The distance filter has specifically been designed for proximity
sensors such as laser range-finders. Distance filters are based on a
simple observation: In proximity sensing, unmodeled obstacles
typically produce readings that are *shorter* than the distance
expected from the map. In essence, the *distance filter* selects
sensor readings based on their distance relative to the distance to
the closest obstacle in the map.

To be more specific, this filter removes those sensor measurements *s*
which with probability higher than (this threshold is set to
0.99 in all experiments) are *shorter* than expected, and which
therefore are caused by an unmodeled object (e.g. a person).

To see, let be a discrete set of possible
distances measured by a proximity sensor. As in
Section 3.2, we denote by
the probability of measuring distance if the robot is at
position *l* and the sensor detects the closest obstacle in the map
along the sensing direction. The distribution describes the
sensor measurement *expected* from the map. As described above,
this distribution is assumed to be Gaussian with mean at the distance
to the closest obstacle along the sensing direction. The dashed
line in Figure 9 represents , for a laser
range-finder and a distance of 230cm. We now can define the
probability that a measured
distance is *shorter* than the expected one given the robot
is at position *l*. This probability is obviously equivalent to the
probability that the expected measurement is longer than
given the robot is at location *l* and thus can be computed as
follows:

In practice, however, we are interested in the probability that is shorter than expected, given the complete current belief of the robot. Thus, we have to average over all possible positions of the robot:

Given the distribution , we now can implement the distance filter by excluding all sensor measurements with . Whereas the entropy filter filters measurements according to their effect on the belief state of the robot the distance filter selects measurements solely based on their value and regardless of their effect on the robot's certainty.

It should be noted that [Fox1998] additionally
developed a *blockage* filter for proximity sensors, which is
based on a probabilistic description of situations in which a sensor
is blocked by an unknown obstacle. We omit this filter here since its
derivation is quite complex and the resulting filter is not
significantly different from the distance filter described here.

Fri Nov 19 14:29:33 MET 1999