# Determination of Navigation System Positioning Accuracy Using the Reliability Method Based on Real Measurements

## Abstract

**:**

## 1. Introduction

_{φ}—standard deviation of the geodetic (geographic) latitude; s

_{λ}—standard deviation of the geodetic (geographic) longitude.

_{φ}= s

_{λ}, p = 0.63, while for the relation s

_{φ}= 10 ·s

_{λ}, p = 0.68.

- The calculations are based on simple Root Mean Square (RMS) determination relationships;
- Gross errors and outliers significantly affect RMS (φ) and RMS (λ), causing a change in the 2DRMS measure;
- Errors are analysed, not as a function of time, but as a function of the subsequent measurement error. The navigation process runs as a function of time. The problem of missing synchronisation with time will emerge in the case of erroneous measurements (recording errors, which have to be removed from the dataset).

- The calculations are quite complex;
- Gross errors and outliers affect the life and failure times in the same way as the other measurements;
- The analysis is carried out as a function of time, similar to the navigation process.

- To propose a new (reliability-based) method to calculate position error values for a navigation system with a probability of 95%;
- To verify which method (classical or reliability) produces results closer to empirical data;
- To check, based on empirical data, the actual measurements of GPS, DGPS and EGNOS systems, whether the distributions of life and failure times for position errors are, in fact, exponential. The other distributions most commonly used in statistics will be tested: beta, Cauchy, chi-square, exponential, gamma, Laplace, logistic, lognormal, normal, Pareto, Rayleigh, Student’s and Weibull.

## 2. Materials and Methods

#### 2.1. Classical Method for Determining the Positioning Accuracy of a Navigation System with 95% Probability

#### 2.2. Reliability Method for Determining the Positioning Accuracy of a Navigation System with 95% Probability

_{n}. Let us choose a specific type (s) of navigation applications for which we intend to check whether the positioning system meets the application requirements in terms of accuracy and availability. These requirements are presented in [16,17,18,19,20,21,22,23,24,26]. Let us run a measurement session of the positioning system of a representative length [37] and calculate position errors as a function of time.

_{n}≤ U for number of measurements (n) = 1, 2,…). When the inverse relationship occurs (δ

_{n}> U), the system is in a failure time.

_{1}, X

_{2},… correspond to the durations of life times and Y

_{1}, Y

_{2},… denote the durations of failure times, which are independent and have the same distributions. Changing the durations of life and failure times results in the change of the operational status of a positioning system (α(t)). Hence, ${Z}_{n}^{\prime}={X}_{1}+{Y}_{1}+{X}_{2}+{Y}_{2}+\dots +{Y}_{n-1}{+\mathrm{X}}_{n}$ become the moments of failure, while ${Z}_{n}^{\u2033}={Z}_{n}^{\prime}{+\mathrm{Y}}_{n}$ are the moments of life (Figure 5) [26,51].

_{i})—expected value of the life time; E(Y

_{i})—expected value of the failure time; V(X

_{i})—variance of the life time; V(Y

_{i})—variance of the failure time.

_{n}and U parameters. Thanks to this, the operational status of a positioning system can be assigned as [26,51]:

_{n}will not be greater than the value of U [51]:

_{n}(t) is a distribution function of the random variable ${Z}_{n}^{"}$.

#### 2.3. Description of GPS, DGPS and EGNOS Measurement Campaigns

- The GPS measurements were carried out at a point with coordinates: φ = 54°32.585029′ N and λ = 18°32.741505′ E (Poland). In March 2013, 168′286 fixes were recorded with a recording frequency of 1 Hz. A typical 12-channel GPS code receiver was used in the study;
- The DGPS measurements were carried out at a point with coordinates: φ = 54°31.756087′ N, λ = 18°33.574138′ E and h = 68.070 m (Poland). In April 2014, 951′698 fixes were recorded with a recording frequency of 1 Hz. 900′000 fixes were used for the analyses, which were the same as for EGNOS. A typical marine DGPS code receiver was used in the study;
- The EGNOS measurements were carried out at a point with coordinates: φ = 54°31.756087′ N, λ = 18°33.574138′ E and h = 68.070 m (Poland). In April 2014, 927′553 fixes were recorded with a recording frequency of 1 Hz. 900′000 fixes were used for the analyses, which were the same as for DGPS. A typical land EGNOS code receiver was used in the study.

## 3. Results

- Do the empirical (actual) distributions of life and failure times for position errors follow an exponential distribution?
- Are there distributions other than exponential with a better fit?
- Depending on the value of the error determining the fitness status (maximum permissible position error for a navigation application), will the statistical distribution of life times change or not?

_{i}), where F(x

_{i}) is the value of the theoretical probability distribution function for respective observation x

_{i}. If the theoretical cumulative distribution is a good approximation of the empirical distribution, then the points on the diagram should be close to the diagonal.

- The analysis of GPS data indicates that the lognormal distribution reflects the course of the PDF of life and failure times determined for navigation system position errors significantly better than the exponential distribution;
- For values above 0.9, the fit between theoretical and empirical distributions (exponential distribution) is very good in all the analysed cases;
- The results obtained from the GPS system also prove that increasing the decision threshold from 1 m to 2 m causes a previously predictable change in the distributions of life and failure times, which does not explicitly prove that this will affect the final results of positioning accuracy calculations;
- Similarly, as in the case of GPS and DGPS systems, EGNOS exhibits similar properties when it comes to fit between the normal distribution and the empirical data.

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Geometric interpretation of the concept of navigation system position error in 2D plane using DRMS and 2DRMS values.

**Figure 3.**Comparison of the classical and reliability methods for assessing the positioning system’s ability to meet the accuracy requirements for a navigation application.

**Figure 4.**The position error as a function of time (

**a**) and three diagrams corresponding to the operational status for: (

**b**) vehicle identification; (

**c**) orders 1a/1b; (

**d**) large ships.

**Figure 5.**The fitness and unfitness statuses of a positioning system in accordance with the reliability method. Own study based on [51].

**Figure 7.**GPS position error as a function of time (168′286 fixes) and two decision thresholds corresponding to the requirements of road transport for vehicle identification and hydrography for special order.

**Figure 8.**Mathcad worksheet for determining life and failure times based on the GPS 2013 measurement campaign.

**Figure 9.**Position error distribution functions of the GPS, DGPS and EGNOS systems calculated using the reliability model.

**Figure 10.**Comparison of three methods for calculating position error values larger than 95% of the population of remaining errors. Analysis of results for the GPS system.

**Figure 11.**Comparison of three methods for calculating position error value larger than 95% of the population of the remaining errors. Analysis of results for: (

**a**) DGPS; (

**b**) EGNOS.

GPS | |||
---|---|---|---|

P-P Plot: Life Time for the Position Error Amounted to 1 m | P-P Plot: Failure Time for the Position Error Amounted to 1 m | ||

P-P Plot: Life Time for the Position Error Amounted to 2 m | P-P Plot: Failure Time for the Position Error Amounted to 2 m | ||

**Table 2.**Statistical analysis of life and failure times for empirical DGPS and EGNOS position errors (1 m).

DGPS | |||
---|---|---|---|

PDF: Life Time for the Position Error Amounted to 1 m | PDF: Failure Time for the Position Error Amounted to 1 m | ||

P-P Plot: Life Time for the Position Error Amounted to 1 m | P-P Plot: Failure Time for the Position Error Amounted to 1 m | ||

EGNOS | |||

P-P Plot: Life Time for the Position Error Amounted to 1 m | P-P Plot: Failure Time for the Position Error Amounted to 1 m | ||

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**MDPI and ACS Style**

Specht, M.
Determination of Navigation System Positioning Accuracy Using the Reliability Method Based on Real Measurements. *Remote Sens.* **2021**, *13*, 4424.
https://doi.org/10.3390/rs13214424

**AMA Style**

Specht M.
Determination of Navigation System Positioning Accuracy Using the Reliability Method Based on Real Measurements. *Remote Sensing*. 2021; 13(21):4424.
https://doi.org/10.3390/rs13214424

**Chicago/Turabian Style**

Specht, Mariusz.
2021. "Determination of Navigation System Positioning Accuracy Using the Reliability Method Based on Real Measurements" *Remote Sensing* 13, no. 21: 4424.
https://doi.org/10.3390/rs13214424