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W I S S E N

T E C H N I K

L E I D E N S C H A F T
Advanced satellite geodesy
Matthias Ellmer, Institut für Geodäsie
28. April 16
u
www.tugraz.at
itsg.tugraz.at 
Variational equations
Goal: Setup a system of linearized obs. Equations for satellite positions.
l  l 0  Ax  e
Unknown parameters
x
Reference
(dynamic) orbit
Design matrix
A
Observed
positions or velocities
r (t ), r (t )
2
Matthias Ellmer, Institut für Geodäsie
28. April 16
r
x
itsg.tugraz.at 
Equation of motion
Equation of motion (2nd order differential equation)
r(t )  f (t , r,)
Specific forces (Force per unit mass)
- Gravity
f (r )  V (r )
with
GM
V (  , , r ) 
R
- Solar radiation pressure
f (t , r )   E p (t ) Cr
3
-
Tides
-
…
A r  rsun
m r  rsun
Matthias Ellmer, Institut für Geodäsie
28. April 16

R
 

n 0  r 
n 1 n
c
m 0
nm
Cnm ( , )  snm S nm ( , )
itsg.tugraz.at 
Integration
Equation of motion (2nd order differential equation)
r(t )  f (t , r,)
Integrate once to get the velocity
t
Solution is unique except 6 integration constants
- Start position and velocity
- Keplerian elements
- Boundary positions
r (t )  r0   f (t , r (t )) dt 
t0
Integrate twice for the position
t t
r (t )  r0  r0t    f (t , r (t )) dt  dt 
t0 t0
t
 r0  r0t   (t  t )  f (t , r (t )) dt 
t0
4
Matthias Ellmer, Institut für Geodäsie
28. April 16
r0
r0
Position has to be known to compute
the position
• Necessitates reference position
• Iterative approach
itsg.tugraz.at 
Integration algorithm
Compute the full acceleration signal at each epoch
r(t )  f (t , r,)
Compute integrated position and velocities, for example with Euler step:
r (t k 1 )  r (t k )  r (t k )t
r (t k 1 )  r (t k )  r(t k )t
(0,0)
Determine inital values, for example through least-squares fit to PODs or
previous iteration of dynamic orbit.
r (t )  r0  r0t  r (t )
r (t )  r0  r (t )
r0
r0
5
Matthias Ellmer, Institut für Geodäsie
28. April 16
itsg.tugraz.at 
Evaluation of force function
r(t )  f (t , r (t ),)
[Mayer-Gürr, 2006]
→ Integration has to be repeated until computed orbit does not change anymore.
6
Matthias Ellmer, Institut für Geodäsie
28. April 16
itsg.tugraz.at 
Result and analysis of orbit integration
The smaller this difference, the
better the integration algorithm.
7
Matthias Ellmer, Institut für Geodäsie
28. April 16
itsg.tugraz.at 
Difference between orbits after convergence
The orbit still changes by ~100 µm
between iterations!
Spectral domain: Large power at 1/rev
8
Matthias Ellmer, Institut für Geodäsie
28. April 16
itsg.tugraz.at 
Encke approach
Split the observed acceleration into two components:
1. An analytically computable component, originating a
reference motion 𝐫𝑟 (𝑡)
2. A perturbing component, causing the perturbation 𝐬(𝑡)
r(t )  g (t )  q(t )
Johann Franz Encke
(1791-1865)
The perturbed acceleration, position, and velocity is given
by the departure from the reference motion.
s(t )  r(t )  rr (t )  q(t )
t
s (t )  r (t )  rr (t )  s 0   q(t ) dt 
t0
t
s(t )  r (t )  rr (t )  s 0  s 0t   (t  t )  q(t ) dt 
t0
9
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝐬 𝑡 , 𝐬 𝑡 , and 𝐬 𝑡 are
the Encke vectors.
itsg.tugraz.at 
Choice of reference motion
g (t )  0
→Results in linear motion
Position and velocity can be derived
from the equinoctial elements with
high precision and efficiency, as no
trigonometric functions are used. In
terms of Kepler elements, the
equinoctial elements are given by:
a=a
λ=M+ω+Ω
h = e sin(ω+Ω)
g (t )  GM
r (t )
r (t )
The choice of reference motion can
reduce the power of the computed
integral significantly.
10
Matthias Ellmer, Institut für Geodäsie
28. April 16
3
k = e cos(ω+Ω)
p = tan(I/2) sin Ω
q = tan(I/2) cos Ω
→Results in Kepler motion
itsg.tugraz.at 
Algorithm
1.
2.
3.
4.
5.
Compute accelerations along reference orbit 𝐠 𝑡 .
Compute perturbing acceleration 𝒒 𝑡 = 𝐫 𝑡 − 𝐠 𝑡 .
Integrate perturbing acceleration to Encke vectors 𝐬 𝑡 and 𝐬 𝑡 .
Add the Encke vectors to the reference motion 𝐫𝑟 𝑡 , 𝐫𝑟 𝑡 to yield 𝐫 𝑡 , 𝐫 𝑡 .
Estimate initial position and velocity 𝐫0 𝑡 , 𝐫0 𝑡 to yield 𝐫 𝑡 , 𝐫 𝑡 .
The reduction of
the integrated
acceleration leads
to improvde
accuracy.
11
Matthias Ellmer, Institut für Geodäsie
28. April 16
Linear
Ellipses
itsg.tugraz.at 
Optimizing the reference motion
g (t )  GM
r (t )
r (t )
3
rr , 0  r0
rr , 0  r0
g (t )  GM
Through smart choice of the
reference ellipses, we can reduce
the power of the integral further. We
perform a least squares fit of the
Kepler ellipses to the dynamic orbit,
thus minimizing the spatial
separation and the power of the
integral.
12
Matthias Ellmer, Institut für Geodäsie
28. April 16
r (t )
r (t )
3
rr , 0  r0
rr , 0  r0
itsg.tugraz.at 
Convergence of the best-fit orbit
The fitted orbit is much closer to the true orbit
•
•
13
Matthias Ellmer, Institut für Geodäsie
28. April 16
The dynamic orbit converges to within ~nm.
The power spectrum of the differences is white
above ~ 4 cycles/revolution.
itsg.tugraz.at 
Variational equations
Goal: Setup a system of linearized obs. equations
l  l 0  Ax  e
Unknown parameters
x
Reference
(dynamic) orbit
Design matrix
A
Observed
positions or velocities
r (t ), r (t )
14
Matthias Ellmer, Institut für Geodäsie
28. April 16
r
x
itsg.tugraz.at 
Unknown parameters
𝐱 = 𝐩, 𝐪, 𝐫𝟎 , 𝐫𝟎
𝑻
Satellite state parameters: 𝐫0 , 𝐫0
Measurement model parameters 𝐪:
• Antenna offsets
• Biases
• …
Force model parameters 𝐩:
• 𝑐𝑛𝑚 , 𝑠𝑛𝑚
• Drag coefficient
•
•
Radiation pressure coefficient
…
[Source: esa]
15
Matthias Ellmer, Institut für Geodäsie
28. April 16
itsg.tugraz.at 
Variational equation
Initial values: Position and
velocity at first epoch:
Satellite state: Position and
velocity at each epoch:
𝐫0
𝛂= 𝐫
0
𝐫 𝑡
y(𝑡) =
𝐫 𝑡
Design matrix 𝐀: Derivation of state y(𝑡) with respect to the parameters 𝐱 = 𝐩, 𝐪, 𝐫𝟎 , 𝐫𝟎
𝐫 𝑡
𝐫 𝑡
𝝏𝐫 𝑡
𝝏𝐩
=
𝝏𝐫 𝑡
𝝏𝐩
Force model
parameter
sensitivity
matrix 𝐒𝐩
16
𝝏𝐫 𝑡
𝝏𝐪
𝝏𝐫 𝑡
𝝏𝐪
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝝏𝐫 𝑡
𝝏𝐫𝟎
Measurement
model parameter
sensitivity matrix
𝐒𝐪
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝐩
𝐪
𝐫𝟎
𝐫𝟎
State
transition
matrix 𝚽
Variational equations:
y(𝑡) = 𝐒𝐩
𝐒𝐪
𝚽
Simplified: 𝐒𝐪 = 𝟎
y(𝑡) = 𝐒
𝚽
𝐩
𝛂
𝑻
𝐩
𝐪
𝛂
itsg.tugraz.at 
The design matrix
r0 ,r0
cnm , snm ,
r (t0 )
S(t0 ) 3m
Φ(t0 ) 36
r (t1 )
S(t1 ) 3m
Φ(t1 ) 36
r (t 2 )
S(t 2 ) 3m
Φ(t 2 ) 36

r (t N )


S(t N ) 3m
Parameter Sensitivity Matrix
Φ(t N ) 36
Simplified: 𝐒𝐪 = 𝟎
y(𝑡) = 𝐒
State Transition Matrix
17
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝚽
𝐩
𝛂
itsg.tugraz.at 
State transition matrix 𝚽
A change in the initial values 𝛂 = 𝐫𝟎 , 𝐫𝟎 𝑻 will affect all positions and velocities along the orbit arc.
This effect is described by the state transition matrix
𝚽(𝑡) =
𝜕𝐲(𝑡)
𝜕𝛂
The change of the state vector is given by
𝐫 𝑡
𝐳 𝑡 = y (𝑡) =
𝐫 𝑡
𝝏𝐫
𝜕𝐳(𝑡)
𝝏𝐫
𝐙𝐲 =
=
𝝏𝐫
𝜕𝐲(𝑡)
𝝏𝐫
𝑡
𝑡
𝑡
𝑡
𝝏𝐫
𝝏𝐫
𝝏𝐫
𝝏𝐫
𝑡
𝑡
𝑡
𝑡
Differential equation of the
state transition matrix:
𝚽 = 𝐙𝐲 𝚽
18
Matthias Ellmer, Institut für Geodäsie
28. April 16
Gravity:
r 
V
 V
r
Gravity gradient:
  2V
 2
 x
  2V
r
 V  
r
 x2y
V
 xz

 2V
xy
 2V
y 2
 2V
yz
 2V 

xz 
 2V 

yz 
 2V 
z 2 
itsg.tugraz.at 
Parameter sensitivity matrix 𝐒
A change in the force model parmaters 𝐩 = 𝑐𝑛𝑚 , 𝑠𝑛𝑚 , ...
velocities along the orbit arc.
𝐒(𝑡) =
𝑻
will also affect all positions and
𝜕𝐲(𝑡)
𝜕𝐩
With 𝐳 𝑡 as before:
𝝏𝐫 𝑡
𝜕𝐳(𝑡)
𝝏𝐩
𝐙𝐩 =
=
𝝏𝐫 𝑡
𝜕𝐩
𝝏𝐩
19
Matthias Ellmer, Institut für Geodäsie
28. April 16
Differential equation of the
parameter sensitivity matrix:
𝐒 = 𝐙𝐲 𝐒 + 𝐙𝐩
itsg.tugraz.at 
Linear ordinary differential equation (ODE)
General form
𝚽 = 𝐙𝐲 𝚽
𝐒 = 𝐙𝐲 𝐒 + 𝐙𝐩
a (t ) y (t )  b(t ) y (t )  c(t )
−𝐙𝐲 𝚽 + 𝚽 = 0
− 𝐙𝐲 𝐒 + 𝐒 = 𝐙𝐩
Solution:
1. find a solution of the homogeneous equation
  (t )
a (t ) y (t )  b(t ) y (t )  0
2. Variation of the constants
Insert into the differential equation
 (t )   c(t )
a (t ) f (t ) (t )   b(t ) f (t ) (t )  f (t )
y (t )  f (t ) (t )
 (t )
y (t )  f (t ) (t )  f (t )
Solution:
 t c(t )


y (t )   (t ) 
dt  C 
 t b(t ) (t )

0

20
 (t )   b(t ) f (t ) (t )  c(t )
f (t )a (t ) (t )  b(t )
b(t ) f (t ) (t )  c(t )
c(t )
f (t ) 
 t 1

b(t ) (t )
S(t )  Φ(t ) Φ (t )Z p dt  C 
t

t
c(t )
0

f (t )  
dt
b
(
t
)

(
t
)
t0
Matthias Ellmer, Institut für Geodäsie
28. April 16

itsg.tugraz.at 
Finding the integration constant
t
𝑡
𝚽 −1
𝐒 𝑡 = −𝚽 𝑡
𝑡′
𝐙𝐩 ⅆ𝑡 + 𝐂
r (t )  r0  r0t   (t  t )  f (t , r (t )) dt 
t0
𝑡0
t
r (t )  r0   f (t , r (t )) dt 
t0
1. Determine the inital values 𝚽0 and 𝐒0 :
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝚽 𝑡 =
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝝏𝐫 𝑡
𝝏𝐫𝟎
=
𝐈 𝐈⋅𝑡
𝟎
𝐈
𝚽0 = 𝐈6,6
𝝏𝐫 0
𝝏𝐩
𝐒0 =
𝝏𝐫 0
𝝏𝐩
= 𝟎6,m
2. Find the value of 𝐂 by using intial values:
𝑡
𝐒0 = −𝚽0
𝑡0
𝑡
𝚽0−1 𝐙𝐩 ⅆ𝑡
+𝐂
0
21
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝐂 = 𝟎6,m
𝚽 −1 𝑡 ′ 𝐙𝐩 ⅆ𝑡
𝐒 𝑡 = −𝚽 𝑡
𝑡0
itsg.tugraz.at 
Integration of parameter sensitivity matrix
Design matrix A can be found by integration:
The state transition
matrix at all epochs:
𝚽 𝑡 =
𝐈
𝟎
𝐈⋅𝑡
𝐈
r0 ,r0
cnm , snm ,
Is this correct?
r (t0 )
S(t0 ) 3m
Φ(t0 ) 36
r (t1 )
S(t1 ) 3m
Φ(t1 ) 36
r (t 2 )
S(t 2 ) 3m
Φ(t 2 ) 36
𝑡
𝚽 −1 𝑡 ′ 𝐙𝐩 ⅆ𝑡
𝐒 𝑡 = −𝚽 𝑡
𝑡0

Dependance of velocity on
force model parameters:
𝝏𝐫 𝑡
𝝏𝐩
𝐙𝐩 =
𝝏𝐫 𝑡
𝝏𝐩
22
𝟎3,𝑚
= 𝝏𝛻𝑉 𝑡
𝝏𝐩
Matthias Ellmer, Institut für Geodäsie
28. April 16
r (t N )


S(t N ) 3m
Φ(t N ) 36
Parameter Sensitivity Matrix
State Transition Matrix
itsg.tugraz.at 
Composition of 𝚽:
We recall that the state
transition matrix is defined as:
𝚽 𝑡 =
𝜕𝐲 𝑡
𝜕𝛂
𝝏𝐫 𝑡
𝝏𝐫𝟎
=
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝑡
𝝏𝐫 𝑡
𝝏𝐫𝟎
𝝏𝐫 𝑡
𝝏𝐫𝟎
We split 𝚽 into two submatrices
for position and velocity:
𝚽 𝑡
𝚽 𝑡 = 𝐫
𝚽𝐫 𝑡
𝑡0
𝑡
𝐟 𝑡 ′ , 𝐫 𝑡 ′ , … ⅆ𝑡′
𝐫 𝑡 = 𝐫0 +
𝑡0
𝑡 − 𝑡′
𝚽𝐫 𝑡 = 𝚽𝐫 𝑡 +
The position appears on
both sides of the equation
Correct derivative for just the position using chain rule:
𝝏𝐫 𝑡
𝝏𝐫0 + 𝐫0 𝑡
=
+
𝝏𝛂
𝝏𝛂
𝑡
𝑡 − 𝑡 ′ ⋅ 𝐟 𝑡 ′ , 𝐫 𝑡 ′ , … ⅆ𝑡′
𝐫 𝑡 = 𝐫0 + 𝐫0 𝑡 +
𝑡
𝝏𝐟 𝑡 ′ , 𝐫 𝑡 ′ , …
𝑡−𝑡 ⋅
ⅆ𝑡′
𝝏𝛂
′
𝑡0
𝑡
𝝏𝐫 𝑡
𝝏𝐫0 + 𝐫0 𝑡
=
+
′
′ 𝝏𝛂
𝝏𝛂
⋅ 𝐓𝐫 𝑡 𝚽𝐫 𝑡 ⅆ𝑡′
𝑡
𝝏𝐟 𝑡 ′ , 𝐫 𝑡 ′ , … 𝝏𝐫 𝑡 ′
𝑡−𝑡 ⋅
ⅆ𝑡′
𝝏𝐫 𝑡 ′
𝝏𝛂
′
0
𝑡0
𝚽𝐫 𝑡
23
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝚽𝐫 𝑡
𝛁𝛁𝐕 𝑡′ ≝ 𝐓𝐫 𝑡 ′
𝚽𝐫 𝑡′
itsg.tugraz.at 
Linearization of 𝚽:
We can do the same for the velocity:
𝑡
𝑡 − 𝑡 ′ ⋅ 𝐓𝐫 𝑡 ′ 𝚽𝐫 𝑡 ′ ⅆ𝑡′
𝚽𝐫 𝑡 = 𝚽𝐫 𝑡 +
𝑡0
We introduce the integral operator
Combined equation system:
𝚽 = 𝚽 + 𝐊𝐓𝚽
𝑡
𝑡 − 𝑡 ′ (⋅) ⅆ𝑡′
𝜅=
𝚽𝐫
𝚽
𝐊 𝐓𝐫 𝚽𝐫
= 𝐫 + 𝐫
𝚽𝐫
𝐊 𝐫 𝐓𝐫 𝚽𝐫
𝚽𝐫
𝑡0
which can be discretized and
represented by an integration matrix 𝐊 𝐫 .
𝚽𝐫 = 𝚽𝐫 + 𝐊 𝐫 𝐓𝐫 𝚽𝐫
Φ3 N 6
24
 I 33

I
  33


 I 33
t0 I 33 

t1I 33 
 

t n I 33 
Solve for the state transition
matrix:
𝚽 = 𝐈 − 𝐊𝐓
−1
𝚽
Gravity gradient:
T3 N 3 N
 f
 diag
 r t0
Matthias Ellmer, Institut für Geodäsie
28. April 16
𝐒:
f
Integrate
f 

r t1
r𝐒t N𝑡 = −𝚽 𝑡
𝑡
𝚽 −1 𝑡 ′ 𝐙𝐩 ⅆ𝑡
𝑡0
itsg.tugraz.at 
Variational equations
cnm , snm ,
r0 ,r0
Goal: Setup a system of linearized obs. equations
l  l 0  Ax  e
r (t0 )
S(t0 ) 3m
Φ(t0 ) 36
r (t1 )
S(t1 ) 3m
Φ(t1 ) 36
r (t 2 )
S(t 2 ) 3m
Φ(t 2 ) 36
Unknown parameters
x
Reference
(dynamic) orbit
Design matrix
A
Observed
positions or velocities
r
x



r (t ), r (t )
r (t N )
25
Matthias Ellmer, Institut für Geodäsie
28. April 16
S(t N ) 3m
Φ(t N ) 36
itsg.tugraz.at 
SST range rate observations 𝜌
Projection of differential
velocity onto baseline.
26
Matthias Ellmer, Institut für Geodäsie
28. April 16
Observation equation can be
derived from variational equations:
𝜌 = 𝐞12 , 𝐫2 − 𝐫1
𝜕𝜌 𝜕𝜌 𝜕 𝐫
𝐀=
=
𝜕𝐱 𝜕 𝐫 𝜕𝐱
This can be realized as simple
matrix multiplication.
itsg.tugraz.at 
Observation equations for SST
27
Matthias Ellmer, Institut für Geodäsie
28. April 16