Second order perturbation solution to modified optimal low thrust escape proble

Jacobson, R. A.

Powers, W. F.

English

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https://ntrs.nasa.gov/search.jsp?R=19710001928 2020-03-12T00:51:06+00:00Z
1%1 -V
May 1970
X(LC ION N
	
0
C;IL
O^jt0. (NASA C
*Performed under NASA Grant NGR 23-005-329.-
1•
02677-T-2
.
a
OPTIMAL LOW THRUST ESCAPE
t.	 VIEWED AS A RESONANCE PHENOMENON*
by
Robert A. Jacobson
and
William F. Powers
The University of Michigan
Ann Arbor, Michigan
sr
_	
+
yr
=	 s	 --;	 r^^;_LL
_	
g	
-	 -	 f	 -
-intf =
.^ {
1	
^`
Abstract
---	 In this study a complete second carder perturbation solution to a modi-
fied optimal low thrust escape problem is presented. The optimal thrust
	
-
direction is shown to be tangential to first order, and oscillatory to sec-
r
- s	 =
_	 -	 -	 -	 -
t
I. Introduction
i
^`	 A
r = radial. distance
6 = polar angle
V = total velocity magnitude
y = flight path angle measured from the local horizontal
= control angle measured from the velocity direction
a = thrust acceleration
as shown in Figure 1.
—>	 P l m ^ 3
M
s-
a
where
tPbqRi
Lt
2
-..
	
^	 -.	 }-.	 #s	 -.	 ^'^'	 ^T	 # £;	 rte_	 ^6 -— _	 _
_—^,
	
4Y'	 a	
-fa	 rx#.£PF 	 -££+-yt?	 £^£	 1^is3.	 ;it	 y	 -
_
z	 z 	 r	 xs
*
- 4
	
N&	 tom' #	 '#	 ^	 •.	 —	 ; 'i
,: _	 •	 .'..	 -	 4i	 a	 `-	 'l	I-nw- - '
}
3
The multiplier equations are
t i ^ X 2 	cosy - 'ZX3 4 siny + ^ 4	 -
r	 r	 r	 r V
_..
X3 = -a I siny - Xz	 cosy - k	 1 +	 cosy+ a	 sink,
_
r	 r	 rV	 V
i4 _ -XIVcosy+X? V siny+A3 ^ COS Y+A4	 V - -- `•—	 singr	 r	 r	 rV
The boundary conditions are
- to - 0	 AV-2 (tf} -	 E fr f
r (0) -- Ro	 X2 {t f} = 0
0 (0) - 0	 X4 (t f) - 0
V(0) _	 = Ro wo	 ,	 H (tf) - l
- RKS (tf)
Y(0) =	 k l (t f) 
-r t V={	 (	 0f}	 f)
where the last four terminal conditions are given by the transversality
conditions, and the particular choice of the initial flight path angle will be
explained later,
III,	 ApRroxi♦mate Analtical Solution
Since 9 does not influence the problem and X ? (t) = 0,, only the sixth
ordE r system defined by (r, v, y #X I : X3 , h4 ) w4_11 be considered.	 A solution
in the form of an expa,risivn in powers of a, the thrust acceleration, will be
assumed as follows:
r = ro
	 + ar i + a3 r3 + .. .
V =
 No + aV + a2V2 # ... -
. XI = ak31 + e X33 + ...	 -
Sin 	 + e.82 -}-
Cod	 i 4r	 + 
•,_^ -_' _
.	 t 3.r	 t	 N	 1
Ai
rM
W.-l	
- -
4
Ib
•
Ilia. Zero order state variable solutions
After substituting the above expansions into the system of equations
and grouping terms in powers of a, the zero girder state equations are
%o = Vo sinyo
V4 = - r^^ sin 1'0
1 (VO2
Yo =^ rfl -	 cos
The solution of these equations subject to the given initial conditions is
ro (t) = RQ
Vo{t} - R,000	 -
YOM - 0
That is the zero order solution is a circular orbit of rad us .
Illb. First order state and multiplier . solutions
The first order system of equations obtained from the expansions is
l".
i31_ ro + r Z ^43
which implies
X11 (t)
1
X3 1 (t) '$
k41 {t) 0
It follows immediately that
&,ft = 1	 cc^s = 1
IT
j	 NP
NO 3.-^	 -41WO	
-	 "tea'	 -	 - ^^'-$	 ^^i ^
	
^ ^ _ zt	 £	 y	 ^	 ' _ •	 -	 -	 -z s- iV
5
x 11 (t) = C;z + a Cos. (wat + ^')
)t31 (t) _	 +	 c 1 cos ,(coot + ^)	 'WO Rowo
X41 (t) = c l sin (wpt + ^)
From the expansion of the terminal conditions in powers of s we find to
first order at t f that
X11 (t f) - WO%31(tf) ' 0
X41(tf) = 0l
k314 f)
# _
AM
Ulc. Second order stag and multiplier solutions
After substitution oZ the zero order solution into the second order
equations, we have
r2 S RowoY2 + Y I V I
' TZ
 = -RjwoZY2 + Zc.)O2 rIY I
Y2 = A r2 + V2	 vi " - rIV I - N r  +	 ^I
i IZ = -	 ' X42 - 2(002Y 1X31
Ro
132 ' 
2^ 
A42 — kIIYI
1 42 = - P	 I2 +	 6ZX32 ` VI X II — ?"W )I I1 rI

`-
mx
_
WIN-
3 et i 	 4A kUk-U
rL
8	 l	 18 - cos wpt f
YY ^ = -1` i- t +	 t cosWO tt _ tf)	
w	
sin cant
In the above expressions it can be seen that "resonance" type terms
of the form t sin t and t cos t have been introduced by the control input
which oscillates at the natural frequency of the second order solution. It
will be shown that it is precisely these terms which make the optimal
"better" than the tangential escape trajectory.
We now have a complete second order expansion for the state variables
and multipliers of the optimal control problem as defined in section H. The
only remaining unknown is the final time which may be found by application
of the terminal energy condition. Clearly, the solution as given will not
hold to escape, is e. , E f = 0, since the various terms in the expansion will
become large and invalidate the assumed solution form. But if E f is near
E0 , then the solution should be accurate.. Furthermore, the control angle
for this energy increase problem should behave like that in the initial J*r-
tior► of the escape trajectory.
IV. Energy Increase Coml?arison
The rate of change of energy is
1 = a v cos
The small parameter expansion form of this equation is
E = a[VO&O + s (Vo&l + Vlao) + (Vt ao + as ve + al Vl } ]
since
V=Vs+aV1 + aivz
cos + = a$ + aa1 + at, az
'' Or
cro _ 1
	
a, = 0
	 (tangential)
Cr?	0
The rates of change of energy on the optimal and the tangential tra-
jectories are then
E 'opt = a[VO + aVi + a2 (VZ + V °aZ )]opt
E
^an ` a
[Vo + aV'1 + az V2 ]tan
Since there is no difference between the optimal and the tangential trajec-
tories in the zero and first order solutions,
Vo lopt = VO Itan
vi 
,opt = V1Itan
The second order expression for the velocity along the tangential trajec-
tory can be determined easily. Since sui = 0 along the tangential trajec-
tory, then P1 s 0. After substituting A, = 0 into the second order equations
of motion above, the second order tangential state solution can be obtained.
The second order tangential velocity is
VZ(tan
	
8 0 - cos wot)
Upon integration of the k-equations, the energy changes along the
two trajectories at any time are
1i	
2
	E^opt = R^wot - i at z + --
	 [
 (33 + 2 c os wo (t - t f) )cant - 3 6 s in wot
+ Z cos wdt f sin c ftt - 4 sin 2wa (t t f) - 4 sin ?.wot fl
tAEkan
 =P 	 - 4a 't  tZ + `-- a 132wot - 32sinw9t1
.
-7Q
9
For the tangential thrust program cos $ 1. which implies
^
_
+,
-
10
t
The energy difference between the two trajectories at' any time is
3
A(AE) = -----a 1 + 2coscoo(t- t )]wot -- 4sinwot+2 cos wot sinwot4RO Of
•	 A s in Zwo (t - tf ) - -I s in Zwotf
At the final time
3
1S (11E)f = 40 - 13wotf + 4sin 2wotf - 4sinwotf]
It can be*shown that A(A-E) f is positive for all wotf.	 In fact for small watf,
the series expansion of the trigonometric expressions can be used to show
t that
A(AE)f -- 40ROwe (wotf)5
Therefore, at the time the optimal trajectory reaches the specified termi-
nal energy, the energy level is higher than the energy level at the corre-
sponding time on the tangential trajectory.	 This- implies that the optimal
trajectory will reach a specified terminal energy level faster than the tan-
gential trajectory.	 It must be noted, however,, that at intermediate times_-
the energy on the' tangential may be greater than on the optimal. 	 In other
words, the tangential trajectory may reach intermediate energy levels
sooner than the optimal.
	
This phenomenon should not be entirely unexpected
since the minimum time trajectory to a given energy level is not the mini-
mum time trajectory to all lower energy levels.	 It is only the minimum =
time trajectory from the initial state to all of the states occurring along
the optimal.
It is well known that the tangential thrust program at each instant =
maximizes the rate of energy increase along a trajectory's .	 Therefore,
it is reasonable to ask how the optimal manages to improve on the tan -
f gential. Since it has been shown that the optimal is tangential to first order,
higher- order terms must produce the difference. In looking at the second
E
f
order solution, we Bind that the difference between the tangential and =__
_^-
II
optimal velocities may or may not be positive at any given time. However,
the difference between their mean values taken over a revolution-to-go,
from t = tf - (2N+ 2)n/wo to t = tf
 - 2Nn/wo , is greater than zero, i, e.
2
V2 1opt - V2 [tan = ©V - 2 - > 0.
We conclude then that on the average the velocity is higher along the opti-
mal trajectory. If the tangential velocity is compared to the component of
the optimal velocity in the optimal thrust direction , the mean value of the
optimal velocity component is also found to exceed the mean value of the
tangential velocity, i, e. ,
V1
	 2
V cos $Io t tan 4R > 0P	 owo
Therefore, not only does the optimal velocity exceed the tangential velocity
	 =_
on the average, but also its component in the direction of thrust exceeds
the tangential velocity. Recalling that the rate of. energy increase depends
only upon the thrust acceleration and the velocity component along the
thrust vector, we see that the optimal improves on the tangential by main-
taining a higher velocity component in the direction of thrust. The key to
the higher velocity on the optimal is the existence of the "resonance" type
terms in the optimal velocity expression which have been introduced by the
control angle oscillations.
Considering the motion from a physical viewpoint, on the escape
spiral the low thrust engine does work on the spacecraft causing its energy
to increase. The vehicle spirals outward increasing its potential energy
and decreasing its kinetic energy. The rate of energy increase depends
highly upon the vehicle's velocity, i, e. , its kinetic energy, and therefore
decreases as the vehicle moves out. The tangential thrust program maxi-
mizes the rate of• energy increase at each point along the trajectory but
12
kinetic energy are reduced. The higher kinetic' energy on the optimal then
gives the vehicle more capability for increasing its energy as it moves out.
The vehicle uses this additional capability in the latter portion of the tra-
jectory to add more energy than it could on a tangential thrust trajectory,
and in this way achieves escape in less time. The oscillation in the opti-
mal control angle is a result of the trade-off between keeping the rate of
energy increase high and the rate of kinetic energy decrease low.
V. Numerical Results
In order to test the accuracy .of the approximate analytic solution, a
comparison was made with an exact optimum energy increase trajectory.
The exact solution was generated by numerically solving the two point
boundary value problem using a secant iteration method. The initial val-
ues of the analytic multipliers were used as first . guesses in the iteration
scheme and seemed to work quite well. The initial values for the state
variables were
r(0) = 6.67817 X 106 meters
V(0) = 7.72580 X 106 meters/sec
y (0) = 2. 19518 X 10 -Z. radians .
These correspond to an initial orbit with the following eccentricity and
energy
eo = 2.0 X 10 -3
E,, '= -2.98440 X 107 newton- meters/ kg
The specified terminal energy was
E f = -Z.86218  X 107 newton- meters /kg
which corresponds to wot f -- 67r in the analytic solution. Since
WO = 1.15687 X 10 -3 , the analytic tf = 1.629353 X 104 seconds. The termi-
nal time found in the numerical solution was t  = 1.629351 X 10 4 seconds.
In Figures 2 and 3 a comparison between the analytical and numerical so-
lutions is made. The state variables are in close agreemnt, as are the
costate variables, with the primary differences appearing as a slight
13
mean value offset in the velocity costate, and as a small period discrepancy
in both the velocity and flight path angle costates.
	 The optimal control
angles are also close with only slight differences in period and amplitude.
A solution was also carried out using a terminal energy of
Ef = -2. 58695 X 10 7 newton-meters/ kg
In the analytic solution this energy level occurs at wotf = 20,r , or t 	 =
_ 5.43117 X 104 seconds.	 Numerical results give a final time, tf =
5.43116 X 104 seconds; and again the analytical and numerical state and,
costate variables differed only slightly.	 The determination of the full -
limitations of this approximation are currently under study..
VI.	 Conclusions
As a result of this analysis we reaffirm the well known fact that for
low thrust spiral escape trajectories, tangential thrust is nearly time
optimal, and in fact, is optimal to first order -in the thrust-acceleration
expansion solution.	 In addition, we now conclude that the observed oscil-
lation in the optimal control angle is a second order resonance type phe-
nomenon which reduces the velocity loss and therefore increases the rate
of energy gain along the trajectory.
Y
11 r
References
1. Tsien, H S. , "Take-Off from Satel.litu Orbit," Journal of the Ameri-
can Rocket Society, Vol. 23, No. 4, July-August 1953.
2. Benney, D. J. , "Escape from a Circular Orbit Using Tangential
Thrust," Jet Propulsion, Vol. 28, No. 3, March 1958.
3. Dobrowolski, A. , "Satellite Orbit Perturbations Under a Continuous
Radial Thrust of Small Magnitude," Jet Propulsion, Vol. 28, No. 10,
October 1958.
4. Lass, H. , J.Lorell," Low Acceleration Takeoff from a Satellite Orbit,"
ARS Journal, Vol. 31 No. 1, January 1961.
S.	 Cohen, M. J. , "Low Thrust Spiral Trajectory of a Satellite of Variable
Mass," AIAA Journal, Vol. 3, No. 10, October 1965.
6.	 Ting, L. , S. Brofman, "On Take-off from Circular Orbit by Small
Thrust," Z. Angew Math. Mech. ,44, 1964.
71	 Zee, Chong-Hung, "Low Constant Tangential Thrust Spiral Trajec-
tories, AIAA Journal, Vol. 1, No. 7 , .July 1963. 
8. Okhotsimskii, D. E. , "Investigation of Motion in a Central Field under
the Inml'luence of a Constant Tangential Acceleration," Cosmic Research,
Vol. 2, No. 6, November-December 1964.
9. Shi, Y. Y. , M. C. Eckstern, "Ascent or Decent from Satellite Orbit
by Low Thrust, "AIAA Journal, Vol. 4, No. 12, 1966.
10. Perkins, F. M. , "Flight Mechanics of Low Thrust Spacecraft,"
Journal of the Aerospace Sciences,. May 1959.
11. Moeckel, W. E. , "Trajectories with Constant Tangential Thrust in
Central Gravitational Fields," NASA Technical Report •R- 53, 1960.
12. Melbourne, W. G. , "Interplanetary Trajectories and Payload Capa-
bilities of Advanced Propulsion Vehicles," JPL Technical Report
32-68, January 1961.
13. Efimov, G. B. , D. E. Okhotsimskii, "Optimal Acceleration of a Space-
craft in a Central Field," Cosmic Research, Vol. 3, No. 6, November-
December 1965.
14. Sherman, B., "Low Thrust Escape Trajectories," Proceedings IAS
Symposium on Vehicle Systems Optimizations, New York, 1961.
9A
s^
•
15. Irving, J. Ii. , "Low Thrust Flight: Variable Exhaust Velocities in
Gravitational Fields," Space Technology, 10-01--10-54, (H. Siefert,
ed.) John Wiley and Sons, New York, 1959.
16. Lawden, D. F. , "Optimal Escape f.roin a Circular Orbit," Astronau-
tics. Acta, 4, pp. 218-233, 1958.
17. Gobetz, Frank W. , "Optimal Variable-Thrust Transfer of a Power-
Limited Rocket between Neighboring Circular Orbits," AIAA Journal,
Vol. 2, No. 2, February 1964.
18. McIntyre, John E. , Luigi Crocco, "Linearized Treatment of the Op-
timal Transfer of a Thrust-Limited Vehicle between Coplanar Circu-
lar Orbits. " Astronautica Acta, Vol. 12, No. 3, 1966. .
19. McIntyre, John E. , Luigi Crocco, "Higher Order Treatment of the
Optimal Transfer of a Thrust-Limited Vehicle: between Coplanar
Circular Orbits," Astronautica Acta, Vol. 13, No. 1, 1967.
20. Edelbaum, T. N. ,, "Optimum Power-Limited Orbit Transfer in Strong
Gravity Fields," AIAA Journal, Vol. 3, No. 5, May 1965.
21. Edelbaum ; T. N. , "An Asymptotic Solution for Optimum Power-
Limited Orbit Transfer ., AIAA Journal, Vol. 4, No. 8, August 1966.
22. Hinz, H. K. , "Optimal Low-Thrust Near-Circular Orbit Transfer,"
AIAA Journal, Vol. 1, No. 6, June 1963.
23. Melbourne, W. G. , Carl G. Sauer, "Optimum Thrust Programs for
Power-Limited Propulsion Systems," JPL Technical Report 32-118,
1961.
s	
tt
ii
I
Appendix
In the solution of the systems of equations it was found easier to con-
_
sider separate expansions of the control angle functions rather than to use
the multiplier expressions directly.
	 The optimal control program is de-
=	 fined by ￿_
tan	 _ - 4k3
 V
By assuming expansions of the form
V = VO +aV I +azVz+...
X3 = a\31 + a2X32 + a3 X33 + ... _
X4 - aX41 + a2X42 + W X43 +
we obtain
tan	 _ q0 + aq	 + a2 Tit + • • • -
where
X41
T10	 =
X31 v0
1
2
X42	 X41 	 +	 3^"'
X31 v0
-
a V0 	 X31
1 X	 X3 A	 ^ 2 • .v.!L	 = .. ^. )lX43 - X42	 + ^	 ) +X41	 +	 +	 "
(X31 VOV31	 0 V	 V2 X	 V0	 31 /	 31	 31	 0/0
Aft
Next consider
-
X	 V
sin 4)-	 4	 -	 cos	 -	 ^'
X4 +X3V
	
X4+ a3 V
which can be written as
cos ^ = [1 + tan2+]
ain • + = tan	 cos
"	 Substituting in the expression for tan	 and expanding in a Taylor's series
about a = 0;
cos +
	
. a►o + a&, + at ay + .. .
where
.ff	 f-S
• J
ao	 - (1 + ria)
-
ai	
- -no'l i (1 + 110 )0
- ^• z' `Zz	 zaz	 = - I3 ^1o'n^ (1 + nto)	 -
zoq2 + 211012) ( 1 + 'no)	 l .
i
s ^
r Using the expansions for tan and cos we obtain
sin 
= Po +aP1 + azPz+...
where
Po ° 110 (1 4- i1o)y z
_ 3
P1 - ?11 (1 
+^^).	
- 111710( 1 + 1b) "^
Pz = 'nz (1 + 'no)- - 4 (- 3,no^1 i + 1 ^1o'nz) (1 + '10) z + i non, (1 + 'no)
0
= ti
6.90
X
6.8
CO
^o
6.7
a^
.12
^v
cd
0
W
c
U
. 11
M
4175	 100175	 160175
Time (Secs)
Figure 2. Radial Distance, Total Velocity, and Their
Corresponding Costates
4175	 10,175
	
16,175
Time (Secs)
.4175	 10,17*
	
16,175
Time (Secs)
..^	 7.70
X
v
a^
7.65
U
> 7.60
H
102.2
0
a^
0 102. 0
H0
0
U 101.8
'*
i• 24+o
x
y
0!
.23
Cd
a
x
w	 .22
4175	 10,175	 16,175
Time (Secs)
Figure 3. Flight Path Angle, its Costate, and the
Control Angle
Time (Secs)
4175	 10,175	 16,175
N^	 -1
0
-4
a^
-7
a 
-10
.w°
-13
0
- 16
v^
U
Exact $
Analytic -8--
Time (Secs) ==
4175	 10,175	 16,175
.00 
-.03 
as
-
-.06 
.09
U =
- • 1 ' Exact $ '
Analytic -$-
F


Optimal low thrust escape viewed as a resonance phenomenon

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