methods of orbit propagation

Pg 1 of 45AGI www.agiuc.com

Methods of Orbit PropagationJim Woodburn


Why are you here?

• You want to use space• You operate a satellite• You use a satellite• You want to avoid a satellite• You need to exchange data• You forgot to leave the room after the last talk


Motivation

• Accurate orbit modeling is essential to analysis • Different orbit propagation models are required

– Design, planning, analysis, operations – Fidelity: “Need vs. speed”

• Orbit propagation makes great party conversation

STK has been designed to support all levels of user need


Agenda

• Analytical Methods– Exact solutions to simple approximating problems– Approximate solutions to approximating problems

• Semi-analytical Methods– Better approximate solutions to realistic problems

• Numerical Methods– Best solutions to most realistic problems


Analytical Methods

Definition – Position and velocity at a requested time are computed directly from initial conditions in a single step– Allows for iteration on initial conditions (osculating to

mean conversion)


Analytical Methods

• Complete solutions– Two body– Vinti

• General perturbations– Method of averaging Mean elements– Brouwer– Kozai


Two-Body

• Spherically symmetric mass distribution• Gravity is only force• Many methods of solution

• Two Body propagator in STK


Vinti’s Solution

• Solved in spheroidal coordinates

• Includes the effects of J2, J3 and part of J4

• But the J2 problem does not have an analytical solution

• This is not a solution to the J2 problem

• This is also not in STK


Interpolation with complete solutions

• Standard formulations– Lagrangian interpolation, order 7 [8 sample pnts]

• Position, Velocity computed separately

– Hermitian interpolation, order 7 [4 sample pnts]• Position, Velocity computed together

• Why interpolate? Just compute directly!


Complete Soln Pros and Cons

• Fast

• Provide understanding

• Capture simple physics

• Serve as building blocks for more sophisticated methods

• Can be taught in undergraduate classes

• Not accurate

• Need something more difficult to teach in graduate classes

Pros Cons


General Perturbations

• Use simplified equations which approximate perturbations to a known solution

• Method of averaging• Analytically solve approximate equations

– Using more approximations


• Central Body Gravity• Defined by a potential function • Express U in terms of orbital elements• Average U over one orbit

– Separate into secular and long term contributions

– Analytically solve for each type of contribution

GP – Central Body Gravity

SPLP UUUU sec

SPLPt 00

rU


GP Mean Elements

• Selection of orbit elements and method of averaging define mean elements– Only the averaged representation is truly mean– Brouwer– Kozai

• It is common practice to “transform” mean elements to other representations


J2 and J4 propagators

• J2 is dominant non-spherical term of Earth’s gravity field

• Only model secular effects of orbital elements– Argument of Perigee– Right Ascension of the Ascending Node– Mean motion (ie orbital frequency)

• Method– Escobal’s “Methods of Orbit Determination”– J2 First order J2 terms– J4 First & second order J2 terms; first order J4 terms

• J4 produces a very small effect (takes a long time to see difference)


J2 and J4 equations

• First-order J2 secular variations:

02

2

20 cos23 ttni

pRJ e

022

2

20 sin252

23 ttni

pRJ e

ie

pRJnn e 222

2

2 sin2311

231

00 ttnMM


SGP4

• General perturbation algorithm– Developed in the 70’s, subsequently revised– Mean Keplerian elements in TEME frame– Incorporates both SGP4 and SDP4

• Uses TLEs (Two Line Elements)– Serves as the initial condition data for a space object– Continually updated by USSTRATCOM

• They track 9000+ space objects, mostly debris– Updated files available from AGI’s website– Propagation valid for short durations (3-10 days)


Interpolation with GP

• Standard formulations– Lagrangian interpolation, order 7 [8 sample pnts]

• Position, Velocity computed separately• Should be safe

– Hermitian interpolation, order 7 [4 sample pnts]• Position, Velocity computed together• Beware – Velocity is not precisely the derivative of position

• Why interpolate? Just compute directly!


GP Methods – Pros & Cons

• Fast

• Provide insight

• Useful in design

• Less accurate

• Difficult to code

• Difficult to extend

• Nuances– Assumptions– Force coupling

Pros Cons


Numerical Methods

Definition – Orbit trajectories are computed via numerical integration of the equations of motion

One must marry a formulation of the equations of motion with a numerical integration method


Cartesian Equations of Motion (CEM)

• Conceptually simplest• Default EOM used by HPOP, Astrogator

...33 srpdragrdBodiesaspherical aaaarra


Integration Methods for CEM

• Multi-step Predictor–Corrector– Gauss-Jackson (2)– Adams (1)

• Single step– Runge-Kutta– Bulirsch-Stoer


Numerical Integrators in STK

• Gauss-Jackson (12th order multi-step)– Second order equations

• Runge-Kutta (single step)– Fehlberg 7-8– Verner 8-9– 4th order

• Bulirsch-Stoer (single step)


Integrator Selection

• Pros– Very fast– Kick near circular butt

• Cons– Special starting procedure– Restart– Fixed time steps– Error control

• Pros– Plug and play– Change force modeling– Change state– Error control

• Cons– Slower– Not good party conversation

Multi-step Single step


Interpolation with CEM

• Standard formulation– Lagrangian interpolation, order 7 [8 sample pnts]


– Hermitian interpolation, order 5 [2 sample pnts]• Position, Velocity, Acceleration computed together

• Integrator specific interpolation– Multi-step accelerations and sums


CEM Pros and Cons

• Simple to formulate the equations of motion

• Accuracy limited by acceleration models

• Lots of numerical integration options

• Physics is all in the force models

• Six fast variables

Pros Cons


Variation of Parameters

• Formulate the equations of motion in terms of orbital elements (first order)

• Analytically remove the two body part of the problem

VOP is NOT an approximation

perturbaM )(0


VOP Process

• Two/three step process– Integrate changes to initial orbit elements– Apply two body propagation– Rectification

1kt tk

1kt tk perturbtkt aMt

k

)( Integrate

11 kt tk

Propagate


VOP Process

Timetk tk+1 tk+2


VOP - Lagrange

• Perturbations disturbing potential• Eq. of motion – Lagrange Planetary Equations

R

iR

iena

sin1

122


VOP - Poisson

• Perturbations expressed in terms of Cartesian coordinates

• Natural transition from CEM

perturbar


VOP - Gauss

• Perturbations expressed in terms of Radial (R), Transverse (S) and Normal (W) components

• Provides insight into which perturbations affect which orbital elements (maneuvering)

Tnav2

2 RrSr

naa 2

2


VOP - Herrick

• Uses Cartesian (universal) elements and Cartesian perturbations

• Implementation in STK

perturbaffrfrr ```0

perturbaggrgrr ```0


Interpolation with VOP

• Standard formulation– Lagrangian interpolation, order 7 [8 sample pnts]


– Hermitian interpolation, order 7 [4 sample pnts]• Position, Velocity computed together

– Danger due to potentially large time steps

• Variation of Parameters– Special VOP interpolator, order 7 [8 sample pnts]

• Deals well with large time steps in the ephemeris• Performs Lagrangian interpolation in VOP space


VOP Pros & Cons

• Fast when perturbations are small

• Share acceleration model with CEM (minus 2Body)

• Physics incorporated into formulation

• Errors at level of numerical precision for 2Body

• Additional code required

• Error control less effective

• Loses some advantages in a high frequency forcing environment

Pros Cons


Encke’s Method

• Complete solution generated by combining a reference solution with a numerically integrated deviation from that reference

• Reference is usually a two body trajectory• Can choose to rectify

• Not in STK (directly)

Prrr

r

3

3

31


Encke Process

Timetk tk+1 tk+2


Encke Applications

• Orbit propagation• Orbit correction

– Fixing errors in numerical integration– Eclipse boundary crossings

• AIAA 2000-4027, AAS 01-223

– Coupled attitude and orbit propagation• AAS 01-428

• Transitive partials


Semi-analytical Methods

• Definition – Methods which are neither completely analytic or completely numerical.

• Typically use a low order integrator to numerically integrate secular and long periodic effects

• Periodic effects are added analytically• Use VOP formulation• Almost/Almost compromise


Semi-analytical Process

• Convert initial osculating elements to mean elements

• Integrate mean element rates at large step sizes• Convert mean elements to osculating elements as

needed• Interpolation performed in mean elements


Semi-analytical Uses

• Long term orbit propagation and studies• Constellation design• Formation design• Orbit maintenance


Semi-analytic in STK - LOP

• Long Term Orbit Propagator

• Developed at JPL• Arbitrary degree and order gravity field• Third body perturbations• Solar pressure• Drag – US Standard Atmosphere


Semi-analytic in STK - Lifetime

• Developed as NASA Langley• Hard-coded to use 5th order zonals• Third body perturbations• Solar pressure• Atmospheric drag – selectable density model


DSST

• Draper Semi-analytic Satellite Theory• Very complete semi-analytic theory

– J2000– Modern atmospheric density model– Tesseral resonances


Semi-analytical Methods – Pros & Cons

• Fast

• Provide insight

• Useful in design– Orbit– Constellations/Formations

• Closed Orbits

• Difficult to code

• Difficult to extend

• Nuances– Assumptions– Force coupling

Pros Cons


Questions?

methods of orbit propagation

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