Methods of Orbit Propagation

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Methods of Orbit Propagation

• Jim Woodburn

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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

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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
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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

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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)

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Analytical Methods

• Complete solutions
• Two body
• Vinti
• General perturbations
• Method of averaging Mean elements
• Brouwer
• Kozai

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Two-Body

• Spherically symmetric mass distribution
• Gravity is only force
• Many methods of solution
• Two Body propagator in STK

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Vintis 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

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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!

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Complete Soln Pros and Cons
Cons
Pros

• 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

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General Perturbations

• Use simplified equations which approximate
  perturbations to a known solution
• Method of averaging
• Analytically solve approximate equations
• Using more approximations

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GP Central Body Gravity

• 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

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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

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J2 and J4 propagators

• J2 is dominant non-spherical term of Earths
  gravity field
• Only model secular effects of orbital elements
• Argument of Perigee
• Right Ascension of the Ascending Node
• Mean motion (ie orbital frequency)
• Method
• Escobals 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)

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J2 and J4 equations

• First-order J2 secular variations

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SGP4

• General perturbation algorithm
• Developed in the 70s, 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 AGIs website
• Propagation valid for short durations (3-10 days)

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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!

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GP Methods Pros Cons
Cons
Pros

• Fast
• Provide insight
• Useful in design

• Less accurate
• Difficult to code
• Difficult to extend
• Nuances
• Assumptions
• Force coupling

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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
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Cartesian Equations of Motion (CEM)

• Conceptually simplest
• Default EOM used by HPOP, Astrogator

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Integration Methods for CEM

• Multi-step PredictorCorrector
• Gauss-Jackson (2)
• Adams (1)
• Single step
• Runge-Kutta
• Bulirsch-Stoer

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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)

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Integrator Selection
Multi-step
Single step

• 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

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Interpolation with CEM

• Standard formulation
• Lagrangian interpolation, order 7 8 sample pnts
• Position, Velocity computed separately
• Hermitian interpolation, order 5 2 sample pnts
• Position, Velocity, Acceleration computed
  together
• Integrator specific interpolation
• Multi-step accelerations and sums

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CEM Pros and Cons
Cons
Pros

• 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

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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
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VOP Process

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

Integrate
Propagate
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VOP Process
tk
tk1
tk2
Time
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VOP - Lagrange

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

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VOP - Poisson

• Perturbations expressed in terms of Cartesian
  coordinates
• Natural transition from CEM

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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)

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VOP - Herrick

• Uses Cartesian (universal) elements and Cartesian
  perturbations
• Implementation in STK

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Interpolation with VOP

• Standard formulation
• Lagrangian interpolation, order 7 8 sample pnts
• Position, Velocity computed separately
• 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

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VOP Pros Cons
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

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Enckes 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)

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Encke Process
tk
tk1
tk2
Time
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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

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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

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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

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Semi-analytical Uses

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

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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

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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

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DSST

• Draper Semi-analytic Satellite Theory
• Very complete semi-analytic theory
• J2000
• Modern atmospheric density model
• Tesseral resonances

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Semi-analytical Methods Pros Cons
Cons
Pros

• Fast
• Provide insight
• Useful in design
• Orbit
• Constellations/Formations

• Closed Orbits
• Difficult to code
• Difficult to extend
• Nuances
• Assumptions
• Force coupling

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Questions?

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RAAN evolution comparison
Evolution of the Right Ascension of the Ascending
Node
45.0
44.9
Two-Body Constant
44.8
44.7
44.6
J2 Secular Only
44.5
44.4
HPOP 2x0 Periodic and Secular
44.3
44.2
44.1
1 Jan 2001 030000.00
1 Jan 2001 000000.00
1 Jan 2001 013000.00
(UTCG)
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SRP boundary mitigation

• Crossing lighting boundaries
• Penumbra event occurs over short time intervals
• Discontinuity in force model sampling
• Without mitigation, propagation is sensitive to
  sampling steps
• STKs Method uses Encke correction to account
  for lighting changes
• More efficient than re-starting integrator at
  boundary
• AGI authored papers (Jim Woodburn)
• AIAA 2000-4027, AAS 01-223

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Numerical integrators (contd)

• Bulirsch-Stoer
• Uses first-order form of equations of motion
• Handles discontinuities gracefully
• Able to use VOP formulation
• Richardson extrapolation with automatic step size
  control

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Comparing integrators

• LEO
• 24 hours
• Nearly circular, 28 deg inclination
• TwoBody vs. J2 vs. RK
• RK vs. BS vs. GJ
• Stepsizes 60, 30, 15 sec
• SRP Mitigation On and Off

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Choosing settings

• Table of ephemeris runs
• Leo 1 day e0.001, 740km altitude, 14 revs per
  day
• Meo 3 days circular, 4 revs per day
• Heo 7 days Molniya, 2 revs per day
• Geo 14 days
• 7 different force model settings
• 3 integrators (RK78, BS, GJ)
• 4 formulations (standard, reg. time, vop, vop
  reg. time)
• 5 different error tolerances used for RK78 and BS

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Choosing settings (contd)

• 1,344 cases total
• Truth model
• RK78, 10 sec max step size
• Table of differences available from Help system

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Take-aways

• MEO
• max difference is 2mm over all 336 cases
• 306 have less than 1 mm difference
• GEO
• max difference is 9mm over all 336 cases
• 289 have less than 1 mm difference
• LEO, HEO
• About half of LEO cases have less than 1 mm
  difference
• HEO cases have the most diversity
• Sensitive to SRP
• VOP may not adequately sample large gravity
  fields
• Boundary mitigation works better in real time,
  not reg. time

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Cowell (based on Vallado)

• Cowells Formulation Specifies a formulation of
  the second order equations of motion in terms of
  Cartesian elements
• Cowells Method Specifies Cowells Formulation
  of equations of motion used with a numerical
  integration scheme based on finite differences
  (Gauss-Jackson)