Robust Trilateration Positioning Method

Robust trilateration positioning method will be introduced in this section in detail. First we need to construct the environment where all RSS information from every T-R can be collected. We assume that there are at least three reference nodes as transmitters T₁, T₂, …, Tn (n≥3) with positions (X1, Y1), (X2, Y2), …, (Xn, Yn), the target as the receiver R1 with the position (X₀, Y₀), and all real T-R distances are d₁, d₂, …, d_n. Following subsections will describe this method with explicit steps by the scenario.

4.2.1 Multi-Frequency Signal Strengths Measurement

In this step, we gather all RSSs from all transmitters to the receiver. At the beginning, every transmitter in the space emits two signals with frequency f1 and frequency f2. And frequency f₁ is smaller than frequency f₂ so it means that wavelength λ₁ with frequency f₁ is greater than wavelength λ2 with frequency f2. Then RSS information what the receiver gets from transmitters T₁, T₂, …, T_n with two frequencies are S_1,1, S_1,2, S_2,1, S_2,2, …, S_n,1, S_n,2 in order.

Based on the characteristic of the radio frequency in normal, RSS from the smaller frequency signal is usually greater than RSS from larger frequency with the same

transmitting power and the same distance. It means that we can observe that S_x,1 is larger

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than Sx,2 where x is the number of the transmitter. But in some conditions, RSS from

frequency f₁ is smaller than RSS from frequency f₂. We will discuss this specific situation in section 4.3.

4.2.2 Dynamic PLE Calibration for Each T-R Path

By section 4.2.1, the information what we get from the receiver are RSS of all T-R paths with every transmitter which provides two data from two signals. In this step, we estimate PLEs for all T-R paths by these RSS information. And after these PLE calibrations, we calculate all T-R distances by equation (8) and use trilateration to count more suitable PLEs for all T-R paths.

At the beginning, we set two default PLE values LEAST_PLE for 1.0 and MOST_PLE for 6.0 which in the sequence are the least value of PLE and the most value of PLE. And we start to calculate all T-R paths’ PLEs, which are ple₁, ple₂, ..., ple_n, by following steps. First in T1 to R1 path, we set ple1 as LEAST_PLE and count two distances by equation (8) with f1, S_1,1 and f₂, S_1,2, then we can get two distances d_1,1 and d_1,2. Next we set the value min as d_1,1 minuses d1,2 and ple1 as ple1 pluses 0.01, then we count two distances and get d1,1 and d1,2

again. We set diff as d_1,1 minuses d_1,2 like above and test whether diff is less than min or not.

If diff is less than min, we set min as diff, ple1 as ple1 pluses 0.01, and count two distances, and do the action like above until diff is greater than min or ple₁ equals MOST_PLE. At last, we can get suitable PLE as ple1 minuses 0.01 or MOST_PLE when ple1 equals MOST_PLE.

As the result, when there are n T-R paths, we have to do n PLE calibrations and get all ultimate values of ple1, ple2, ..., plen. These steps of PLE calibration are also showed in Figure 6.

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Figure 6: Steps of dynamic PLE calibration for each T-R path

The range of all PLEs in this step is from LEAST_PLE to MOST_PLE. However, few conditions show that PLE equals LEAST_PLE or MOST_PLE which means that the attenuation of signal strengths is very slow or very fast. Or on the other hand, one of two signals with different frequencies from the same transmitter through the same T-R path is confused so the attenuation rates of two signals may be large different with each other. For avoiding this problem, we will propose some methods for enhancing our positioning

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method in section 4.3 to solve it.

4.2.3 Check the Intersection Existence by Trilateration

In this step, we check that all T-R distances calculated by above PLE calibration can intersect an area with at least one position by trilateration. And after trilateration, if there is the intersection existing, we next iterate changing all PLEs of all T-R paths and calculating all T-R distances with new PLEs again for eliminating the intersection size to the smallest.

Otherwise, if there is no intersection existing, we also repeatedly change all PLEs of all T-R paths and calculate all T-R distances with new PLEs until there is at least one position in the intersection by trilateration. The following is the steps of this stage.

At the beginning, we first calculate every T-R distance by equation (8) with above PLEs and their frequency f1 and RSS Si,1 of the transmitter i. And then we use every transmitter’s position as the center and its estimated T-R distance to do trilateration for checking the existence of the intersection. After this step, if the intersection exists then we store all PLEs, take every PLE to add the value w which is counted by equation (12), calculate new T-R distance, and do trilateration. We repeat to do this step again and again until the intersection vanishes. And when the intersection vanishes, we restore forward PLEs and then go to the next step. On the other hand, if the intersection is not in the first check then we take every PLE to minus the value w which is also counted by equation (12), calculate new T-R distance, and do trilateration. We also do this step again and again until getting the intersection. And we go to the next step when the intersection appears, too. The flowchart of this section is shown in Figure 7.

w = ^wi_w

n k k=1

(12)

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Figure 7: Steps of check the intersection existence by trilateration

4.2.4 Trilateration Positioning

By above steps, we can get all n T-R distances which are d1,1, d2,1, …, dn,1 by equation (8) with their PLEs and RSSs of the signal with lower frequency f₁. And there is at least one

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point in the intersection from trilateration. The purpose what we want in this step is finding the best proper point in the intersection. And in section 2.3, we introduce the basic

trilateration positioning method and its maximum likelihood method for finding the best point. We take this method for getting the point in this step, too. And after this method, the point what we get is the result of our robust trilateration positioning method.

However, some exceptions may be occurred in reality because of the fluctuation of the signal. For example, one of two signals with different frequencies through the same T-R path may be detrimentally changed hence we may estimate false PLE and calculate wrong distance of the T-R path. Moreover, the positioning result usually increases the error rate after the mistake. Therefore, in the next section we add and introduce three methods in our proposed scheme for reducing some special cases.