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.