Computational and analytical techniques which simplify the solution of complex problems in orbit mechanics, Astrodynamics and Celestial Mechanics were developed. The major tool of the simplification is the substitution of transformations in place of numerical or analytical integrations. In this way the rather complicated equations of orbit mechanics might sometimes be reduced to linear equations representing harmonic oscillators with constant coefficients

Szenbehely, V.

English

NASA Technical Reports Server

ADVANCEDALGORITHMFOR ORBIT COMPUTATION
i By
VictorSzebehely
L.B.MeadersProfessor
Departmentof AerospaceEngineering
Universityof Texas
Austin,Texas
; ABSTRACT "
Th}s paper is a reporton the continuationof the author'swork with
Y V. Bondof NASA-JSCperformedin 1981-82. The subjectis the formulation
z of computationaland analyticaltechniqueswhich simplifythe solutionof
_4
complexproblemsin orbitmechanics,Astrodynamicsand CelestialMechanics.
The major toolof the simplificationis the substitutionof trans-
formationsin placeof numericalor analyticalintegrations.In thisway
the rathercomplicatedequationsof orbitmechanicsmioht sometimes be
reducedto linearequationsrepresentingharmonicoscillatorswith con-
stantcoefficients, i
i
= The first partof thiswork was reportedin severalpapersand reports
: by V. Bondand V. Szebehely,which are listedand discussedin the body
of this paper. One outcomeof the previouswork was the derivation
of an equationfromwhich the transformationsmay be computedfor a given
problem. Thisequationis knowntoday in the literatureas the "Szebehely-Bond-
EqL._ion."
The recentlyperformedwork r_nortedhere,generalizesthm p _inus
resultsto multi-dimensionalproblems,investigatesthe roleof in_egrals
in conjunctionwith the transformationsand discussessome of the,as yet
unsolvedproblems.
_ CenterResearchAdvisor: VictorR. Bond,
24-1 _,_.;J. •
1986004609-353
https://ntrs.nasa.gov/search.jsp?R=19860004632 2020-03-20T16:33:01+00:00Z
zi
INTRODUCTION
i The method of transforming non-linear differential equations of
,_ orbit mechanics into linear differential equations is one of the major
I
: problems in celestial mechanics. Hamiltonian canonical transformations
in the extendedphase-spaceattack this problem and the relation of this
approach to regularization is discussed in detail in several reference "
(
books, see for instance Szebehely, 1967. Regularization is the technique i
_':: to eliminate the si_Igularitiesof the differential equations of motion
_* and the associated transformations often lead to linearization, see for
.-
: instanceLevi-Civita, 1903 or Stiefel and ScheiTele, 1971.
" One of the first general investigation of regularization was performed
; by Sundman, 1912 who introduced a new independent variable and regularized
binary collisions in the general problem of three bodies. The purpose of
Sundman's work was not to linearize the equations of motion but to show
the existence of solutions of the non-linear bu_ regular differential
equations of motion. For this reason Sundman's work was not generally
accepted and was seldom used by workers in orbital mechanics until close-
• I
approach trajectories had to be computed in connection with lunar and planetary
missions. Regularization and linearization were rediscovered and were
described usually as "transformations"since, as it will be shown, new
independent and dependent variables are to be introduced to linearize.
24-2
1986004609-354
.: |
,E
:+-_: THEORY
+{
I The basic ideaof this projectis deceivinglysimple. The execution
is, on the other hand,extremelycomplicatedand difficult.
" " The equationsof motion in Astrodynamicsare secondorderdifferential
equationswith non-lineartermsand with singularities.All threedifficultiesi
" (secondorder,non-linear,singular)werealreadyknown to Sir I._aacNewton;
theyare due to Newton'ssecondlaw of motion (accordingto which the
ol
accelation,x is relatedto the force),and are connectedwith Newton'slaw
of gravity(accordingto which the force is inverselyproportionalto the
squareof the distance). The simplestdemonstrationof the problemuses
a one-dimensionalexamplewhich is representedby the equation
- Ux: , (1)
where dots denotederivativeswith respectto time, x is the distance
betweenthe participatingbedies_nd U is a constantdependingon the masses
of the participatingbodiesand on the constantof gravity. Equation(1) is _.
a secondorder,non-lineardifferentialequationwith a singularity
q
at x=O. In this,simplesto,=all _ases,linearizationis easilyaccomplished
by introducinga new independentvariable and consequentlymeasuring
time (t)with a new clock. The two times {sand t) are connected(Sundman,Igl2)
, by:
ds : dt (2)
X
I
24-3
', _lmmmmmv_m - - -
1986004609-355
: It might be seen that this equation introduces a new time s. As x _.
; becomes small and the usual time step _t is reduced during numerlcal
integration, the new time step As remains approximately constant. This
might be looked upon as a built-in time step control, popular in numerical
_" integration techniques. If we rewrite Equation (1) in terms of s instead ,
of t we obtain another non-linear equation, which, however, with the use
of the integral of the energ#, might be written in a linear form. The
analysis is simple, nonetheless, it reveals some fundamental aspects of i
the problem and, therefore, it will be reproduced here. The two
_ "time derivatives", i.e. the two velocities
_= dx X' dx
_-_ and = d'-_
* X !
; are related by x = -_- (3)
: and the two accelerations by
X n
---¢ - (x')Z/x3 (4)
,.
So the new equation of mot4on becomes
X II -- " _-J _../_.i_/¥1'_x (s)
which is just as unpleasant as the original equation was, see Equation (1).
s
So the famous Sundman transformation (Equation 2) neither re.qularizes nor
lineariz_s the equation of motion. Nevertheless, it might be shown that
_ 24-4 _,
._ -, t
1986004609-356
r$
., the term
,. (x')21x
becomes linear when the principle of energy conservation is used.
Equation (1) has an energy integral which might be written as
..
. ;_2=2u + 2h__
x (6)_ v
_-_; or as
(x,)2
_ x - 2p + 2 h x . (7) I
T
Consequently, Equation (5) becomes
• x" - 2 h x = p (8)
{
; This simple example reveals several difficulties of fundamental importance . !
in the theory of linearization, some of which have still not been over- I
come in the case of the n-dimensional perturbed motion.
I
As we have seen, the use of Sundman's transformation was not sufficient
to linearize the equation of motio, and an integral of the system had to be
y
used to accomplish linearization. Furthermore, the dependent variable x
i was not transformed. The fact is that with a proper x:f(y) transformation,
t,
i Equation (1) may be linearized without the use of the energy integral.
..I
"_' i _t _
tt
1986004609-357
iConsequently, in principle, non-linear equations without energy conservation
might be linearized when the proper dependent variable is selected. This
is discussed in considerable detail in Szebehely(176a,b) ; Schrapel (1978);
Szebehely and Bond (1982 and 1983); Bond and Szebehely (1982)_ Mittleman and
Jezewski (1982); Belen'kii (1981), etc.
Now that the basic approach and some fundamental problen,s have been
presented, we are ready to increase the dimensionality from one to two. One
: of the recent papers on this problem is by Szebehely and Bond (1983) in
_: which the Szebehely-Bond equation is derived in the form:
_" Ap + B = "/2H-_ [(__.)2 G], (9)
I
; where r=F(_) is the transformation of the old radial cordinate r to the
new one p, g(r) is the function controlling the time transformation which ,
now becomes
ds--g- rt), (lO)
) the functioFJ
2
G=2h + _ _ c
r r_ (ll)
represents the,energy integral and
dF _
do" (12)
Those functions (F and g) which satisfy Equation (9) will linearize
)
the two-dimensionalequatioas of motion. Various combinations of these
functions were and are discussed in the literature (beginning with Kepler).
24-6
1986004609-358
"}" "I
:i The most recentis by Ferrer,to be publishedin CelestialMechanics,
,,_ [Ferrer,1983].
"d
Similartechniquesare availableand applicableto accurate
: f_rdockingof 'orbit calculations for relative motion of satellites ,
" space probes,etc. [Nacozyand Szebehely,1976;Szebehely,1975and 1976c;
-. etc.]
_I ° Transformationsleadingfrom unsolvedr._r_-lineardifferentialequations
_i to solvednon-linearequationsare also popularin the mathematicalliterature.
_ Thesetransformationsdo not transformthe independentvariableand,
.=
L consequently,might not be idealfor problemspertinentto celestialmechanics,
=. nevertheless,theyare mentionedhere sincetheymay open up new avenues
of research[Dasarathyand Srinivasan,1968;de Spautzand Lerman,1967
? and 1969].
- 1
24-7 i _,_
W
1986004609-359
" RESULTS
The generalizationto n-dimensionalmotionwas performedduring
I
the periodof May g - July 15, 1983. The resultswill be summarizedin
this sectionusinganalyticaldescription.The verbalevaluationof these
resultsis in the sectionentitJed CONCLUSIONS.
The two-prongedattackmay be describedas using the directand 4
; the inverseapproaches.
L
The analyticalformulationmay be representedby matrixnotation
_J
_ or by subscriptnotation.
_Z Consequently,fourbasic equationsrepresentthe results.
m
The directapproachstartswith givennon-lineardifferentialequations
and attemptsto findthe transformationswhich resultin lineardifferential
equations. The transformationsof the independentand dependentvariables
are __.
ds:dt (13)g
and
Yi = Fi (xj);xi = Fi (Yj) (14)
Using matrix notation these become
(
_; 'I t
,l P = F (R) ; R : @ (P) . (IS) , .
I
i
]986004609-360
& " i
; i %
Here g=g(xi) or g=g(R) is the function which controls the transformation
, of the independent variable t. Furthermore xi is the i-th component
of the pjsition vector appearing in the original, nonlinear differential
equation. The corresponding vector in matrix notation is R. The dependent
• variable of the transformed linear equation is Yi or in matrix notation P.
• Consequently, Equations(14) repre._entthe c_ordinate transformations in
subscript notation and Equations (15) in matrix notation. All symbols
represent vectors (R, F, R, _ ) or components of vectors (Yi'Fi'xi'fi)
_-: excepting the function g which is a scalar depending on the original *
• dependent variable. The function g in the literature is often called a
scalar- vector function.
; The original nonlinear equation to be transformed is
,T-
: Ri + Hi(xj, xj, t ) = 0 (16)
or R + H (R, _, t) = 0 (17) i
>
4
where dots represent derivaC'ves with respect to time.
The d)sired result of the transformation is
P
,i
ss e
Yi + aij Yj + bij Yj = 0 (18)
_:*' I or
:_: i. P"+AP' + BP = 0 ,
, (19)
*: 24-9 _._._-._. _, d
1986004609-361
where primes are derivatives with respect to the transformed time s
and A(aij), B(bij) are matrices with cc-stant or s -dependent elements.
Equations (16) and (17) are the equations to be transformed and are
• ?
represented here in their most general forms. After the transformations,
Equations (16) and (17) become
,Jg-1 g2
_ . , ,
_: Yl + g F1 i (fi )'k YkYj- F1 iHi = 0 (20)
_.; , ,
.._"
_ and
p,,+ (@.)-I [(_**__ I dg _*)p,]p,_g2(¢*)-iH = 0 (21)g d@
o i
Here
y0 : d_ , p, _ dP
1 dT ds ' _;_
: * d_
_Fl and ¢ -
Fl,i @x dP
1
Consider Equations (Ig) and (21). In order to accomplish linearity
the two last terms of Equation (Ig) must be equal to the two last terms
of Equation (2!). Similarly for Equations (18) and (20). These are the
conditions to be _.atisfiedby the transformation fui_ctiens,g and F in order
to obtain linearizaticn.
!
"1
1986004609-362
• !'
' The problemsassociatedwith these requirementswill be discussedin
the sectionentitledCONCLUSIONS. Severalexampleswere investigatedand
• interestingand unexpectedresultswere obtained.
++ The inverse approachstartswith the linearEquations(18)and (19),
then the transformations,givenby Equations(14)and (15),are appliedand ,
the followingresultsare obtained:
,
_ . /
+_;: Rl + C+ljk x.] xk + 81k x k + Yl = 0 (22) _'
_+ and C
'- +. i
! R + [A R ] _ + BR + I"= 0 (23) !
where J
+'- _ljk ' Blk ' Yl' A, B and r dependon
xi, R, F, g, aij,A, bij, B .
Equations(22)and (23)describethe type of non-linearequations._:hich
might be linearizedby properlyselectedtransformations.Once again,
the requirementsplacedon the transformationswill be discussedin the next
section. It is notedhere thai Equations(22)and (23) _,orin otherwords
l
B y,A,B,r) are availablein formssimilarto the detailsgivenin
Equations(20)and (21).
+
,+
24-11
_n
1986004609-363
CONCLUSIONS
(I) The two major approaches,using two differentnotationswere
formulatedand a thoroughstudy of the comparisonrevealedcomplete
agreement. Fromthis it may be concludedthat the main results,i.e.
I
Equations(20)- (23)are reliable.
(2) Specificexamplespertinentto orbit mechanicshave shownthat
-
_- ir,tegralsof the motionplay an importantrole in additionto the tran'-
formationsselected. _,
_- (3) Transformationsgiven in the literaturewere substitutedand the :
requirementsmentionedin the previoussectionwere satisfied. !
(4) Itwas foundthat the linearizedsystemsdid not necessarily; j
representth:_finalsolutionsof the problemsand presentlydiagonalization
and triangularizationrequirementsof the matricesA and B are investigated.
(5) The literatureconcerningtransformationsof nonlineardifferential
7
equationsis impressive,to say the least,and the numberof references ,,.,j
. given here couldbe easilytripled. I
(6) The transformationsdescribedIn thisreportare restrictedand
their generalizationsmight be of considerableinterest.
(7) Lineardifferentialequationsdo not necessarilyhave Lyapunov-stable
C
sulutions. Thisshouldinfluencethe selectionof the transformationfunctions.
(8) Thereare severaldynamicalsystemsof considerableimportance
_ in orbit mechanics which represe_,_, so-called non-tntegrable systems. If
24-12 *
- ®
]986004609-364
L¢
thesesystemsare investigatedin the lightof the presentrepo,.,we arrive
' ' at one of the followingconclusions:
(i) Non-integrabledynamicalsystemscannotbe transformed
to linearsystemssince linearsyst,;r,,sare integrableand _;._r
integrals,transformedback intothe systemof the original
_" variables would produceintegralso; the system. The con-
trad_ction,_ht be resolve_by claimingthatthe transformations "
do not exist.
¢7
_ (li) Anotherresolutionis th,ar.non-inte9rables_stemsare in reality
_ -
_ i not-integratedsystems,n:e_ningthatthe non- ._._grability
i conditionexistsonly undercertainconditions,see Pol_care's
'_ and Brur_s"assumptionsconcerningthe non-integrabilityof
the restrictedand of the generalproblemsof threebodies.
: Accordingly,transcendentaltransformationfunctionsmight
resultin linearizationand corsquentlyJr.showingintegr&bility
of these famous"non-lntegrable"dynamicalsystemssincesomeof _
the above-quotedconditionsclaim non-inteT..aoilltyin termsof
,, algebraicfunctions.i
_, (
(lii) Furthermore,it i_ knownthatcertainnon-lntegrableSystems
I
, have 1ocallyvalid Integrals.Theseshouldcorrespondto locally
f
valid transformationswhich s,,ouldsatisfythe requirementsmentioned
i in the previoussection.
I
e
............................. L , _ ._ ., t
]986004609-365
J(9) The two, seemingly most significant conclusions are left ._
to items (9) and (I0). From a practical point of view, transformations
which reduce or eliminate numerical integrations are of the utmost
importance. Numerical accuracy is increased and the time requirement
for computations is reduced. Autonomous operations require such im- _
provements and their executions are associated closely with the success
!
of establishing the proper transformations.
" i
i 1
(lO) Establishing transformationseither to linear systems or to
_j integrated non-line_ systems might be considered one of the greatest ii
accomplishments of modern celestialmechanics. Accurate long-time
predictions would be possible for any length of time. This is intimately !
: associated with the study of the stability of the solar system and of the }
origin and evolution of the Universe. To integrate "non-integrable"
!
systems would show that these systems should have been called "not-integrated"
systems to begin with and would challenge the foundation and the famous
t and classical results of celestial mechanics.
1
• 24-14
]986004609-366
, .. w,,,_. ,j "Z,,;,."',i _',,,,w"' " _ , '_" " .4 " ",,w,v'w, ,, • '. ....,_," , ' .,Pa " , , _ ,, , _ • " _ ._,_ , , _....... .*z,,_Q_ _z,_
_ r
.\
_ ',
i
" Ack edg :, ' nowl ements
i This work could not have been performed without the cooperation :
i
of two eminent NASA-JSC scientists, Victor R. Bond and Donald d. dezewski
•. of the Mission Planning and Analysis Division.
Considerable amount of technical support was received from two
visiting faculty members, Dr. A. ,Jackson and Dr. R. Sartain whose
s , _ _ .L
cooperation proved to be most beneficial.
T
_. _ [
:,-_ Part of this Report was typed by Judi Gilbreath of the Summer Faculty
._ ; Program of the Johnson Space Center, NASA and part by Mariou Barr of the
- : Department of Aerospace Engineering of the University of Texas a.tAustin.
Their wonderful and most valuable cooperation is gratefully acknowledged.
-. il
4
1/
|
•I._ .,_"
, _
. i
]986004609-367
REFERENCES
I. Belen'kii, I.M., "A Method of Regularizing the Equations of Motion
in the Central Force-Field," Celestial Mechanics, Vol. 23, pp 9-23,
1981.
2. Bond V. and Szebehely V., "Transformationsof the Two-Body Problem
to Harmonic Oscillators," NASA-JSC-17933, March 1982.
3. Dasarathy, B.V. and Srinivasan, P., "On the Use of Certain Trans-
i formation Techniques in the Study of Non-linear Systems," International
Journal of Control, Vol. 7, No. 4, pp 361-364, 1968.
4. de Spautz, J.F. and Lerman, R.A., "Equations Equivalent to Nonlinear
Differential Equations," Proceedings of the American Mathematical
Society, Vol. 18, pp 441-444, 1967.
5. de Spautz, J.F. and Lerman, R.A., "Nonlinear Differential Equation._
and the Terrestrial Brachistochrone," American Institute of Aeronautics
and Astronautics, Journal, Vol. 7, No. 6, pp I173-I174, June 1969.
6. Ferrer, S. and Sein-Echaluce, M.L., "On the Szebehely-Bond Equation,"
to be published in Celestial Mechanics, July, 1983.
7. Levi-Civita, T., "TraiettorieSingulari ed Urti nel Problema Ristretto
" Ann Math , [3] Vol 9 p. l 1903dei Tre Corpi, . . , . ., ., •
8. Mittleman, D. and Jezewski, D.J. "A Class of Problems in Celestial
Mechanics Reducible to Perturbed Harmonic Oscillator," NASA-JSC-18344,
September 1982.
9. Nacozy, P. and Szebehely, V., "The Computation of Relative Motion
" Celestial Mechanics, Vol 13, pp 449-453,with Increased Precision,
1976.
%
lO. Schrapel, H.D., "Lineare und nichtlineare Hauptschwingungen','Zeitschrift
fur Angewandte Mathematik und Mechanik, Vol. 58, pp.T. 172-173, June 1978.
II. Stiefel, E. and Scheifele, G., "Linear and Regular Celestial Mechanics,"
Springer Verlag, Berlin and N.Y., 1971.
12. Sundman, K. "Memoire sur de Probleme des Trois Corps," Acta Math.,
Vol. 36, p. I05, 1912.
13. Szebehely, V., "Accurate Computation of Relative Motion by Orbit
Smoothing," NASA-JSC-09834, August, 1975.
14. Szebehely, V. and Bond, V., "Equations for Autonomous Orbit Calculations,"
NASA-JSC-18280, June 1982.
I
24-16
i
1986004609-368
.!
_; 15. Szebehely, V. and Bond, V., "Transformationsof the Perturbed Two-
:".- Body Problem to Unperturbed Harmonic Oscillators, CELESTIAL MECHANICS
VoI. 30, pp. 59-69, 1983
"_ 16. Szebehely, V., "Lectures on Linearizing Transformations of Dynamical
Systems" in Long-Time Predictions in Dynamics (V. Szebehely and
•: B. Tapley editors), pp 17-42. D. Reidel Publishing Company,
Dordrecht-Hol land, 1976a.
",e I
t
; 17. Szebehely, V. "Linearizationof Dynamical Systems Using Integrals !
of the Motion," Celestial Mechanics, Vol. 14, pp 499-508, 1976b.
L
18. Szebehely, V. "Theor$ of Orbits," Academic Press, N.Y. Chapters
.. , 2,3, and I0, 1967. 4
!
_ Ig. Szebehely, V., "Time Transformations for Relative Motion," Celestial
;_ Mechanics, Vol 13, pp 465-470, 1976c.
"' t
¢
!
_ f
!
I
t,
24-17
1986004609-369


Advanced algorithm for orbit computation

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