Database Reference
In-Depth Information
Algorithm 6
Kalman smoothing algorithm
Input
:
μ
t
,
Σ
t
,ν
t
+1
,
Φ
t
+1
// predict state using filtered estimate
m
+
=
Aμ
t
P
+
=
A
Σ
t
A
T
+
Q
// compute Kalman smoother gain
J
=(Σ
t
A
T
)
P
−
1
+
// apply correction to filtered estimate
ν
t
=
μ
t
+
J
(
ν
t
+1
− m
+
)
Φ
t
=Σ
t
+
J
(Φ
t
+1
− P
+
)
J
T
smoothed parameters for the
x
T
to be the same as the filtered estimate
for the state at that time, then proceeds backwards through the chain
starting at
t
=
T
1 updating each belief state and covariance matrix
using the update equations in Eq. 10.16 and 10.17.
−
ν
t
=
μ
t
+
J
(
ν
t
+1
−
Aμ
t
)
(10.16)
J
Σ
t
A
T
Φ
t
=Σ
t
−
(10.17)
Lastly, algorithm 6 provides the update algorithm (for a single time
step) of the basic Kalman smoother. The algorithm takes as input the
filtered parameters of the current time step as well as the smoothed
estimates from the
t
+1
st
estimate and produces a smoothed estimate
for state
t
. Similar to the filtering algorithm, we see that the updates
are quite ecient, requiring only a few matrix operations.
To conclude this section, we provide a tracking example in figure 3.2.3.
The red line represents the true trajectory, the blue points are the obser-
vations at each time step. The figure also plots the filtered and smoothed
estimates of the trajectory along with a dashed line at a distance of
1
σ
2
to represent the estimate uncertainty. It is clear that the filtered
trajectory is a substantial improvement over simply connecting the ob-
servations. Furthermore, the smoothed estimate improves upon the fil-
tered estimate, resulting in a very close match to the actual trajectory
with significantly reduced uncertainty (showing more confidence in the
smoothed estimate).
This figure illustrates the importance of applying tracking algorithms
when sensors provide noisy data. Simply using the raw sensor data may
result in inaccurate trajectories which, if used in a MOD or STDB, will
subsequently result in erroneous query results.
Unfortunately, the assumptions upon which the Kalman filter is based
(i.e. linear dynamics, Gaussian measurement and system noise) may be
too restrictive for some applications and therefore more general tech-
niques are required. In these situations, exact inference becomes in-
