JKalman
Java Kalman Filter
Petr Chmelar
FIT, Brno Univesity of Technology
Teoretical Background
The Kalman Filter was originally designed to represents information about a moving objects. Moving objects
are modeled using a discrete time dynamic system. An object has spatio-temporal state x
t
. The spatio-
temporal state is represented by location [x, y, t] and velocity [d
x
, d
y
]. Here [x, y] is referred to as 2D position
[x, y] of the object at the time t.
Data is supposed to be uncertain – noisy, some states might be missing and other are biased. At the
moment, the object state x
t
is characterized by a state vector x, containing position and velocity vector [x, y,
d
x
, d
y
]. In that way, the trajectory is constructed (in 2D). However it cannot be observed directly, because it is
encumbered by hidden (Gaussian) noise w. The system produces visible output vector y that is a simple linear
observation (x evolves first-order Markov process), encumbered by noise v. At time t, the model can be written
as in the Figure 1.
C
Ad
t
w
x
t
y
t
v
(1) ystem dynamic model S
(2) easurement model M
wBuAxx
ttt
++=
−− 11
),0(
QNw
≈
vCxy
tt
+= ),0(
RNv
≈
Figure 1. The dynamic process illustration.
In the Figure 1, d
t
is a delay, A is a state transition matrix; B the optional control matrix with input u, C is the
observation or measurement matrix. N is a normal distribution of mean 0, the (covariance) matrices of w and v
are Q and R, respectively. Based on Bayesian probability P(x
t
| y
t
) ≈ N(xˆ
t
, P
t
), as described below.
An efficient computational (recursive) tool to estimate the state of a process, in a way that minimizes the mean
of error, is the Kalman filter.
