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

To assess the possibility of using a positioning system in a specific navigation application, a mathematical model based on the general theory of reliability has been proposed. Navigation has long used elements of reliability theory in studies related to the assessment of availability, continuity, integrity and reliability, but so far they have not been applied to assess whether a positioning system meets a certain level of accuracy and to determine the navigation system positioning accuracy [49].

Let us consider a positioning system that determines position as a function of time with an error defined as *δn*. 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 [16–24,26]. Let us run a measurement session of the positioning system of a representative length [37] and calculate position errors as a function of time.

Figure 4 (top graph) shows the curve presenting the position error value as a function of time for any positioning system, which should be evaluated in terms of its usability in three exemplary navigation applications. Such applications include: road transport for vehicle identification, with the maximum allowable position error being 1 m with 95% availability, hydrography for orders 1a/1b, with the maximum allowable position error being 5 m with 95% availability and harbour entrance and approach phase for large ships, with the maximum allowable position error being 20 m with 99.7% availability.

Please note that in the presented graph (Figure 4a) the position error value varies as a function of time. Because, near the starting point in this plot, the position error exceeds 20 m, it does not provide the accuracy required for the harbour entrance and approach phase for large ships. This also means that a system with such an error cannot ensure the safe positioning of this process. As a result of exceeding the maximum permissible position error, the system's fitness changes into an unfitness status, as reflected by the "0" values in Figure 4d. After some time, the position error (Figure 4a) decreases to less than 1 m, which means that, for some of the time, the system can be used in all applications. As a result of the reduction in the position error value, for all its applications, the working state of the system changes into a fitness status (Figure 4b–d). The graph (Figure 4a) presents the maximum permissible error values for all three navigation applications (1 m, 5 m and 20 m). If the position error exceeds any of these values, the working state changes. If the position error is smaller than this set value, the system enters the fitness status, and if it is larger, the system is in the unfitness status. Thus, the position coordinate determination can be treated as a two-status stationary renewal process, in which the life and failure times will become random variables, and not the position error as before (a classical approach) [50].

**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.

To be able to determine whether the system was fit or not, a *U* variable was introduced which corresponds to the maximum allowable value of the position error. Let us write it for the three applications under consideration as:

$$\mathcal{U} = \begin{cases} & \text{1 } m \text{ ( $p = 0.95$ ) for road transport-vehicle identification} \\ & \text{5 } m \text{ ( $p = 0.95$ ) for hydrography-orders 1a/1b} \\ & \text{20 } m \text{ ( $p = 0.997$ ) for hardware entrance and approach phase -- large shifts} \end{cases} \tag{3}$$

Assuming that the positioning process varies with time, it can be assigned two states. The first is the life time for which the position error is less than the maximum permissible position error corresponding to the given navigation application (*δ<sup>n</sup>* ≤ *U* for number of measurements (*n*) = 1, 2, ... ). When the inverse relationship occurs (*δ<sup>n</sup>* > *U*), the system is in a failure time.

Let us assume that *X*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 <sup>n</sup>* = *X*<sup>1</sup> + *Y*<sup>1</sup> + *X*<sup>2</sup> + *Y*<sup>2</sup> + ... + *Yn*−1+X*<sup>n</sup>* become the moments of failure, while *Z <sup>n</sup>* = *Z <sup>n</sup>*+Y*<sup>n</sup>* are the moments of life (Figure 5) [26,51].

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

For the reliability method, it is necessary to introduce a number of additional assumptions and designations [50]. It should be assumed that the Cumulative Distribution Functions (CDF) of life (*F(x)*) and failure (*G(y)*) times are right-continuous:

$$P(X\_i \le x) = F(x) \, , \tag{4}$$

$$P(\boldsymbol{Y}\_{i} \le \boldsymbol{y}) = G(\boldsymbol{y}) \text{ for } i = 1, 2, \dots, \tag{5}$$

and that the expected values and variances will take the form:

$$E(X\_i) = E(\mathbf{x})\,. \tag{6}$$

*E*(*Yi*) = *E*(*y*) , (7)

*V*(*Xi*) = *σ*<sup>2</sup> <sup>1</sup> , (8)

$$V(Y\_i) = \sigma\_2^2 \text{ for } i = 1, 2, \dots,\tag{9}$$

where: *E(Xi)*—expected value of the life time; *E(Yi)*—expected value of the failure time; *V(Xi)*—variance of the life time; *V(Yi)*—variance of the failure time.

Moreover, it should be noted that:

$$
\sigma\_1^2 + \sigma\_2^2 > 0. \tag{10}
$$

Based on the above assumptions, it is possible to determine the relationship between the *δ<sup>n</sup>* and *U* parameters. Thanks to this, the operational status of a positioning system can be assigned as [26,51]:

$$\mathcal{U} = \begin{cases} -1 \text{ for } Z\_n' \le t < Z\_{n+1}' \\ \quad 0 \text{ for } Z\_{n+1}' \le t < Z\_{n+1}' \end{cases} \text{ for } n = 0, 1, \dots \tag{11}$$

Let us define the navigation system positioning availability (*A(t)*) as the probability that at any moment of *t*, *δ<sup>n</sup>* will not be greater than the value of *U* [51]:

$$A(t) = P[\delta(t) \le \mathcal{U}],\tag{12}$$

$$A(t) = 1 - F(t) + \int\_0^t [1 - F(t - x)] dH\_\Phi(x),\tag{13}$$

where:

$$H\_{\Phi}(\mathbf{x}) = \sum\_{n=1}^{\infty} \Phi\_{n}(\mathbf{x}) \tag{14}$$

is a function of the renewal stream made up of the renewal moments of the navigation system complying with a specific operation type, while Φ*n(t)* is a distribution function of the random variable *Z*" *n*.

For the purposes of navigation applications, the distributions of life and failure times are exponential. Therefore, their CDFs and Probability Density Functions (PDF) can be calculated using the following formulas [50]:

$$f(t) = \begin{cases} \ \lambda \cdot e^{-\lambda \cdot t} \text{ for } t > 0\\ \ \ \ 0 \text{ for } t \le 0 \end{cases} \tag{15}$$

$$\mathcal{g}(t) = \begin{cases} \; \mu \cdot e^{-\mu \cdot t} \text{ for } t > 0\\ \; \quad 0 \text{ for } t \le 0 \end{cases} \; \prime \tag{16}$$

$$F(t) = \begin{cases} \ 1 - e^{-\lambda \cdot t} \text{ for } t > 0 \\ \ \ 0 \text{ for } t \le 0 \end{cases},\tag{17}$$

$$G(t) = \begin{cases} \ 1 - e^{-\mu \cdot t} \text{ for } t > 0 \\ \ 0 \text{ for } t \le 0 \end{cases} \tag{18}$$

where: *f(t)*—PDF of the life time; *g(t)*—PDF of the failure time; *λ*—failure rate; *μ*—renewal rate. When these assumptions are adopted, the final form of the availability can be noted

as [51]:

$$A\_{\rm exp}(t) = \frac{\mu}{\lambda + \mu} + \frac{\lambda}{\lambda + \mu} \cdot e^{-(\lambda + \mu)\cdot t} \,, \tag{19}$$

as for the limit value:

$$\lim\_{t \to \infty} \left[ A\_{\text{exp}}(t) \right] = A\_{\text{exp}} = \frac{\frac{1}{\overline{\lambda}}}{\frac{1}{\overline{\lambda}} + \frac{1}{\mu}} = \frac{\mu}{\mu + \lambda}. \tag{20}$$

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

Three different navigation positioning systems, commonly used in world navigation, were used to study the reliability method. They include GPS (measurements from 2013), DGPS (measurements from 2014) and EGNOS (measurements from 2014).

GPS is a space-based radionavigation system owned by the United States Government (USG) and operated by the United States Space Force (USSF). The GPS provides two services, or levels of accuracy: the Precise Positioning Service (PPS) and the Standard Positioning Service (SPS). The PPS is available to authorized users and the SPS is available to all users. SPS is the standard specified level of positioning, velocity, and timing accuracy that is available, without restrictions, to any user on a continuous worldwide basis. It provides a global average predictable positioning accuracy of 8 m (*p* = 0.95) horizontally and 13 m (*p* = 0.95) vertically and time transfer accuracy within 30 ns (*p* = 0.95) of Universal Time Coordinated (UTC) (Figure 6a) [52].

DGPS is an enhancement of the GPS, carried out through the use of differential corrections to the basic satellite measurements performed within the user's receiver. The DGPS is based on accurate knowledge of the geographic location of a reference station, which is used to compute corrections to GPS parameters and the resultant position solution. These differential corrections are then transmitted to DGPS users, who apply the corrections to their received GPS signals or computed position. For a civil user of SPS, differential corrections can improve navigation accuracy to better than 5 m (*p* = 0.95) (Figure 6b) [53,54].

EGNOS is Europe's regional Satellite-Based Augmentation System (SBAS), which is used to improve the performance of GNSS systems, such as GPS and Galileo. It has been deployed to provide the safety of life navigation services to aviation, maritime and land-based users over most of Europe. According to [55], the positioning accuracy of the Open Service (OS) should be smaller than 3 m (*p* = 0.95) horizontally and4m(*p* = 0.95) vertically (Figure 6c).

**Figure 6.** Principles of: (**a**) GPS; (**b**) DGPS; (**c**) EGNOS. Own study based on [56].

The following empirical data were used for the numerical analyses:

