Results Obtained with Three ANs

In this section, we show some results relative to the scenario described in Section 5.2.

As in Figure 5.1, the values of the parameters L and V in (5.3) are set to 2 m and 0.5 m, respectively, so that the coordinates of the ANs, expressed in meters, are

s₀= [0, 2] s₁= [−0.5,0] s₂= [0.5, 0]. (5.10) All the results are obtained with the distance threshold dTh set to 0.5 m and the vPeakthreshold vPeak_Th set to 5000. The choice of these parameters is due to ex-perimental calibration.

We assume that range measurements from the three ANs are used to obtain the position estimates of a static TN. We consider 66 different TN positions which are shown in Figure 5.2.

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

x[m]

(a.)

y[m]

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

x[m]

(b.)

y[m]

0≤ d

avg≤ 0.2 [m]

0.2≤ d

avg≤ 0.4 [m]

0.4≤ d

avg≤ 0.6 [m]

0.6≤ d

avg≤ 0.8 [m]

0.8≤ d avg≤ 1 [m]

d avg≥ 1 [m]

Figure 5.3: Values of d_avgrelative to different TN positions for (a.) Algorithm 1 and (b.) Algorithm 2.

For each of the TN positions in Figure 5.2, Algorithm 1 described in Listing 5.1 is applied with Nc= 100. Once the 100 position estimates are given, the knowledge of the true TN position allows evaluating the average distance error davgdefined in (5.2).

Then, Algorithm 2 is applied to the same range estimates used with Algorithm 1 and the average distance error davgis evaluated for each TN position.

The results, in terms of davg, are shown in Figure 5.3. In particular, Figure 5.3 (a.) is relative to Algorithm 1. Each TN position is associated with a different colour, de-pending on the corresponding value of d_avg. More precisely: red squares correspond to values of davgsmaller than 20 cm; orange squares correspond to values of davg be-tween 20 cm and 40 cm; yellow squares correspond to values of d_avgbetween 40 cm and 60 cm; green squares correspond to values of davgbetween 60 cm and 80 cm; blue squares correspond to values of davgbetween 80 cm and 1 m; and violet squares cor-respond to values of d_avglarger than 1 m. Figure 5.3 (b.) can be similarly interpreted but for the fact that it refers to the results obtained with Algorithm 2.

From the results in Figure 5.3 one can deduce that Algorithm 2 is, indeed, an improvement of Algorithm 1. As a matter of fact, Figure 5.3 (a.) shows that when using Algorithm 1, the value of davgis below 20 cm only in 3 (over 66) TN positions, corresponding to 4.5% of the cases. When using Algorithm 2, instead, the value of d_avg is below 20 cm in 11 TN positions, corresponding to 16% of the cases, namely 4 times the percentage obtained with Algorithm 1.

Furthermore, davgobtained with Algorithm 1 is below 40 cm in 28 TN positions, as shown in Figure 5.3 (a.). This corresponds to a percentage of 42%, which is nearly 10 times the percentage relative to d_avg smaller than 20 cm. However, in more than a half of the considered TN positions, the average distance error davgis greater than 40 cm. At the opposite, when using Algorithm 2, the value of d_avgis below 40 cm in 40 TN positions, namely in 60% of the cases.

Moreover, when using Algorithm 1, the value of d_avgis larger than 1 m in 4 TN positions, while if Algorithm 2 is used this happens only once.

Even if, generally speaking, Algorithm 2 improves the performance of Algo-rithm 1, a comparison between Figure 5.3 (a.) and Figure 5.3 (b.) shows that there is one TN position where the value of d_avg obtained with Algorithm 2 is greater than that obtained with Algorithm 1. This happens when the TN coordinates, expressed in meters, are u= [−2,−2]. This is due to the fact that, in this case, some range estimates are accurate even if they are associated with small values of ^vPeak. Al-gorithm 2 ignores such range estimates, leading to a performance degradation with respect to Algorithm 1.

However, in many TN positions, the value of d_avgobtained with Algorithm 1 is analogous to that obtained with Algorithm 2 (for instance, in the TN positions on the left of the metal obstacle) and in 16 TN positions the value of d_avgobtained with Al-gorithm 2 is smaller than that obtained with AlAl-gorithm 1. The improved performance of Algorithm 2 is particularly evident in some TN positions.

As an illustrative example, let us consider the TN position whose coordinates, expressed in meters, are u= [−3,8]. In this case, Figure 5.3 shows that the improve-ment of Algorithm 2 is particularly noticeable as it allows reducing d_avgbelow 20 cm while, according to Algorithm 1, davgis larger than 1 m.

−5 0 5

−2 0 2 4 6 8 10

x[m]

(a.)

y[m]

−5 0 5

−2 0 2 4 6 8 10

x[m]

(b.)

y[m]

Figure 5.4: TN position estimates with (a.) Algorithms 1 and (b.) 2.

All the 100 position estimates are shown in Figure 5.4, where (a.) shows the position estimates obtained with Algorithm 1 and (b.) shows the position estimates obtained with Algorithm 2. From Figure 5.4 (a.) it can be observed that the position estimates with Algorithm 1 can be very inaccurate. To be precise, 16 position esti-mates (among the obtained 100) are nearly 9 meters from the true TN position. This is due to the fact that, in this scenario, AN₂is in NLoS with the TN, leading to wrong estimates of ˆr₂which can have a big impact on the final position estimate.

An illustrative example is shown in Figure 5.5, where the circumferences{Ci}²i=0

centered in{si}²i=0 with radii{ri}²i=0 are shown (dashed lines) together with the cir-cumferences { ˆC⁽²⁹⁾

i }²i=0 obtained with the range estimates {ˆri⁽²⁹⁾}²i=0 in the 29−th iteration. It can be observed that the estimates of r0 and r1 are sufficiently accu-rate and the TN is near one of the two points in ˆC⁽²⁹⁾

0 ∩ ˆC⁽²⁹⁾

1 . At the opposite, r₂ is underestimated by nearly 1 m. Due to this fact, the two points in which Cˆ⁽²⁹⁾

2

intersects Cˆ⁽²⁹⁾

0 are far from the true TN position and, moreover, one of them is near to the wrong intersection between ˆC⁽²⁹⁾

0 and ˆC⁽²⁹⁾

1 . Referring to the procedure findTargetPosition, we are in the case shown in line 5. When looking for the

−10 −8 −6 −4 −2 0 2 4 6 8 10

−8

−6

−4

−2 0 2 4 6 8

x[m]

y[m]

Figure 5.5: The circumferences {Ci}²i=0 (dashed lines) and their estimates obtained from the range estimates in the 29−th iteration { ˆC⁽²⁹⁾

i }²i=0 (solid lines) are shown.

The TN position estimate in this case, expressed in meters, is[5.73, 5.63].

two nearest points of I and J we find that they are the ones on the right of the figure, which turn out to be the wrong ones. The procedurefindTargetPositionVPeak used in Algorithm 2 can overcome this problem. As a matter of fact, in the 29−th iteration the value of vPeak2is below the threshold vPeakTh, so that the localization estimate is performed according to the procedurefindTargetPositionNoS2, i.e., ignoring the range estimate from AN2and choosing as a final position estimate the point in ˆC⁽²⁹⁾

0 ∩ ˆC⁽²⁹⁾

1 with the smallest abscissa, namely the one which is actually nearer to the TN.

The previous example shows how the values of{vPeaki}²i=0can be useful to im-prove the localization performance. These values in the considered TN positions are

x[m]

(a.)

y[m]

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

x[m]

(b.)

y[m]

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

x[m]

(c.)

y[m]

−5 0 5

−8

−6

−4

−2 0 2 4 6 8 10

Figure 5.6: The values of (a.) vPeak₀, (b.) vPeak₁, and (c.) vPeak₂are shown.

shown in Figure 5.6. More precisely, Figure 5.6 (a.) refers to vPeak0, Figure 5.6 (b.) refers to vPeak₁, and Figure 5.6 (c.) refers to vPeak₂. Figure 5.6 (a.) shows that, as expected, if we consider TN positions with ordinate greater than 2 m (so that the TN is in LoS with AN0), the values of vPeak0are large and they gradually decrease as the distance r₀between the TN and AN₀increases. At the opposite, if the obstacle is between the TN and AN0, the values of vPeak0are small because of the signal atten-uation due to the presence of the metal obstacle. Concerning vPeak₁, Figure 5.6 (b.) shows that its values are often large, since the majority of the TN positions are on the same side of the metal obstacle with respect to AN1. At the opposite, Figure 5.6 (c.) shows that the values of vPeak₂are often small, especially in correspondence to TN positions with ordinate greater than 2 m. Once again, this is due to the presence of the metal obstacle between the TN and AN2.

Figure 5.6 shows that three ANs are not sufficient to have a good coverage, in terms of vPeak, in all the considered TN positions. For this reason, aiming at im-proving the results in Figure 5.3, in next sections we consider a similar scenario with four ANs.