INTRODUCTION TO KALMAN FILTER
Rudolf Emil Kalman
- Born 1930 in Hungary
- BS and MS from MIT
- PhD 1957 from Columbia
- Filter developed in 1960-61
- Now retired
Why not the deterministic system and control
theories do not provide a totally sufficient means of
performing the analysis and design?
- No mathematic model is perfect, only a limited
number of parameters are used in modeling.
- Deterministic systems are driven not only by our
own signals, but also by disturbances/noises. These
factors are beyond our control and capability of
analysis (Example: measure the temperature, pilot
control the fly angle, etc.).
- Sensors themselves do not provide perfect and
complete date about a system. Sensors may behave
according to their own dynamic characteristics,
and may be corrupted by noises.
What is a Kalman Filter??
•- Just some applied math.
•- A linear system:: ff((a+b)) = ff((aa)) + ff((b))..
•- Noisy data in hopefully less noisy out.
•- But delay is the price for filtering......
•- Pure KF does not even adapt to the data.
What is it used for??
••- Tracking missiles
••- Tracking heads/hands/drumsticks
••- Extracting lip motion from video
••- Lots of computer vision applications
• - Economics
• - Navigation
A really simple example
Review: conditional probability density.
It indicates that for the given values of all
measurement taken up through time instant i,
what probability would be of x(i) assuming a
certain particular value or range of values.
What will happen if we suppose the
probability distributes in Gaussian pattern
(normal distribution and white)?
Description:
Suppose you are lost at sea at night and want to obtain your
location x by star sighting. At time instant t1, your
measurement is z1. Due to some device inaccuracies and other
uncertainties, you decide the precision is such that its variance
is 
This way, you can set up your conditional probability
of your position x(t1) at time instant t1, conditioned on the
measurement (observation) of z1 .
2
z1 .
Based on this CPD, your best (optimal) estimation of your
position is

x(t1 )  z1
 (t1 )  
2
x
2
z1
Now suppose an more experienced navigator takes another
measurement right after you do at time t2= t1 (so the ship
potion is not changed), similarly, obtained z2
2

and z 2 .
Note

2
z2 <

2
z1 .
Likewise, the CPD is as follows:
At this point, you have two data points for estimating your
position.
Question: How to combine these data?
Based on the assumptions made, the conditional density of
your position x(t2) at time t2= t1 , given z1 and z2, is a Gaussian
density 
below.
and variance

2
, as indicated in the figure
  [ /(   )]* z1  [ /(   )]* z2
2
z2
2
z1
2
z2
2
z1
2
z1
2
z2
1/   (1/  )  (1/  )
2
2
z1
2
z2
Note that,  is less either 
or 
which is to say
that the uncertainty in your estimate of position has been
decreased by combining the two pieces of information.
2
2
z1
2
z2 ,
Given this density, the best estimate of your position is

x(t2 )  
with associated variance

2
.
^
x(t2 )
The equation of estimate

can be rewritten as
x(t2 )  [ /(   )]* z1  [ /(   )]* z2
2
z2
2
z1
2
z2
2
z1
2
z1
2
z2
 z1  [ z21 /( z21   z22 )]*[ z2  z1 ]
In final form that is used in Kalman filter implementation, it is
expressed as



x(t2 )  x(t1 )  K (t2 ) *[ z2  x(t1 )]
^
Note that
x(t1 )  z1 , and
K (t2 )   /(   )
2
z1
2
z1
2
z2
The above equations say that the best estimate at time t2,
^
x(t2 )
, is equal to the best prediction of its value before z2 is
^
taken, x(t1 ) , plus a correction term of an optimal weighting
value times the difference between z2 and its best prediction
^
x(t1 ) .
So the filter takes the structure of “prediction correction”.
From the equation of the variance expressed afore, let us
rewrite it as
 (t2 )   (t1 )  K (t2 ) * (t1 )
2
x
2
x
2
x
^
This indicates that the value of
x(t2 )
and
f x (t2 )|z (t1 ), z (t2 )
 x2 (t 2 ) embody
( x| z1 , z2 )
all the information in
. State
differently, by propagating these two values, the conditional
density of your position at time t2, given z1 and z2, is completely
specified.
Thus far, we have solved the static estimation problem, next
time let us incorporate the dynamics in the system.
(To be described).