r o c r u i s e maneuvers. A z e r o t h and f i r s t o r d e r
s o l u t i o n s a r e developed h e r e , and a r e compared t o
t h e z e r o t h and t h e f i r s t o r d e r s o l u t i o n s of t h e
a e r o g l i d e maneuver.
The c r i t e r i a f o r t h e
comparison a r e t h e f u e l consumption, t h e h e a t i n g
r a t e , t h e d u r a t i o n o f t h e maneuver and t h e
s e n s i t i v i t y t o atmospheric d e n s i t y v a r i a t i o n s .
2
*
**
dB - Vcosr cosy
dt r cos4
YfZ":Y
.
Graduate S t u d e n t , Department of Aerospace
Enginering, Technion,
if^, ~
~presently
~
with Rafael.
R a f a e l , Haifa. I s r a e l .
Copright
@
1990 by t h e American I n s t i t u t e of
A l l rights
Aeronautics and A s t r o n a u t i c s , I n c .
reserved.
3-
Vcosr sinY
dt -
r
dV =
dt
Tcosa
dr =
-
(Tsina + L)cosp
-
D
m
mv
dt
sinr
2
-
V
+ ( r- - Y 2
r) Vc o s r
(5)
where r , 9 and 4 a r e t h e d i s t a n c e from t h e c e n t e r
o f e a r t h , t h e l o n g i t u d e and t h e l a t i t u d e ,
r i s t h e f l i g h t path angle and l is
respectively.
t h e heading.
a i s t h e angle of a t t a c k , p i s t h e
r o l l a n g l e , T i s t h e t h r u s t , L is the l i f t , D i s
t h e d r a g , Isp i s t h e s p e c i f i c impulse of t h e r o c k e t
e n g i n e and m i s t h e mass of t h e s p a c e c r a f t . 5 i s
t h e g r a v i t a t i o n a l constant.
The guidance s t r a t e g y t h a t is implemented
d
u
r
i
n
g~
t h e atmospheric
pass is t h a t t h e v e h i c l e
~
l
.
performs a "zero a x i a l acceleration" t h r u s t law,
.
-
1.e:
Tcosa
-
D = 0
The motivation for this strategy is to reduce
the sensitivity to atmospheric density variations
and to extend the region of plane change angle.
The thrust is then converted from a control
variable to a function of the state variables.
We assume that the vehicle performs a plane
change maneuver close to the equatorial plane, i.e.
+=0,and at small flight path angles, i.e. sinr
r
and COST
1.
In the equations for j and p, there is the
expression (L + Tsina), which can be written as:
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L+ Tsina = L
+
Dtana
=
The state variables are (r, r, m) and the
controls are CL and p.
We assume an exponential atmospheric density
profile :
where f3 is
height.
the
exponential
atmospheric scale
Next, we define non-dimensional variables, as
follows:
(L +Da) = qs(CL+CDa)(9)
w
A psCL*B
= -
2mi
(non-dimensional altitude)
(20)
For the relevant range of angles of attack CDa
is small relative to CL.
Hence we neglect this
q A= Qn
term.
m
i
+ 'e
= !!-
m
i
2
u =A Q n -V-+re U = )1
Now the equations of motion become:
(non-dimensional mass)
"-2
P
(21)
(non-dimensional (22)
velocity)
where mi is the initial mass.
We assume that the vehicle follows a parabolic
lift drag relationship:
where K is the parabolic drag model constant.
shall also define the non-dimensional lift as:
dl = L sinp
dt
We
where CL* is the lift coefficient for which the
Equations (10) and (11) can be integrated
exactly to form the energy equation:
ratio C /C is maximum
L D
Next we define a small parameter
= const.
r
as
(15)
E, the constant of integration, is the energy per
unit mass.
where RE is the mean earth radius.
Now, equations (lo), (12), (13) and (14)
consist of the dynamical equations. By changing
the independent variable from t (time) to l
(azimuth) and using the expressions
L = %pv2scL, D = %pv2scD for the lift and drag, we
get:
dr =
dl
2mr
pSCL sinp
1
3 ,-
P
-dm= -
dP
tanp
+
('
The velocity during the maneuver is almost
constant, so we define
Using the above definitions, the equations can
be written as follows:
-
2m
P
T
)
rpscL sinp
V r
CD m V
CL sinp V
eq
where p is the density, CL and OD are the lift and
drag coefficients, respectively.
-dw
= - -
dl
'e 7
X sinp
and a test f o r minimum gives:
where :
The Zeroth Order S o l u t i o n
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By studying t h e above e q u a t i o n s , w e can d e f i n e
two c o n t r o l v a r i a b l e s
6 = XCOSP
( l i f t i n g force)
(31)
o = Xsinp
(side force)
(32)
F o l l o w i n g Speyer and crues7 a new independent
v a r i a b l e z , i s introduced:
The z e r o t h o r d e r s o l u t i o n is obtained by
n e g l e c t i n g t h e i n e r t i a l terms with r e s p e c t t o the
aerodynamic t e r m s , i . e . c=O and t a k i n g ep i n eq.
(34) a s a c o n s t a n t ( k , ) .
This implies t h a t the
e f f e c t of t h e change of mass on t h e a l t i t u d e i s
small.
Solving t h e optimal c o n t r o l problem defined
a b o v e , by u s i n g s t a n d a r d technique1',
the zeroth
o r d e r c o n t r o l s oo and 60 a r e obtained from:
where
The f i n a l dynamical equations become:
A
A
A
3
2
A
6
where (u, w, 7 , p a r e t h e s t a t e v a r i a b l e s . 6 and o
are the controls.
4
0
0
=
=
=
=
=
where Pw and C a r e given by:
The Optimal Control Problem
The optimal
follows :
control
problem
is
stated
as
Find t h e c o n t r o l s A, p which minimize:
S u b j e c t t o eqns. (33) t o ( 3 6 ) , t h e i n i t i a l
c o n d i t i o n s Yi, w , ' y i , pi a n d t h e t e r m i n a l
c o n d i t i o n s if.wf and r f .
and z f s i s found from:
The Hamiltonian of t h e system is:
A1
Zfs =
2
The F i r s t Order Solution
The o p t i m a l
Hamiltonian a r e
controls
obtained
from
the
Based on t h e zeroth order s o l u t i o n . h i g h e r
o r d e r c o r r e c t i o n terms can be computed and added t o
AS i n Appendix A, t h e
the zero order controls.
Lagrange m u l t i p l i e r s and c o n t r o l s can be expanded
i n t h e small parameter c :
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s u b s t i t u t i n g t h e above expansions i n t o
(40) we o b t a i n :
(39) and
o r b i t determines t h e i n i t i a l conditions of t h e
atmospheric t r a j e c t o r y .
The terminal h e i g h t was
a l s o taken a s 60960 m and f i n a l f l i g h t p a t h a n g l e
a s t h e n e g a t i v e of t h e i n i t i a l f l i g h t path angle.
A f t e r t h e optimal maneuver i n t h e atmosphere, t h e
OTV e x i t s t h e atmosphere and i s boosted ( i f needed)
t o t h e f i n a l o r b i t a l t i t u d e (300 km) and then
another impulse i s given i n o r d e r t o e n t e r t h e
final circular orbit.
It i s pointed o u t t h a t no
o p t i m i z a t i o n was made on t h e e x o a t m o s p h e r i c
trajectories.
The t o t a l f u e l consumption is then
c a l c u l a t e d i n c l u d i n g t h e exoatmospheric impulses.
To g e t a more r e a l i s t i c assessment of t h e
possible trajectories,
t h e aerodynamic h e a t i n g
i s s u e should be considered.
The h e a t i n r a t e was
estimated u s i n g an approximate expression'lo.
.
where p , t h e d e n s i t y i s i n kg/km3 and V , t h e
v e l o c i t y i n km/sec.
The h e a t i n g r a t e computed i s
i n Watt/cmz.
The following i s s u e s were i n v e s t i g a t e d :
Po? a r e obtained from t h e z e r o t h
22: 2L32 and
and:
-
t.
Comparison between t h e proposed a e r o c r u i e
m a n e u v e r a n d t h e a e r o g l i d e maneuver
The e f f e c t of t h e f i r s t o r d e r c o r r e c t i o n .
The s e n s i t i v i t y t o a t m o s p h e r i c d e n s i t y
variations.
Discussions of t h e R e s u l t s
A t f i r s t , t h e assumption about t h e e f f e c t of
t h e change of mass on t h e a l t i t u d e was checked.
D i f f e r e n t v a l u e s f o r kz ( t h e mass r a t i o i n t h e
equation f o r w ) were taken f o r t h e same i n i t i a l and
terminal c o n d i t i o n s f o r d i f f e r e n t t u r n i n g angles
(10". 20'. 30' and 4 0 " ) .
The c o s t f u n c t i o n ( f u e l u s e d ) was t h e n
c a l c u l a t e d and found n o t t o be s e n s i t i v e t o t h e
d i f f e r e n t v a l u e s t h a t w e r e c h o s e n f o r k,.
where :
Yd.
The e x p r e s s i o n s f o r t h e p a r t i a l s of R1 a r e presented
i n Appendix B.
Numerical Example
Methodology
A numerical s i m u l a t i o n of t h e atmospheric
t r a j e c t o r y , was w r i t t e n .
The s i m u l a t i o n s i m u l a t e s
t h e e x c t e q u a t i o n s o f motion ( 1 + 7 ) and d e n s i t y
profi*
A t y p i c a l O r b i t a l T r a n s f e r Vehicle (OW)
was used t o model t h e s p a c e c r a f t ( s e e Table 1 ) . A t
every i n t e g r a t i o n i n t e r v a l , t h e c o n t r o l s were
computed from t h e z e r o t h and f i r s t o r d e r s o l u t i o n s
a s found above.
Generally. t h e s c e n a r i o simulated ( s e e Fig. 1)
The OTV is assumed t o be c i r c l i n g
t h e e a r t h a t an a l t i t u d e of 300 km. It i s given a
t h r u s t impulse i n o r d e r t o change i t s t r a j e c t o r y
and e n t e r t h e atmosphere a t an a l t i t u d e of 60960 m
(200,000 f t ) .
The impulse given a t t h e c i r c u l a r
i s a s follows:
Simulation runs were performed i n o r d e r t o
compare between t h e guidance law derived above
a n a e r o g l i d e g u i d a n c e law t h a t was d e r i v e d 6
This law i s a l s o a feedback expansion law obtained
from
the
perturbed
Hamilton-Jacobi-Bellman
equation.
The comparison of t h e two s t r a t e g i e s was made
f o r p l a n e changes of l o 0 , 20" 30' and L O D . A t each
plane change, seven d i f f e r e n t i n i t i a l c o n d i t i o n s
were i n v e s t i g a t e d .
The above set of runs was
performed f o r a nominal d e n s i t y p r o f i l e and f o r a
perturbed d e n s i t y p r o f i l e .
Each r u n , a s described above, was made f o r
t h e z e r o t h and f i r s t o r d e r c o n t r o l law.
In the
r e s u l t s o b t a i n e d , i t was found t h a t t h e a e r o g l i d e
maneuvers f o r t h e plane change of 40' (both c o n t r o l
laws) and 30" f o r t h e z e r o t h o r d e r c o n t r o l law d i d
n o t meet t h e t e r m i n a l c o n d i t i o n s . The OTV l o s t t o o
much energy d u r i n g t h e maneuvers and could n o t
climb o u t of t h e atmosphere, o r i t e n t e r e d too deep
i n t o t h e atmosphere and t h e h e a t r a t e s were very
high.
I n c o n t r a s t ( s e e Fig. 2 ) , a l l of t h e
a e r o c r u i s e p l a n e c h a n g e s met t h e t e r m i n a l
conditions.
F i g s . 3 and 4 show t h e f u e l
consumption v s . maneuver d u r a t i o n , f o r plane change
angles of 10' and 20". The h e a t i n g r a t e values are
marked on t h e curves.
The aeroglide maneuver using the first order
control law, gave some improvement over the zeroth
order control.
However, on the aerocruise
maneuver, the improvement was negligible.
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Comparing the two strategies, it seems
generally that for the same amount of heating rate
the aerocruise maneuver is slightly more economic
than the aeroglide maneuver which follows a zeroth
order control law, but slightly less fuel economic
than the first order solution.
Figs. 5 and 6 show the sensitivity of each
maneuver type to atmospheric density variations,
for a plane change angle of 20'.
As anticipated,
the aerocruise maneuver is significantly less
sensitive to atmospheric variations.
In the simulation runs where the density was
perturbed by 25% and :25%, the aerocruise strategy
handled successfully all of the maneuvers. On the
other hand, the aeroglide strategy did not
accomplish every task.
order desired.
We assume also that all of the
states are observable.
The optimization problem is:
Find u(.)cU which minimizes J=+(yf) subject to
(A-1) and to the terminal constraints Y(yf) = 0.
The optimal return function, i.e. the cost of
going from state x to the terminal mainfold
Y(yf) = 0 using the optimal control uoPt is
A
P(x,t) = minJ = min [+(yf.tf)] on (y,t)(A-2)
ucu ucu
P can be found from the Hamilton-Jacobi-Bellman
equation
where HoPt, the optimal Hamiltonian is given by:
ap
(f+cg)l = ax
[fopt +
Conclusion
An approximate guidance law that is applied
during an aerocruise plane change maneuver, has
been proposed. The guidance law performance was
compared to a parallel aeroglide guidance law.
From a simulation of a typical case, several points
in favour of the proposed maneuver are evident.
cgl
(A-4)
and
fopt = f(x,uOPt,t)
The optimal control uoPt is found by the optimality
condition:
The main conclusions are:
For this example negligible improvement in
fuel comsumption was obtained using a first
order control law with respect to use of a
zeroth order control law.
The fuel comsumption in the aerocruise zeroth
order control law is slightly less than in the
aeroglide zeroth order control law.
Large turning angles were always accomplished
in the aerocruise strategy. These angles were
usually not accomplished by the aeroglide
maneuver (for the boundary conditions that
were investigated here).
P and u can be expressed as power series in c,
m
P(x.t) = minJ =
ucu
1
i
P.c
i=o
f(y,u,r) can be expressed as power series about the
zeroth order control uOoPt .
The aerocruise guidance law sensitivity to
atmospheric density variations is much smaller
than the aeroglide guidance law sensitivity.
Appendix A
The Perturbed Hamilton-Jacobi Equation
The dynamic system to be controlled is given
by:
Using (A-8) and (A-9), we get:
f(y.u.r) = f(Y.ugOPt.r)
+
aa uf
-
.. ..
opt 2
( U ~ O P ~ ~ r+ +.
U ~ )+.
Substituting (A-7) and
equation, we obtain:
y is an nxl state vector, u is an mxl control
vector, x is the initial state, and t is the
initial time. All the functions are assumed to be
differentiable to all their arguments so that the
asymptotic expansion can be carried out to any
+
(A-10)
(A-10) into the HJB
a R ~R~ [ -l-e-u
e'1-+-ee
ari - wr
E ~ *
ari
By e q u a t i n g t h e c o e f f i c i e n t s of equal powers
o f c , we g e t a s e t o f p a r t i a l d i f f e r e n t i a l
e q u a t i o n s . The s o l u t i o n o f t h e s e e q u a t i o n s g i v e u s
t h e v a r i o u s Pi's.
The e q u a t i o n f o r Po y i e l d s :
-aPo
= - -
at
aPo
ax
6o (l-e-u)e"l
Ev*
ar,
+
a R ~= R~
- [ -l-e-u
e'l-+-ee
ari w r E ~ * ari
opt
R1 [B
w
r
+
-11
+
aW
ar,
0
'
p -u -au+
E ~ *
ar,
fo
+
and f o r P
1
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+
0
'
q -u a~
E ~ *
api
,
'0
( ~ - e - ~ ) e *I
'
ar,
Ev
aR1
R
asi
wr E ~ *
6
-= C O
and i n general
R1 B
[;
-11 2~
ar,
(1-e -U)e'l
The above
following ones:
ar,
ari
+
are
dependent
ac
TI
where :
IKZz
au - -2e -
u -Ew Ear,
K
ari
The s o l u t i o n o f t h e s e p a r t i a l d i f f e r e n t i a l
e q u a t i o n s i s d o n e by u s i n g t h e method o f
characteristics.
The c h a r a c t e r i s t i c s o f t h e s e
systems a r e given by t h e z e r o t h o r d e r s o l u t i o n :
aPow
ari = -Ev*
+
ac
-ari
1
P
au -2e - u E E x
ari
- w ari
U
and P. i s given by:
f
These p a r t i a l s depend on t h e following:
The p a r t i a l s o f P . a r e given by:
(A-18)
Appendix B
The P a r t i a l s o f R,
The r i g h t - h a n d s i d e o f t h e d i f f e r e n t i a l
e q u a t i o n (A-13), f o r t h e f i r s t o r d e r expansion is:
The p a r t i a l s of R1, w i t h r e s p e c t t o t h e
current s t a t e variables, are:
(B-3)
(B-4)
partials
- -- -Ev* [KZz aPow
(A-14)
+
on
the
181 Vinh, N.X., "Optimal Trajectories
Atmospheric Flight". Elsevier. 1981.
in
[9] "U.S. Standard Atmosphere, 1976". National
Oceanic & Atmospheric Administration. National
Aeronautics & Space Administration, U.S. Air Force,
Oct. 1976.
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2832
[lo] Mease, K.D.. and Vinh. N.X., "Minimum Fuel
Aeroassisted Coplanar Orbit Transfer Using Lift
Modulation" , Journal of Guidance Control &
Dynamics, Vol. 8. No. 1. Jan.-Feb. 1985, pp. 134141.
[Ill Bryson, A.E., & Ho, Y.C., "Applied Optimal
C ~ n t r o l ~ Hemisphere
~,
Publishing Corporation.
Washington. New York, London, 1975.
Table 1. Characteristics of the O W
References
Initial mass:
Reference area:
Maximum CL/CD:
Specific impulse:
m =5443 kg
S =16.48m2
E*= 2.36
Is, = 295 sec
[I] Walberg. G.D., "A Survey of Aeroassisted Orbit
Transfer", Journal of Spacecraft and Rockets, Vol.
22, No. 1, pp. 3-18, Jan.-Feb. 1985.
[2]
Mease, K.D., "Optimization of Aeroassisted
Orbital Transfer: Current Status", The Journal of
the Astronautical Sciences, Vol. 36, Nos. 1/2,
Jan.-June 1988. pp. 7-13.
CIRCULAR
ORBIT
\
[2] Hull, D.J.. Giltner, J.M., Speyer, J.L. and
Mapar, J.. "Minimum Energy-Loss Guidance for the
Aeroassisted Orbital Plane Change", Journal of
Guidance, Control and Dynamics. Vol. 18, No. 4, pp.
487-493, Ju1.-Aug. 1985. .
[4]
Miele, A., Lee. W.Y., and Mease, K.D..
"Optimal Trajectories for Aeroassisted. Noncoplanar
Orbital Transfer, Part 2: LEO to LEO Transfer", IAF
Paper, 87-328, 38th Congress of the IAF, Brighton,
England, Oct. 1987.
[5] Calise, A.J., "Singular Perturbation Analysis
of the Atmospheric Orbital Plane Change Problem",
The Journal of the Astronautical Sciences, Vol. 36,
No. 1-2, pp. 35-43, Jan.-Jun. 1988.
[6] Mishne. D. and Speyer, J.L., "Optimal Control
of Aeroassisted Plane Change Maneuver Using
Feedback Expansions", AIAA Paper, 86-2136,
Atmospheric Flight Mechanics Conference.
Williamsburg. Va., Aug. 1986.
[7] Speyer, J.L. and Crues, E.Z., "An Approximate
Guidance Law for Aeroassisted Plane Change
Maneuvers", AIAA Paper, 88-4174. Guidance and
Control Conference, Minneapolis. Minn.. Aug. 1988.
INTERFACE
Fig. 1. The Aeroassisted Orbital Transfer
Maneuver.
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dpsi d e q
Fig. 2.
148
\
'.'.
Heat
Fuel consumption v s . plane change a n g l e
aeroglide,
------ .- .- .
zeroth order
aeroglide, f i r s t order
a e r o c r u i s e , z e r o t h & first orde
Rate
'.
~ ~ r nS e~
L'
Fig. 3 . Fuel consumption
C
v s . maneuver time, A l v = l O O
Heat
Rate
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(watt/crn 2 )
------
-.-.-.
200
250
300
aeroglide, zeroth order
aeroglide, f i r s t order
aerocruise, zeroth & f i r s t orde
35C
400
450
time s r c
Fig. L . Fuel consumption
v s . maneuver time, A1=20°
Fig. 5.1. E f f e c t of d e n s i t y v a r i a t i o n s on t h e
f u e l consumption f o r t h e a e r o g l i d e
maneuver, z e r o t h o r d e r c o n t r o l .
500
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t i m e sec
Fig. 5.2. E f f e c t of d e n s i t y v a r i a t i o n s on t h e
f u e l consumption f o r t h e a e r o g l i d e
maneuver, f i r s t o r d e r c o n t r o l .
Fig. 6 . 1 . E f f e c t o f d e n s i t y v a r i a t i o n s on t h e
f u e l consumption f o r t h e a e r o c r u i s e
maneuver, z e r o t h o r d e r c o n t r o l .
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-2832
t i m e sec
Fig. 6.2. E f f e c t of d e n s i t y v a r i a t i o n s on t h e
f u e l consumption f o r t h e a e r o c r u i s e
maneuver, f i r s t order c o n t r o l .
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