As described in Section 3.2, the perception
model 
for proximity sensors only depends on the distance
to the closest obstacle in the map along the sensor beam. Based
on the assumption that the map of the environment is static, our
approach pre-computes and stores these distances for each
possible robot location l in the environment. Following our sensor
model, we use a discretization 
of the possible
distances . This discretization is exactly the same for the
expected and the measured distances. We then store for each location
l only the index of the expected distance in a
three-dimensional table. Please note that this table only needs one
byte per value if 256 different values for the discretization of
are used. The probability 
of measuring a distance
if the closest obstacle is at distance (see
Figure 6) can also be pre-computed and stored in a
two-dimensional lookup-table.
As a result, the probability of measuring s given a
location l can quickly be computed by two nested lookups. The first
look-up retrieves the distance to the closest obstacle in the
sensing direction given the robot is at location l. The second
lookup is then used to get the probability 
. The
efficient computation based on table look-ups enabled our
implementation to quickly incorporate even laser-range scans that
consist of up to 180 values in the overall belief state of the robot.
In our experiments, the use of the look-up tables led to a
speed-up-factor of 10, when compared to a computation of the distance
to the closest obstacle at run-time.