Fuzzy Beta-coefficient System

To use the beta-coefficient system with fuzzy numbers, we simply perform the calculations described in the previous section using the fuzzy operators defined above. However, because of the nature of fuzzy operators, some landmark configurations may not be feasible (the matrix inversion used for computing the $\beta$-vector - Equation 3.1- may produce a division by 0), so not all configurations can be stored in the network.

When using the network to compute the position of a landmark, we obtain a fuzzy polar coordinate $(r,\phi)$, where $r$ and $\phi$ are fuzzy numbers, giving us qualitative information about its location. An advantage of working with fuzzy coordinates is that it gives us information about how precise the location estimate is, since it represents the location not as a crisp coordinate, but as a spatial region where the landmark is supposed to be.

Another difference with Prescott's model is the criterion used to select among different estimated locations for the same landmark. In our extended system, instead of looking at the size of the $\beta$-vectors, we use the imprecision of the estimated location itself. The imprecision of a landmark location, $I(l)$, is computed by combining the imprecision in the heading and in the distance as follows. $I_h(l)$ is the imprecision in heading, and it is defined by taking the interval corresponding to the 70% $\alpha$-cut of the fuzzy number representing the heading to the landmark (see Figure 3.3). This imprecision is normalized dividing it by its maximum value of $2\pi$. Similarly, $I_d(l)$ is the imprecision in distance, and it is defined as the 70% $\alpha$-cut of the fuzzy number representing the distance. It is normalized by applying the hyperbolic tangent function, which maps it into the $[0,1]$ interval. Finally, the two imprecisions are combined according to:

$$I(l)=\lambda \cdot \tanh(\beta \cdot I_d(l)) + (1-\lambda) \cdot \frac{I_h(l)}{2\pi}$$ (3)

where $\lambda$ weighs the relative importance of the two imprecisions, and $\beta$ controls how quickly the transformed $I_d$ approaches 1. In our experiments, we set $\beta=1$ and $\lambda = 0.2$. When an object-unit receives a new location estimate, it computes the imprecision of this estimate, compares it with the imprecision of the current location estimate, and keeps the least imprecise location.

Figure 3.3: Computation of the imprecision of the heading toward landmark $l$ as a fuzzy number