HIDROGRAFÍA

The location affects the performance in two ways. First, a signal can be lost due to absorp-tion if a beacon is placed too far away. Second, the geometric diluabsorp-tion of precision may be too high, which makes that a position estimate must be discarded.

11.4.1 Beacon placement

The placement of the beacons influence the quality of signals received at the mobile sta-tion, and consequently, the obtained performance for a position estimate. Several factors limit an optional placement of the beacons. First, the skip distance limits reception close to a beacon, as was proven in section 6.3.2. Second, the models obtained in chapter 10 are only valid in the range for which a model fit could be made from the simulation data. This range is lower bounded by the skip distance, and upper bounded by some maximum distance c.f. fig-ure 6.8. Third, the limits on the signal strength dictate when a signal is lost due to low signal strength. Section 6.4.2 proved that the absorption increases with distance. Hence, an up-per bound exists, defined by the maximum allowed absorption. All these bounds themselves depend on many factors, like the ionospheric composition and the operational frequency which is used.

Furthermore, the previous sections have shown that the performance can be poor for certain scenarios or localization technologies. This leads to the question if beacons can be placed such, that the performance is enhanced under all scenarios. To examine this, several beacon arrangements are defined. An arrangement contains information about the location of beacons. So, each arrangement contains a certain number of beacons, placed at specific locations. It is expected that the arrangements with more beacons yield a better position es-timate, as there are simply more beacons to provide positioning information. The following arrangements are defined:

• RECT-S: 4 beacons, placed in a square with sides of 5^◦

• RECT-L: 4 beacons, placed in a large square with sides of 10^◦

A FEASIBILITY STUDY ON GROUND-BASED LOCALIZATION FOR MARS EXPLORATION 90

• GRID-S: 12 beacons, placed in a rectangular grid. Beacons are spaced by 10^◦

• TRIA-S: 14 beacons, placed in a mesh of equilateral triangles with sides of 5^◦

• TRIA-L: 14 beacons, placed in a mesh of equilateral triangles with sides of 10^◦

A graphical representation of the five beacon arrangements is given in figure 11.4. The RECT-L arrangement is considered the standard; it has been used in this research so far.

A comparison of performance for different beacon arrangements is shown in figure 11.5.

In the figure, only the performance for the nominal scenario (fig. 11.5a) and the min scenario (fig. 11.5b) is shown. It is clear that the beacon arrangements with more beacons yield a better performance. These are the GRID-S, TRIA-S and TRIA-L arrangements. The beacons in the TRIA-S arrangement, however, are packed closely together. As can be seen from fig-ure 11.5, this only increases the performance marginally. In other words, the GRID-S and TRIA-L arrangement cover a lot more surface for a comparable performance to the TRIA-S arrangement. This is considered a downside for the TRIA-S arrangement, as a closely packed arrangement needs a lot more beacons to cover a certain area. When more beacons are needed, the mission cost rises. The mission complexity rises, too, as more beacons need

RECT-S RECT-L GRID-S TRIA-S TRIA-L

Figure 11.4: An illustration of the five beacon arrangements. Each triangle represents one bea-con.

Figure 11.5: An overview of the performance using a selection of different beacon arrange-ments. The operating frequencies are 2.5MHz and 4.5MHz. The performance for the nominal scenario is shown in figure 11.5a, while figure 11.5b shows the performance under a minimal scenario. The measures of precision MP , used to obtain these results, are found in appendix B.

to be deployed. Therefore, the GRID-S and TRIA-L arrangements are superior to the TRIA-S arrangement.

The lowest geometric errors are obtained by using a TRIA-S beacon arrangement. The GRID-S and TRIA-L arrangements, compared in the figure, yield similar results. The TRIA-S arrangement provides a precision of at least 2m for the nominal scenario, and 10m for the min scenario, as is evident from figure 11.5. The TRIA-L arrangement, on the other hand, provides a precision below 4m and below 20m, respectively, which is about twice as high as the TRIA-S arrangement. This precision can be obtained for both an operating frequency of 2.5MHz and 4.5MHz. Discarding the RSS -approach, a precision below 10m is even possible. In the min scenario, both these arrangements and the GRID-S arrangement provide a sub-10-meter precision for all technologies, except the RSS -approach at an operating frequency of 4.5MHz.

Considering that the mission cost and complexity is too high for the TRIA-S arrangement, it can be concluded that a precision of less than 4m and 10m are possible in the nominal and min scenario, respectively. Note that for the nominal scenario, this arrangement attains a precision twice as low for the TOA -approach.

The malfunction of beacons has so far not been considered. It is likely that a beacon will malfunction at some point in time. Hence, beacons should be placed such, that there are always three beacons within range of the mobile station. As was seen in section 8.1, three beacons are minimally required, and a fourth one is redundant. It is assumed that all arrangements contain a sufficient number of beacons to provide some level of redundancy.

However, it is recommended to study the effects of beacon malfunction in more detail.

11.4.2 Dilution of precision

The geometric performance of a localization system is related to the geometry of the bea-cons in that system. To quantify the coverage, the GDOP is calculated for each technol-ogy, frequency and scenario. The GDOP has previously been explored in section 8.3. The GDOP value is a dimensionless variable, indicating the geometric error caused by the place-ment of the beacons in a system. It was found in section 8.3 that values around 1 indicate a low error, whereas GDOP values above 6 are generally discarded.

The GDOP has been evaluated for the nominal and min scenario in figure 11.6. Re-sults for different frequencies and technologies have been grouped together per beacon arrangement. The standard deviations in the results are caused by this grouping. It was expected that the GDOP metric would be equal for any frequency or technology, as it is purely a geometric measure. However, it has been shown in section 11.3, that beacons are sometimes unavailable due to a weak signal strength. This poses an interesting problem, as the number of remaining beacons are reduced. Ultimately, a reduction in beacons increases the geometric error, and this is what causes the deviations in figure 11.6. Nevertheless, the deviations in GDOP are quite small compared to the results obtained for the beacon ar-rangements.

In all cases, however, the GDOP metrics show a good quality of the estimates. Refer to section 8.3 for a description of the numerical values. It remains to be seen what the impact is of mountainous terrain which can severely degrade the GDOP metric. This is left for future work.

A FEASIBILITY STUDY ON GROUND-BASED LOCALIZATION FOR MARS EXPLORATION 92

Nominal Min 1

1.5 2 2.5

GDOP

GRID−S TRIA−S TRIA−L

Figure 11.6: A comparison of GDOP results for the nominal and min scenario. Error bars indicate the standard deviation of the GDOP caused by distinct frequencies and localization technologies.

11.5 Summary

Several limitations persist on the use of a ground-based localization network on Mars.

First, ionospheric effects cause that some localization technologies cannot always be used. It is shown that the TOA and AoA -approaches are stable under any scenario, whereas the RSS -approach is not.

Second, the performance degrades unambiguously under an increase in frequency. The best results are obtained when lower frequencies are used. From the frequency analysis it is further apparent that the TOA -approach can yield a more stable performance than the AoA approach. Again, the performance using a RSS -approach is significantly worse than the other two.

Third, the antennas which are used limit the reception of signals, as some signals are too weak to be received. A blackout factor is used to measure the occurrences of loss of signal. Generally, an increase in frequency induces a higher fraction at which signals are lost.

Furthermore, the SEP and min ionospheric scenarios are especially susceptible to blackout.

Last, the locations at which the beacons are placed have a profound effect on the perfor-mance of a localization system. The most optimal arrangements are the GRID-S and TRIA-L arrangements, where beacons are placed in a rectangular and triangular grid, respectively, and with distances of 10^◦between the beacons.

The results indicate that a precision of less than 4m can be obtained with the use of any technology in the nominal scenario with a TRIA-S arrangement, for frequencies of 2.5MHz and 4.5MHz, Moreover, a precision of less than 10m for these frequencies is achieved, when the beacons are arranged in either an GRID-S or TRIA-L arrangement. using either an TOA or AoA approach. The TRIA-S arrangement is discarded, as it is found to be suboptimal from a mission cost & complexity perspective.

The limitations have shown that some approaches to a localization system on Mars are to be avoided, in order to guarantee a stable system with a good performance. Ultimately, a localization system using a TOA -approach with frequencies of 2.5 or 3.5MHz, and a beacon arrangement with either rectangular or triagonally placed beacons, spaced 10^◦apart, yields the best and most stable performance.