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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 U=GM/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

)1cos3()1(4

3

)cos51()1(4

3

cos)1(2

3

22/322

2

20

2222

2

20

222

2

20

Iea

naCn

dt

dM

Iea

naC

dt

d

Iea

naC

dt

d

e

e

e

Ref: Seeber p 84

C20 not normalised, n: mean motion

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Classification of orbits

• Sun synchronous orbits: runs as fast as the Earth’s 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 determinationKepler’s 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:

1a : sin

0a : cos)(cos,

,1

,,

m

mPY

YCr

a

a

GM

r

GMV

FVX

nmnma

nmanma

n

amn

e

e

i

i

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This is Y200

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Y300

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Y210 and Y211

10

Y320

11

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.

Example gun bullet physics

),( tyFy

agy drag

function f = bullet( t,state )

% implements bullet dynamics

xp = state(1); yp = state(2);

xv = state(3); yv = state(4);

g = 9.81;

dia = 0.44*2.54; length = 1.5*2.54;

dens = 8000; area = pi*(dia/2).^2;

mass = dens*area*length;

cd = 1;

h = yp;

f = exp(-h/6000*log(2));

rho = 1.2 *f ;

v = sqrt(xv*xv+yv*yv);

ad = 0.5*rho*(area/mass)*v*v*cd;

nx = xv/v;

ny = yv/v;

xa = -nx*ad;

ya = -ny*ad - g;

f = [state(3) state(4) xa ya]';

i

i hityhtyhty ).()()(

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.^2+yp.^2);

factor = mu/r/r/r;

xa = -factor*xp;

ya = -factor*yp;

f = [state(3) state(4) xa ya]';

)(),(

with

tGtyFyr

UUy

Example Kepler orbit physics:

In reality

i

i hityhtyhty ).()()(

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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:

)( ),(),( 00 tyytgtyFy

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 MATLABspan = [0 14500]; state = [ 1e7 0 0 7e3]; option = odeset('RelTol',1e-10);

[t,y] = ODE45('satdyn',span,state,option); plot(y(:,1),y(:,2))

0)0(

)(),(

with

yty

tGtyFyr

UUy

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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

X

0X

Y

0Y

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Function model (2)

Xe

Ye

Ze

BS

Ri

Rj

Rij

jiij RRR

)(XvY

)( 000 XvY

Axvy

XXx

YYy

0

0

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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

rxN

yQAxAQA

xAyQA

xAyQAxxAyQy

xAyQxAy

vQvvxAy

vT

vT

vT

vTT

vT

vT

vT

equations) (normal

0

min

min

min : isour task ,

11

1

zero put to topossible

11

1

1

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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

minimum

)(

1

0

vQv

X

XA

vT

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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 bull’s eye

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Variational equations

parameter nal variatio:

1a : sin

0a : cos)(cos,

,1

,,

2

m

mPY

YCr

a

a

GM

r

GMV

F

X

VX

FVX

nmnma

nmanma

n

amn

e

e

i

i

i

i

i

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: JPL’s GPS data processing method