16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 1 of 8
Lecture 9
Last time: Linearized error propagation
1s
eSe=
Integrate the errors at deployment to find the error at the surface.
11
1
T
sss
TT
T
Eee
See S
SE S
=
=
=
Or Φ can be integrated from:
, where (0)
()
FI
xfx
df
F
dx
Φ= Φ Φ =
=
=
&
&
where F is the linearized system matrix. But this requires the full Φ (same
number of equations as finite differencing).
n
t =time when the nominal trajectory impacts.
1
21
() ()
()
nn
rn r
et t e
et e e
=Φ
==Φ
where
r
Φ is the upper 3 rows of ()
n
tΦ .
Covariance matrix:
21
T
rr
EE=Φ Φ
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 2 of 8
() () ()
() () () () ()
() () () ()
() ()
T
TT
TTT
T
eFe
Et etet
Et etet etet
Fe t e t e t e t F
FE t E t F
=
=
=+
=+
&
&
&&
You can integrate this differential equation to t
n
from
1
(0)EE= . This requires the
full 66× E matrix.
2
()
upper left 3 3 partition of ( )
TT
rr rv
n
TT
vr vv
n
ee ee
Et
ee ee
EEt
??
=??
??
=×
For small times around t
n
,
() () ()( )
() ( () ()( )
() ()( )
nnn
nnnvn n
nnn n
et et vt t t
et v t e t t t
et v t t t
=+ ?
=+ + ?
=+ ?
2
2
1()1 1 ()( )0
1
()
1
TTT
vvvnn
T
v
in T
vn
et e v t t t
e
tt
v
=+ ?=
?=?
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 3 of 8
3
2
2
2
position error at impact
1
1
1
"projection matrix"
1
T
v
n T
vn
T
nv
T
vn
e
e
ev
v
v
Ie
v
=
=?
??
=?
??
??
[]
32
123
[]
cos ... ...
... ... ...
... ... ...
cos
111
ij
ij ij
eRe
R
R
R
θ
θ
′ =
??
??
=
??
=
=
1
j
= unit vectors along the j
th
axis of the 2 frame expressed in the coordinates of
the 3 frame.
32
32
T
eRe
ERER
′=
′ =
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 4 of 8
3
3
3
() cos ( )
()
() sin ( )
RRn n
TT
H Hn n
et e v tt
et e
et e v tt
γ
γ
=+ ?
=
=? ?
Impact:
3
3
33
33
3
() sin( ) 0
1
()
sin
cos
()
sin
cot
()
Hi H n i n
in H
n
n
Ri R H
n
RH
Ti T
et e v t t
tt e
v
v
et e e
v
ee
et e
γ
γ
γ
γ
γ
=? ?=
?=
=+
=+
=
The transformation which relates R,H,T errors at the nominal end time to R and T
errors when H=0 is:
33
3
4
33
()
()
cot
10cot
01 0
Ri
Ti
RH
T
et
e
et
ee
e
ePe
γ
γ
??
=
??
??
+
??
=
??
??
??
′′=≡
??
??
If the
s
e defined earlier, based on integration of perturbed trajectories, is
measured in R,T coordinates, then the sensitivity matrix defined at that point is
equivalent to
1s
r
eSe
SPR
=
=Φ
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 5 of 8
43
2 22
2
T
RRT R RT
RT T R T T
EPEP
RRT
TR T
σ μσρσ
μσ ρσσ σ
′=
??
??? ?
== =??
??? ?
??? ?
??
If all the original error sources are assumed normal, R and T will have a joint
binormal distribution since they are derived from the error sources by linear
operations only. This joint probability density function is
()
()
22
2
2
21
2
1
,
21
RRTT
rrtt
RT
frt e
ρ
σσσσ
ρ
πσ σ ρ
??
?? ??????
???+
?? ??????
?? ??????
?
?
??
=
?
where ,
RT
σ σ and ρ can be identified from
4
E . Recall that we are considering
unbiased errors.
Contour of constant probability density function is
22
2
2
RRTT
rrtt
cρ
σσσσ
?? ??????
?+=
?? ??????
?? ??????
cos sin
sin cos
rx y
tx y
θ θ
θ θ
=?
=+
Get:
{
222
0
() () ()x xy y cθθθ
=
++=
Coefficient of ,x y equals zero for principal axes.
22 22
tan 2
RT RT
RT RT
ρσσ μ
θ
σ σσσ
==
??
Use a 4 quadrant tan
-1
function.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 6 of 8
Once θ is found, can plug into pdf expression, get
x
σ and
y
σ .
222 2
222
222
()(2)
1
()
2
1
()
2
RT RT
xRT
yRT
h
h
h
σσ ρσσ
σσσ
σσσ
=?+
=++
=+?
cos
sin
ixi
iyi
xc
yc
σ φ
σ φ
=
=
May want to choose c to achieve a certain probability of lying in that contour.
In principal coordinates, the probability of a point inside a “cσ ” ellipse is
2
2
1
c
Pe
?
=?
People often choose c to find what is called the circular probable error (CPE).
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 7 of 8
1
()
2
xy
σ σσ=+
Choosing P=0.5, c=1.177
0.588( )
xy
CPE σ σ=+
This approximation is good to an ellipticity of around 3.
Random Processes
A random process is an ensemble of functions of time which occur at random.
In most instances we have to imagine a non-countable infinity of possible
functions in the ensemble.
There is also a probability law which determines the chances of selecting the
different members of the ensemble.
We generally characterize random processes only partially.
One important descriptor – the first order distribution.
This is the classical description of random processes. We will also give the state
space description later.
16.322 Stochastic Estimation and Control, Fall 2004
Prof. Vander Velde
Page 8 of 8
1
()x t is a random variable.
[ ]( , ) ( ) , where ( ) is the name of a process and is the value taken
(,)
(,)
Fxt Pxt x xt x
dF x t
fxt
dx
=≤
=