Integration of Perturbed Motionpsingla/Teaching... · 2006. 11. 8. · 03/05/2006 Integration of Perturbed Orbits Slide 4 Relative Strengths of Forces Acting on a Typical Satellite

03/05/2006 Integration of Perturbed Orbits Slide 1

Integration of Perturbed Motion

John L. Junkins


OutlineIntegration of Perturbed Motion

COWELL AND ENCKE METHODSVARIATION OF PARAMETERSGRAVITY MODELING & OBLATENESS PERTURBATIONS

State Transition & Related Sensitivity Matrices for Perturbed Motion

The Three Body ProblemEQUATIONS OF MOTIONTHE RESTRICTED THREE BODY PROBLEM

Jacobi’s integral & other miraclesThe libration points, stability & the zero velocity surfaces

∂(current state) / ∂(initial state)∂(current state) / ∂(force model parmeters)

Initial Value and Two-Point Boundary Value Problems

03/05/2006 Slide 3

Integration of Perturbed MotionThree Quasi-Independent Sets of Issues Must be Addressed:

What physical effects will be considered?

Which set of coordinates will be integrated?

What integration method will be used?

Gravitational perturbation due to non spherical earth

Gravitational perturbation due to attraction of non-central bodies

Aerodynamic forces

Thrust

Solar radiation pressure

Relativistic effects

Rectangular coord. in nonrotatingref. Frame (Cowell’s Method)

Departure motion in rectangular coordinates (Encke’s Method)

Variation-of-Parameters; slowly varying elements of two-body motion: - classical elements

- other elements

Regularized VariablesK.S. transformed oscillatorsBurdet transformed oscillators

Canonical CoordinatesDelunay Variables

Numerical (“special”) Methods:Single Step Methods:

Analytical continuationRunge-Kutta methods

Multi Step Methods:Adams-Moulton methodAdams-Bashford methodGaussian second sum methodSymplectic Integrators

Analytical (“general”) Methods:Pedestrian asymptotic expan.Lindstedt-Poincare methodsMethods of averagingMultiple time scale methodsTransformation methods

Questions: What is the solution needed for? How precise must the solution be? What software is available?


Relative Strengths of Forces Acting on a Typical Satellite(“Junkins with 10 m2 solar panels” at 350 km above earth)

1.0.0010.000 070.000 0050.000 000 20.000 000 080.000 000 04

Source of Perturbing Force 2| perturbing force |

| / |GMm r

inverse square attraction

dominant oblateness (J2)

in-track drag (B = 0.35)

higher harmonics of gravity field

cross-track aerodynamic force

attraction of the Moon

attraction of the Sun


Gravity Modeling OverviewPotential of a “Potatoe”:

( )0 0

sin cos sinnn

m m mn n n

n m

RGMU P C m S mr r

φ λ λ∞

⊕

= =

⎛ ⎞ ⎡ ⎤= +⎜ ⎟ ⎣ ⎦⎝ ⎠∑ ∑

Acceleration:

Problems: (1) “The more you learn, the more it costs!”(2) ∞ is a painful upper limit(3) For n > 3, convergence is very slow.

1South:

1East: cos

Radial:

S

E

R

UGr

UGrUGr

φ

φ λ

∂= −

∂∂

= −∂

∂=∂

Spherical Rectan

gular

x

y

z

UGxUGxUGx

∂=

∂∂

=∂∂

=∂


During 1975 – 76, J. Junkins et al developed a (“finite element”) gravitymodel based upon the starting observation “horse-sense”:

( ), ,REFU U r θ λ= ( ), ,U r θ λΔ“Everything Else”Dominate terms

. . . Use global modelfor these . . .

. . . Use global family of local, piecewise continuousfunctions to model these. . .

+

Thesis: It may take a >500 term spherical harmonic series to model Uglobally, but URεF can be modeled using 2 or 3 terms and ΔU can be locally modeled with ~ 10 terms computational efficiency results. This is the genesis of earliest version of the GLO-MAP piecewise continuous approximation methods published by JLJ et al during the mid 1970s.

Gravity Overview…


Investigation of Finite-ElementRepresentation of the Geopotential

RADIAL DISTRUBANCE ACCELERATION ON THE EARTH’S SURFACE(contour interval is 5 x 10-5 m/sec2)

Gravity Potential GM Ur

= + Δ ( )2Radial Acceleration -GM Ur r

∂= + Δ

∂


FINITE ELEMENT MODELING OF THE GRAVITY FIELD: THE BOTTOM LINES

• Basic tradeoff is storage versus runtime

• Factors of ~ 50 possible increased speed to calculate local acceleration

• In one example, a global 23rd degree and order spherical harmonic expansion has been “replaced” by 1500 finite elements

• RMS of acceleration residuals ≈ 0.000, 002 m/sec2

• Max acceleration error ≈ 0.000, 008 m/sec2

• Mean acceleration error ≈ 0.000, 000, 03 m/sec2

• 1500 local functions 20 coefficients each 30,000 coefficients total

See: Junkins, J.L., “Investigation of Finite Element Representations of the Geopotential”, AIAA, J., Vol. 14, No. 6, June. 1976.


Encke’s Method: Integrate Departure Motion from an Osculating Reference Orbit

The parenthetic term is a small difference of large numbers,It is profitable to re-arrange it to avoid numerical difficulties...

From which it follows that:

3 3osc

osc osc doscr r

δ δ μ⎛ ⎞

= + → = − = − +⎜ ⎟⎝ ⎠

r rr r r r r r a

( ) ( )( ) ( )

osc o o

osc o o

t t

t t

=

=

r r

r r

Osculation Condition at t0

Note that: Also note:

osc

osc

δδ

= +

= +r r rr r r

3

3osc

d

oscosc

r

r

μ

μ

= − +

= −

rr a

rr

( )tδ r


Encke’s Method: Re-arrangement of Departure MotionDifferential Equation to Avoid SDOLN

(small differences of large numbers!)

On the previous chart we developed the departure differential equation:

3 3 , osc

osc d oscoscr r

δ μ δ⎛ ⎞

= − = − + = +⎜ ⎟⎝ ⎠

r rr r r a r r r

This equation can be arranged into a more computationally attractive form:

( )3 3 dosc osc

f qr rδδ μ μ= − − +r rr a [note, no small differences of large #’s!]

where

The development of the above form is given on the following 3 pages.

( )( )

2

3/ 22

2 3 3, , 1 1

oscq qq f q q

r qδ δ δδ

⎛ ⎞⋅ − ⋅ + += + = = ⎜ ⎟

⎜ ⎟+ +⎝ ⎠

r r r rr r r


The actual motion is governed by

The osculating orbit satisfies

So the departure (“pertubative”) acceleration is

Making use of

Introduce some useful alternatives since

From which

3 drμ

= − +r r a

3osc oscoscrμ−

=r r

3 3osc

osc doscr r

δ μ⎛ ⎞

= − = − +⎜ ⎟⎝ ⎠

r rr r r a

,osc δ= −r r r I get

( )

3

3 3 3 1osc

dosc osc

f q

rr r rμ μδ δ

−

⎛ ⎞−= + − +⎜ ⎟

⎝ ⎠r r r a

2 2 2 osc osc osc oscr rδ δ δ δ= − → = ⋅ = − ⋅ + ⋅r r r r r r r r r

2

2 2

21 , oscr q qr r

δ δ δ⋅ − ⋅= + ≡

r r r r

Encke Manipulations ….


( ) ( )

( ) ( )

2

2 2

3 312 2

3

3 32

3

2 1 ,

1 1

thus

1 1 1

this can be further manipulated to more attractive forms --here's one of the

osc

osc osc

osc

r q qr rr rq qr r

rf q qr

δ δ δ⋅ − ⋅= + =

= + → = +

⎛ ⎞ ⎡ ⎤= − − = − − +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

r r r r

( ) ( )( )

( )

( )( )

( ) ( )

32

32

32

32

33 22

m:

1 1 1 1

1 1

1 1 3 3 1 11 1

qf q q

q

q q qf q qqq

⎡ ⎤+ +⎢ ⎥⎣ ⎦⎡ ⎤= − − +⎢ ⎥⎣ ⎦ ⎡ ⎤+ +⎢ ⎥⎣ ⎦⎡ ⎤ ⎛ ⎞− + + +⎣ ⎦ ⎜ ⎟= =

⎜ ⎟⎡ ⎤ + ++ + ⎝ ⎠⎢ ⎥⎣ ⎦



So, finally, we get the (exact!) departure motion differential equationwhich lies at the heart of Encke’s Method.

( )

( ) ( )

( )( )

3 3

0 0

2

2

3/ 2

0where

2

3+3q+q 1+ 1+q

when , gro

dosc osc

osc

f qr r

t t

qr

f q q

μ μδ δ

δ δ

δδ δ δ

δ δ

= − − +

= =

− +⋅ − ⋅

=

⎛ ⎞= ⎜ ⎟

⎜ ⎟⎝ ⎠

r r r a

r r

r r rr r r r

r r w too large "rectify the orbit"!→

is computed from a 2-body solution(e.g. the & functions), isusually done by numerical methods (e.g., Runge-Kutta).

osc

F G δ δ δ→ →r

r r r



Rectification of the Reference Orbit in Encke’s MethodOriginal osculatingreference orbit (kissesactual motion at time t0)

“Rectified” (new) osculatingReference orbit (kisses the actual motion at time t1).

Whenever exceeds some preset tolerance,The position and velocity at time t1 are used to calcualte aNew “rectified” reference two-body orbit. Note that thisHas the effect of re-setting the “initial” departure positionand velocity to zero. Since rectification can be done as often as we please (as long as we pay the “overhead”!),the departure motion can be kept as small as we please.

Updated reference orbitOsculates at time t1

Original reference orbitosculates at time t0:

( ) ( )( ) ( )

0 0

0 0

osc

osc

t t

t t

=

=

r r

r r

δ δε+ =

r rr r 1( )tδ r


Continuous limit of osculating orbits: Variation-of-Parameters

( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ){ }1 2 3 4 5 6

It is evident that given (t) and (t), I can compute the transformationto determine the elements of the instantaneous osculating orbit:

, , , , , ,t t e t e t e t e t e t e t⇔

The Essence of Variation

r r

r r

3Knowing the equations of motion and the above transformmation,

drcan I determine differential equations for the element

μ= − +

- of - Parameters lies in the affirmative answerto the following question :

rr a

( )

( )1 2 3 4 5 6

, , , , , , , , 1, 2, …,6?

Note that the elements are "slow variables" (since they are constants ofunperturbed motion).

i

ii d

s e t in the formde f t e e e e e e idt

= =a


Effects of Earth Oblateness on the Osculating Orbit ElementsEight Revolutions of a J2 – Perturbed Orbit*

These results were computed by Harold Black of the Johns Hopkins Applied Physics Lab using

{ }0 0 0 0 0 30 27378 , 0.01, 30 , 45 , 270 , 90 , 1.0827 10a km e i M Jω −= = = Ω = = = = − ×Least square fit of Ω & ω above gives 5.207 deg/day, 8.449 deg/dayωΩ = − =

The first order (EQS 10.94, 10.95) secular terms give 5.184 deg/day, 8.230 deg/dayd ddt dt

ωΩ= − =


Variation of Parameters Tutoring

Consider the two problems

• The forced linear oscillator

( )2 , , ,dx x a t x xω= − +• The perturbed two-body problem

3

3

a

d

dx

r

x xr

μ

μ

−

−=

= +

+

r r a∼ },x y z→

We’ll look first at the linear oscillator to illustrate the essential ideas.

(1)

(2)


( )

( )

( )

2

00

0 0

0

The solution of

For the unperturbed 0 case is well-known -- I write it in two forms FORM 1:

cos sin

sin cos

d

d

x x aa

xx t x

x t x xt t

ω

ωτ ωτω

ω ωτ ωττ

= − +

→ = ←

⎫= +

= + ⎬≡ −

( ) ( )( ) ( )

( )2

2 2 0 00

0

FORM 2:

cos

sin

where

= + , tan =

x t Ax t A

x xA xx

ωτ φω ωτ φ

ωφω

⎪⎪

⎪⎪⎭

= + ⎫⎪⎬= + ⎪⎭

⎛ ⎞⎜ ⎟⎝ ⎠

(1)`

(3)

(4)

(5)


( ) ( )

( ) ( )1 2

1 2

In general, the un-perturbed solution can be written

, , , , "elements"

, , ,

The element are constants of the un-perturbed motion.

The essence of the variation-of-p

i

i

x t f t e e efx t t e et

e

= = ⎫⎪⎬∂

= ⎪∂ ⎭

( )( )

arameters idea is to considerEqs. (6) to be a coordinate transformation for the perturbed problem

and ask the question: How can we "vary the constants" i.e.,

in Eq. (6) so that the homogenous soi ie e t=

lution form of Eq. (6) becomes the solution for the perturbed motion?

(16)


( ) ( ) ( ) ( )

22

2

1 2 1 2

22

2

Developments:The unperturbed motion satisfies

and the solution is

, , , , ,

we seek to solve

with a solution of the form

d

d xx xdt

fx t f t e e x t t e et

d x x adt

ω

ω

≡ = −

∂= =

∂

= − +

( ) ( ) ( )( )( ) ( ) ( )( )

( ) ( ) ( )( )

1 2

1 2

1 2

2

1

, ,

, ,

For , , , the chain rule gives the velocity expression

ii i

x t f t e t e t

dx t f t e t e tdt t

x t f t e t e t

dedx f fdt t e dt=

⎫=⎪⎬∂

= ⎪∂ ⎭

=

∂ ∂= +∂ ∂∑

(1)``

(6)`

(1)```

(7)

(8)


( ) ( ) ( )( )

1 2

1 2

1 2

Comparing (7) & (8), we obtain the "osculation" contraint

0

So the velocity solution for the perturbed case is

, ,

Taking the time deriv

de dedx f f fdt t e dt e dt

f t e t e tdx tdt t

∂ ∂ ∂= ⇒ + =∂ ∂ ∂

∂=

∂

2 2 22

2 21

2

2

ative of (8)`, the acceleration is

Substituting (7) & (10) (1) gives

i

i i

ded x f fdt t t e dt

ft

=

∂ ∂= +∂ ∂ ∂

⇒

∂∂

∑

222

1

i

i i

def ft e dt

ω=

∂+ = −

∂ ∂∑

1

1 22 2

2

1 2

(cancellation due to Eq. (1)``)

Equations (9) & (11) can be combined as

0

d

d

a

f f dee e dt

adef fdtt e t e

+

∂ ∂⎡ ⎤ ⎧ ⎫⎢ ⎥ ⎪ ⎪∂ ∂ ⎧ ⎫⎪ ⎪⎢ ⎥ =⎨ ⎬ ⎨ ⎬⎢ ⎥∂ ∂ ⎩ ⎭⎪ ⎪⎢ ⎥ ⎪ ⎪⎩ ⎭∂ ∂ ∂ ∂⎣ ⎦

(9)

(8)`

(10)

(11)

(12)


21

1 22 2

1 2

1

2

Now, consider FORM 1:

cos sin ,

1 cos , sin

sin , cos

Equation (12) is then

1cos sin

sin cos

oef e t t

f fe e

f ft e t e

dedtdedt

ωτ ωτ τω

ωτ ωτω

ω ωτ ωτ

ωτ ωτω

ω ωτ ωτ

= + ≡ −

∂ ∂= =

∂ ∂

∂ ∂= − =

∂ ∂ ∂ ∂

⎧⎡ ⎤ ⎪⎪⎢ ⎥ ⎨⎢ ⎥−⎢ ⎥⎣ ⎦ ⎩

( )1 2

0

This is easy to invert for

1 sin , cos

d

i

d d

a

dedt

de dea adt dt

ωτ ωτω

⎫⎪ ⎧ ⎫⎪ =⎬ ⎨ ⎬

⎩ ⎭⎪ ⎪⎪ ⎪⎭

⎛ ⎞= − =⎜ ⎟⎝ ⎠

(12)`

(13)


idedt

Of course, the justification for variation-of-parameters “runs deeper”Than solving linear ODE’s! However, the essence of the ideas is easy to illustrate for this case.

The inversion for is typically “more significant” for the higher dimensioned case. Lagrange developed an elegant process “Lagrange’s Brackets” and applied it to the perturbed 2-body problem (Ch. 10 of RHB). We now consider this material.

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Integration of Perturbed Motionpsingla/Teaching... · 2006. 11. 8. · 03/05/2006 Integration of Perturbed Orbits Slide 4 Relative Strengths of Forces Acting on a Typical Satellite