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.