Satellite observation systems and reference systems (ae4-e01)

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Satellite observation systems and reference
systems (ae4-e01)

• Orbit Mechanics 2
• E. Schrama

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Contents

• Perturbed Kepler orbits
• Linear C20 perturbations and classification of
  orbits
• Orbit determination, solve the equation of
  motions
• Effects of other acceleration models
• Numerical implementation
• Example 1 Bullet physics
• Example 2 Kepler and higher order physics
• Orbit determination
• Parameter estimation
• Parameters in function model
• Parameter estimation procedure
• Variational equations
• Organisation parameter estimation

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Perturbed Kepler Orbits

• Please remember that the Kepler problem assumes a
  central force field with UGM/r
• In reality the gravity potential U is more
  difficult than that and spherical harmonics are
  involved.
• Moreover there are other conservative and
  non-conservative forces that determine the motion
  of a spacecraft

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Linear perturbations by C20
C20 not normalised, n mean motion
Ref Seeber p 84
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Classification of orbits

• Sun synchronous orbits ? runs as fast as the
  Earths rotation around the Sun. This is possible
  by tuning the a, e and I.
• Golden inclination Perigee is frozen in time
• Repeating Ground tracks reoccupy the same
  geographic locations after a certain time (a
  cycle)
• Polar orbits the orbit plane is fixed in
  inertial space despite the presence of
  gravitational flattening.

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Orbit determination
Keplers theory happens to be a very good
approximation to describe the motion of small
particles in a gravity field as a result of the
presence of a large body like the Earth or the
Sun. In reality there are higher order multipoles
in the gravity field and other accelerations play
a role. The more complete equations of motion
are therefore
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This is Y200
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Y300
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Y210 and Y211
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Y320
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Y330
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Solution equations of motion

• Analytic
• Lagrange planetary equations
• Gravity Potential in Kepler elements
• Isolate first order solution
• Approximate higher order perturbations
• Numeric
• Conversion to system of first order ODE
• Integration of system of equations

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What other accelerations?

• Tidal forces cause by Sun and Moon
• Gravity effect of air, water in motion etc
• Radiation pressure as a result of sun light and
  light reflected from Earth (Albedo)
• Heat radiating away from the spacecraft
• Atmospheric drag
• Relativistic mechanics

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Effect of perturbing accelerations
The table below lists various acceleration terms
that act on the orbit of a GPS satellite,
gravitational flattening is by far the largest
contributor.
Ref Seeber table 3.4
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Hard to model perturbations

• The remaining perturbations always result in
  oscillating functions. There are cos/sin series
  from which the amplitudes and phases are defined
• Numerical integration is the way to go, all orbit
  determination s/w uses this method.
• Required is an initial state vector and an
  acceleration model for the satellite.
• To classify satellite orbits a first-order
  analytical solution can be used.

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Numerical implementation

• Keplerian physics is easy to understand,
  essentially follows from a central force field
  with a point-mass potential
• The real world is more difficult, essentially
  because there are higher order terms in the
  potential and because there are other
  accelerations
• Orbit dynamics can be described in the form of
  ordinary differential equations. You should
  formulate the problem as a system of first-order
  ODEs
• There are efficient numerical tools to solve
  ODEs, in particular single-step and multi-step
  integrators

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Demonstration numerical solution ordinary diff.
eq.
function f bullet( t,state ) implements
bullet dynamics xp state(1) yp state(2) xv
state(3) yv state(4) g 9.81 dia
0.442.54 length 1.52.54 dens 8000 area
pi(dia/2).2 mass densarealength cd 1 h
yp f exp(-h/6000log(2)) rho 1.2 f v
sqrt(xvxvyvyv) ad 0.5rho(area/mass)vvcd
nx xv/v ny yv/v xa -nxad ya -nyad
- g f state(3) state(4) xa ya'
Example gun bullet physics
In reality
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Demonstration Numerical Implementation (2)
function f satdyn( t,state ) implements
Kepler dynamics xp state(1) yp state(2) xv
state(3) yv state(4) mu 4e14 r
sqrt(xp.2yp.2) factor mu/r/r/r xa
-factorxp ya -factoryp f state(3)
state(4) xa ya'
Example Kepler orbit physics
In reality
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Orbit prediction (1)
During orbit perdiction one needs to integrate
the equations of motion. Suitable numerical
techniques are used to treat differential
equations of the following type
There are numerical procedures like the
Runge-Kutta single step integrator and
Adams-Moulton-Bashforth multi step integrator
that allow the state vector y0 to be propagated
from y0 till yn. In this case a state vector at
index j coincides with the time index t0(j-1)h
where h is the integrator step size.
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Example in MATLAB
span 0 14500 state 1e7 0 0 7e3 option
odeset('RelTol',1e-10) t,y
ODE45('satdyn',span,state,option)
plot(y(,1),y(,2))
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Orbit prediction (2)

• The orbit prediction problem is entirely driven
  by the choice of the initial state vector y0, the
  definition of F(y,t) and g(t).
• The basic question is of course, where does this
  information come from?
• F(y,t) and g(t) fully depend on the realism of
  your mathematical model and its ability to
  describe reality
• However, knowledge of the initial state vector
  should follow from 1) earlier computations or 2)
  launch insertion parameters
• The conclusion is that it is desirable to
  estimate initial state parameters from
  observations to the satellite.

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Parameter estimation

• Terminology
• Here, a problem refers to an interesting case to
  study.
• Problems in satellite geodesy
• Type of problem
• does it contain orbit parameters?
• does it contain gravity field parameters?
• does it contain any other geophysical parameters?
• How do you organize parameter estimation?
• it is a batch or a sequential least squares
  problem?
• can you solve it from one observation set or are
  more sets involved?
• Is preprocessing of observations involved or is
  it in the problem?

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Function model (1)

• The function model aims to relate observations
  and parameters to another
• The unknowns are gathered in vector
• The observations are in vector
• Usually we begin to approximate reality by a
  priori estimates and

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Function model (2)
Ze
S
?Rij
B
Rj
Ri
Ye
Xe
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Function model (3) Examples

• The over-determined GPS navigation solution for
  one receiver
• VLBI observations of phase delay
• Two GPS receivers double difference processing
• SLR network station, orbit parameters, earth
  rotation parameters
• DORIS with orbit and gravity field improvement
• Spaceborne GPS receiver on a LEO

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Implementation

• From our function model we conclude that
• it is by definition a non linear problem
• it depends on a priori information
• it almost always depends on orbit dynamics
• orbit predictions are used to correct the raw
  observations and to set-up the design matrix
• the orbit prediction model is not necessarily
  accurate the first time you apply it

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Least squares
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Minimize cost function

• The way the A matrix is computed completely
  depends on the type of observations and
  parameters in your problem.
• We will distinguish between problems that contain
  orbit parameters and problems that do not.
• Our first task will always be to model an orbit
  in the best possible way given the existing
  situation
• This task is called orbit prediction

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Example Initial state vector estimation in POD
Task determine the size, orientation and
position of the arrow, it determines whether you
hit the bulls eye
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Variational equations
Example ? initial state vector component, terms
in force model etc
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Set-up parameter estimation program

• In reality orbit parameters are estimated from
  observations like range, Doppler or camera to the
  satellite or inbetween satellites
• Orbit prediction method
• Numerically stable schemes are used
• Choice initial state vector
• Definition satellite acceleration model
• Variational method
• Define parameters that need to be adjusted using
  least squares
• Iterative improvement of these parameters
• Use is made of the variational equations

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Parameters in POD

• Station coordinates
• Station related parameters (clock, biases)
• Initial state vector elements of satellite orbits
• Parameters in acceleration models satellite
• Other satellite related parameters (clock,
  biases, etc)
• Signal delay related parameters
• Earth rotation related parameters
• Gravity field related parameters

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Organization parameter estimation

• For large scale batch problems
• separation of arc -- and common parameters
• combination of normal matrices and right hand
  sides
• choice of optimal weight factors for combination
• example development of earth models like EGM96
• Sequential problems
• apart from the adjustment procedure there is a
  state vector transition mechanism
• During transition state vector and variance
  matrix are advanced to the next time step
  (normally with a Kalman filter)
• Example JPLs GPS data processing method