The classical problem of determination of the rocket trajectory which minimizes mass expenditure during motion between two points in the field of a single gravitating body is analyzed. The known integrals of the system are incorporated into the adjoint equations resulting in a reduction from a seventh-order adjoint system to a third-order adjoint system. The first case which is treated in that of planar motion under specific end conditions. In this case a regularization of the recently derived equations is achieved. The general three-dimensional case is also reduced from a seventh-order adjoint system to a third-order adjoint system. In this case a regularization has not been found

Burns, R. E.

English

NASA Technical Reports Server

AN
NASA TECHNICAL NOTE NASA TN D-7592
CN
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I -
S (BNASA--TN-D-7592) INCLUSIG Of K OWN V74'-1653
INTEGRALS IN THE OPTIMAL TRAJECTORY
PROBLEM (NASA) -a p BC $2.75 CSCL 22C Unclas
_ H1/30 29215
/ ,
INCLUSION OF KNOWN INTEGRALS
IN THE OPTIMAL TRAJECTORY PROBLEM
by Rowland E. Burns
George C. Marshall Space Flight Center
Marshall Space Flight Center, Ala. 35812
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION * WASHINGTON, D. C. • FEBRUARY 1974
https://ntrs.nasa.gov/search.jsp?R=19740008424 2020-03-23T12:52:29+00:00Z
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 3. RECIPIENT'S CATALOG NO.
TN D- 592
4 TITLE AND SUBTITLE 5. REPORT DATE
February 197Inclusion of Known Integrals in the Optimal Trajectory PERFORMING ORGANIZATION CDE
Problem 6. PERFORMING 
ORGANIZATION CODE
7. AUTHOR(S) 8. PERFORMING ORGANIZATION REPORrT
Rowland E. Burns M 125
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. WORK UNIT NO.
George C. Marshall Space Flight Center 1. CON TRACi OR GRANT NO.
Marshall Space Flight Center, Alabama 35812
13. TYPE OF REPORT & PERIOD COVERED
12. SPONSORING AGENCY NAME AND ADDRESS
National Aeronautics and Space Administration Technical Note
Washington, D.C. 20546 14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
Prepared by Aero-Astrodynamics Laboratory, Science and Engineering
16. ABSTRACT
The classical problem of determination of the rocket trajectory which minimizes mass
expenditure during motion between two points in the field of a single gravitating body is analyzed.
The known integrals of the system are incorporated into the adjoint equations, resulting in
a reduction from a seventh-order adjoint system to a third-order adjoint system. The first case
which is treated is that of planar motion under specific end conditions. In this case a regularization
of the recently derived equations is achieved.
The general three-dimensional case is also reduced from a seventh-order adjoint system to a
third-order adjoint system. In this case a regularization has not been found.
17. KE~ WORDS 18. DISTRIBUTION STATEMENT
Cat . 3o
19. SECURITY CLASSIF.(of this report) 20. SECURITY CLASSIF. (of this page) 21. NO. OF PAGES 22. PRICE
Unclassified Unclassified , $ 2.75
MSFC - Form 3292 (Rev December 19 72) For sale by National Technical Information Service, Springfield, Virginia 2215 I
I
TABLE OF CONTENTS
Page
INTRODUCTION ..................................................
KNOWN RESULTS .................................................
THE PLANAR CASE ......................... 3....................
11
THE NONPLANAR CASE....................... 11...................
15
CONCLU SION S .... ............................................... 15
16
REFERENCES ...... .............................................. 16
PRECEDING PAGE BLANK NOT FILMED
ii
INCLUSION OF KNOWN INTEGRALS IN THE OPTIMAL
TRAJECTORY PROBLEM
INTRODUCTION
The optimal trajectory problem concerning a single attracting center - which is
usually taken to be that trajectory which minimizes mass expenditure - has been treated
in numerous publications.
Two important integrals of the problem were quickly isolated in the early
literature. One of these integrals, vectorial in nature, corresponds to a sort of "angular
momentum" conservation. The other, a scalar integral, is a sort of conservation of
"energy." It has long been realized that, if full use could be made-of the information
contained in these integrals, a great reduction in the number of numerical integrations of
the adjoint equations would be achieved. Thus, the iteration required would be faster.
The transformation which yields such a reduction has been elusive. In this paper
the transformation is explicitly demonstrated. The transformation exacts a price, however,
in that the resultant equations are ill-behaved from the numerical point of view.
In the case of a planar trajectory which has the magnitude of the radius and
velocity vector, as well as the flight path angle, constrained at the terminal point, a
regularization is achieved. In the general three-dimensional case, such a regularization has
not been found.
The first section below deals with known results. Subsequently, the planar case is
discussed in depth and the formulation of the three-dimensional case follows. A few
concluding remarks on other unsuccessful lines of approach to achieve a general
regularization are given.
KNOWN RESULTS
The general equation for rocket motion about a single attracting center is
ST Ar
r - (1)
m r
3
where r is the position vector of the rocket from the attracting center, T is the vectorial
thrust, m is the mass of the rocket, and a is the gravitational parameter of the attracting
center. Derivatives with respect to time are shown by the dot notation. The rate of
chang of the mass is related LUto tLL magnitudeU t0 le IIruSt, T, and the exhaust velocity,
c, via
rh = - T/c (2)
If one defines X as the solution of the vectorial differential equation
S3r r (3)
and further defines
X oS - (4)
m c
where
TX
S- - ,(5)
m
then it is well known [1] that an optimal trajectory (in the sense of minimizing mass
expenditure) will result if the direction and magnitude of thrust are chosen according to
T = T- (6)
Tmax if S > 0
T = (7)
Tmin if S < 0
The case where S - 0 for a nonzero interval of time will not be considered here (see
Reference 2).
2
Two important integrals of the system of equations represented by equations (1)
through (6) are given by
x r- x r = C (8)
and
TS - - = C 4  (9)
where C = (C1 , C2 , C 3 ) and C 4 are integration constants.
If IrI, I rl, and 0 = cos-ir are fixed at the final point and only these
quantities are fixed, then it may be shown that [1]
C = (C, C 2 , C) = 0 = (0, 0, 0) (10)
Furthermore, if the final time is unspecified, it is known that [1]
C4 = 0 (11)
THE PLANAR CASE
The case represented by equations (10) and (11) will be discussed first. Assuming
that equation (10) is valid, take the dot product of r with equation (8) and interchange
the order to yield
X(r x r) = 0 (12)
Geometrically, this result can be interpreted to indicate that the X vector will
always remain orthogonal to a vector which is orthogonal to the plane formed by ? and '.
Thus, one can write
3
r=  + q v (13)
where ( and 71 are scalar functions of time. The degrees of freedom represented
by and 7r are utilized by requiring that equations (8) and (9) be satisfied identically.
In Reference 1 it is shown that
r S
(- )= T( (14)
r-v r.v
and
T
r + -_ 2 = -2 (15)
mX
where
v =r (16)
and
X =  2 2 + 7 2 V2 + 2 ir . (17)
The numerical application of the theory developed to this point is now clear.
From appropriate starting values of the required quantities, equations (5), (14), and (15)
could be integrated forward in time. Equation (13) then yields the X vector as a
function of r and v, so that equations (6) and (7) give the direction and magnitude of
the thrust vector. Equation (1) may then be integrated, numerically, to yield updated
values of r and v.
The advantage of using equations (14) and (15) in lieu of integrating equation (3)
directly is that of dimensionality. Specifically, equation (3) represents three second-order
equations which are not necessarily simple to integrate. Equations (14) and (15) are two
first-order equations - i.e., a reduction from 7 to 3 in the number of integrations necessary
to determine the steering program has been achieved. Furthermore, the transversality
conditions [equation (8)] are identically satisfied; likewise, the fact that the Hamiltonian
[equation (9)] will remain identically zero is assured.
4
The advantage of reducing the number of required integrations becomes even
more important in light of the two-point boundary value problem associated with the
trajectory optimization procedure. The satisfaction of the four equations represented by
equations (8) and (9) is guaranteed no matter what initial values are chosen for the
variables and 7 . Thus, a vast reduction in iteration accompanies the reduction in
integration.
The difficulty in using equations (14) and (15) is that the occurrence of r v in
the denominator of equation (14) is a near-singularity. r. v will become sufficiently small
that overflow is almost certain to be encountered during the optimal trajectory
calculations. Therefore, the next process is the regularization of equation (14).
Set
= (18)
Then
X =  4 r2 + V2 + 2 4 r v (19)
Define, for convenience,
+* = V2 2 2 + 2r v . (20)
Using equations (18) and (20) in equation (15) gives
+ T 7) (21)
Next, differentiate equation (18) to yield
n + ' = 
- 2 2  m ) (22)
where equation (21) was used for 7.
5
Substituting equation (22) into equation (14) and employing equation (18) to
eliminate gives
v) ( - 2 ) - = -- a = ir . (23)
r m c
The expression on the right side of the first equality sign comes from separating S into
its elements via equation (4). Subsequently, equation (2) was employed.
To simplify the algebraic calculations, abbreviate
- (r ( - 2 2 ) - v2 (24)
so that equation (23) can be written as
n = n a (25)
Since the singular arc case has been excluded via choice, the thrust magnitude
must be either Tmax or Tmin . In either case rh is a constant. Thus, equation (25) can.
be differentiated to give
T X* 77
r4) + t47 =  h = i - (26)
m
2
from equation (5).
Eliminating /7 by use of equation (21) and assuming that
?? 0 0 (27)
allows equation (26) to be written as
( m T) = h (28)
6
The next task is to calculate < from equation (24). This calculation is somewhat
complicated but the result is
= 2 m X ( r 2 + - 2 ( - 242) + (r'-.) ( - 4 )
T v + v Tm *
_ 2 q - 42 ( r-v + v ) - + m 2
m r r2 (-v) + 4 ) + V2 _ T rv) 2  -mrr3
(29)
Inserting equations (29) and (24) into equation (28), gathering, and simplifying yields
( .V)$ _ ( .7) ) 6 + -3+ 4 3 -- ) 3 r v + 24 r 2 *
2T X* 2Tv 2  r T pr ) r (30)
m m x* m * r
Further simplifications of equation (30) are possible. Notice that
2T 2T 2 2T 2 -
-* v- (4r2 + 2r v) (31)
m mX* m**
Then equation (30) may be written- as
) + 43 + T T 0
- 4 + m* m * 3 m1 * r
3  )
(32)
7
In general r - v 0 so the first factor of equation (32) must vanish. Then
equation (32) becomes
- 5 + 33 + mT -+ + 0  + =0 . (33)
This is the equation which was originally sought. There is no longer a singularity
involving r - v.
The substitution represented by equation (18) also converts the state equation (1)
into a very convenient form. Using equations (18), (19), and (20) in equation (1) yields
-4 T - r
r - ( 4 r + v) 3 (34)
m X* r
Equations (33), (34), and (5) now constitute a complete set of differential equations for
the optimal trajectory problem.
Equation (34) may be put into an interesting form as follows:
4 -+ T - -+
r r xr=- v x r (35)
m x*
but
rxr - (v x 0 (36)
dt
Writing
L = rx v , (37)
8
equation (35) becomes
- T-
L - L (38)
m x*
This equation easily integrates to yield
t
T dt
L = Lo e 0 (39)
where Lo is the initial value of the angular momentum. Thus, L is always parallel to Lo
(which was known anyway) and equation (39) can be written in scalar form as
t
f T dt
L = Lo e 0 mX* (40)
A modification of equation (33) which makes the problem more numerically
tractable follows [3]
Set
= 0 (41)
Then, from equation (33),
p= = 50p - 303 + 0 + T* - 0) - . (42)
Thus
(5)p - 33 + ( +m T )(p - 02
dt dp m* r 3 (43)
(do 'do 
Pdt
9
or
dp 3 3 T 2  /1
S=5 - + + 1 (44)
d p m * p p r3
Next, set
0 = ev  (45)
and
p = w e 2 v  (46)
Then
dp dw e3v  ( T e2 v
-= e +2w e = 5 e - 3 + e+ T we2
do d v w e2v m * w e 2 v  w e 2 v r
(47)
Clearing and collecting yields
w + 2 (w - (w - 1)= 1 + T-w)( -1 e -2 (48)
dv 2 mX* r3
For the trivial case of field-free coast, equation (48) solves to yield
(w - 1) 1
v = In C (w (49)
( w 2),X
where C s is an integration constant.
10
THE NONPLANAR CASE
Although the initial set of equations, equations (1) through (9), apply equally
well to either two- or three-dimensional trajectories, equation (12) involves the
assumption of planarity. The next goal is to relax this assumption by choosing a nonzero
value for Z. Even so, the time-open problem (C4 = 0) will be assumed.
In order to allow for nonplanar steering, write
X = r + v+ (r x v) (50)
where , r7, and { are scalar functions of time. The variables t and 77 as used in this
section agree with the variables and r1 of the last section if - 0. From equation (50),
-= r+ v + q-v + 7v + (Tx v) + x ( v) (51)
and, from equations (1) and (50),
S T p rr (52)
v - - + 17v + (rx) 3.5
m X r
Calculating r x and inserting the result along with equation (52) into equation (51)
yields
+ T - +T T 2
T+ +~- r - v r+ + r+-  v
mX r mX mX mX
+ + 2- q (r x V) (53)
It is now possible to calculate i x r as
4xr= + + - 2 2 ) X + + 2---) [ rv .-)7],
mX mX mX
(54)
11
while from equation (50)
x =  (rx v) + [( r -)- v2 r1  (55)
Using equations (54) and (55) in equation (8) now gives
( T T 2 r2 )) [( _) 2
+ 2t + 2VX r) + +2T r 2 _ (T --
mX mX mX • v
2T -- 1 -+
+ + - 7 (r v) + v2  r = C (56)
mX
If the dot product equation (56) with -vx 7r is calculated, one finds
T C (v x r
n + 2 + -- 2 r (2 2 ) = (57)
mX2
Dotting equation (56) with v yields
2T C - v
S+ - = (58)
m 2ir x V 12
Finally, dotting equation (56) with r" yields
C r
S X (59)ir x vi2
12
Since two expressions which govern the variable " have been found [equations
(58) and (59)], it is necessary to show that they are compatible. This is most easily
accomplished as follows. From equation (52)
-* 4 d - T -
rx - ( x v) - [- (r(x ) + rx (rx v)] (60)
dt m
so that
d - T 2(rx v) -(r x v) - - (rx 2 (61)
dt mX
Then calculate
d 1 2 d -2T
- (rx d) (rx =v (62)
dt dt 2dt x )2  (rX v) 4  mX V ( )2
Assuming equation (59), one finds
C • v d 1 C v 2TqTC r ( S+ C r-- - (63)
(r x v)2 dt [x )2 (rx ) m X 0xv)2
Substituting for the combination of variables represented by equation (59) gives
2T C • v
+ - -+
mX (r x )2
Thus equations (58) and (59) are compatible.. From this point onward the discussion
deals only with equation (59) since equation (58) is redundant.
It is still necessary to derive the equation governing . To do this, return to
equation. (9). Calculating - . from equation (53).yields
13
S T , T t. - . T T 2 \
X v + - ( + + ~ r2 ' 2 V2
mX r mX mX m
(64)
From equation (50),
• r = r2 + r v (65)
Then equation (9) may be written as
. T C • (rx v) TS
m X mX Ir x v 12 r
(66)
where equation (57) was used to eliminate 7 . Assuming that r - v = 0 one finds
-. V_ -- 2 T -- mX mX
r(r•v) r x v = v
(67)
Equations (57), (59), and (67), along with equations (1) and (5) now define an
optimal three-dimensional trajectory. All of the advantages that were iterated in the
planar case are still valid; i.e., there are but two adjoint equations which need be
integrated. The nonplanar component of the steering vector, given by equation (59), is
carried algebraically, so no further integration is involved in the three-dimensionalization
of the problem.
The previously stated disadvantage of r -v appearing in the denominator of
equation (67) has returned. The obvious need is for a regularization akin to equation
(18). But in this case such a regularization has not been found. The primary reason for
this is, simply, the algebraic complexities which result when one attempts to deal with
the three-dimensional equations.
A fairly natural approach to avoid the singularity without the difficulties of
excessive algebraic manipulations is to modify the fundamental representation of
the X vector, i.e., equation (50). X could be assumed to be written with a vector basis
14
formed by the sets [, r x (rx v), (rx v)], [r, vx (rx v), r xv, [rx (rx v), (rx )], or
[v, x (r x v), (rx v)] just as equation (50) assumed (r, v, r x v) as a spanning set. In each
case the reciprocal r occurs, however, so this approach was not successful.
It is hoped that the difficulty of the singularity will be removed in future research.
CONCLUSIONS
It has been known for many years that integrals of the minimum mass
expenditure optimal trajectory about a single gravitating center exist. These integrals have
existed as a separate part of the literature and no direct use has been made of them.
In this paper it has been demonstrated that the use of these integrals yields a
reduced set of adjoint equations which is to be integrated. This reduced the time required
for two-point boundary value p_ ulem iteratiors.
In the planar case, a regularization has been found which improves numerical
properties of the equations after the inclusion of known integrals. Such a regularization
has not been found in the three-dimensional case.
George C. Marshall Space Flight Center
National Aeronautics and Space Administration
Marshall Space Flight Center, Alabama, November 1973
15
CA/ J0
REFERENCES
1. Bums, Rowland E: A Study on the Optimal Trajectory Problem. Dissertation,
University of Alabama, University, Alabama, 1971.
2. Bums, Rowland E: An Explicit Solution for the Lagrange Multipliers on the
Singular Subarc of an Optimal Trajectory. NASA TN D-4119, National
Aeronautics and Space Administration, Washington, D.C., Sept. 1967.
3. Siefert, H.: Deutche Mathematiker Vereinigung Jahresbericht. vol. 5, 1942, pp.
75-79.
16 NASA-Langley, 1974


Inclusion of known integrals in the optimal trajectory problem

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