Transcript Conjunction Analysis Fundamentals

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
– Brouwer
– Kozai
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Mean elements
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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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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
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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
• Not accurate
• Provide understanding
• Need something more
difficult to teach in
graduate classes
• Capture simple physics
• Serve as building blocks
for more sophisticated
methods
• Can be taught in
undergraduate 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



U
r
• Defined by a potential function
• Express U in terms of orbital elements
• Average U over one orbit
– Separate into secular and long term contributions
U  Usec  U LP  U SP
– Analytically solve for each type of contribution
  0  0t   LP   SP
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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 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)
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J2 and J4 equations
• First-order J2 secular variations:
 3 Re2
3 2 
2
n  n 1  J 2 2 1  e 1  sin i 
p
 2

 2
 3 Re2


  0   J 2 2 cosi n t  t0 
2 p

 3 Re2  5 2  
  0   J 2 2 2  sin i  n t  t0 
p  2

2
M  M 0  n t  t0 
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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)
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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
• Less accurate
• Provide insight
• Difficult to code
• Useful in design
• 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

r 




a    3  aaspherical  a3rdBodies  adrag  asrp  ...
r
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Integration Methods for CEM
• Multi-step Predictor–Corrector
– 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
• Pros
Single step
• Pros
– Very fast
– Kick near circular butt
–
–
–
–
• Cons
–
–
–
–
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Special starting procedure
Restart
Fixed time steps
Error control
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
• Physics is all in the force
models
• Accuracy limited by
acceleration models
• Six fast variables
• Lots of numerical
integration options
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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

 
 0  M ( ) a perturb
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

 
 tk tk   M ( t ) a perturb

t tk 1 
Integrate
t tk 1 
k
Propagate
k
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
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
t
k 1
tk 1 
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VOP Process
tk
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tk+1
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tk+2
Time
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VOP - Lagrange
• Perturbations
disturbing potential
R
• Eq. of motion – Lagrange Planetary Equations
 

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R
2
2
na 1  e sin i i
1
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VOP - Poisson
• Perturbations expressed in terms of Cartesian
coordinates
• Natural transition from CEM
 
   a perturb
r
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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)
2v
2
a  2 r S  r R   2 T
na
na
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VOP - Herrick
• Uses Cartesian (universal) elements and
Cartesian perturbations
• Implementation in STK
  

r0 ` r g `r g ` g a perturb

  

r0 ` r f `r f ` f a perturb
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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
• Additional code required
• Share acceleration model
with CEM (minus 2Body)
• Loses some advantages in
a high frequency forcing
environment
• Error control less effective
• Physics incorporated into
formulation
• Errors at level of numerical
precision for 2Body
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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
    3  
 
r  3 1  3 r  r   P
  r 

• Not in STK (directly)
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Encke Process
tk
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tk+1
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tk+2
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
• Closed Orbits
• Provide insight
• Difficult to code
• Useful in design
• Difficult to extend
– Orbit
– Constellations/Formations
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• Nuances
– Assumptions
– Force coupling
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Questions?
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