2.1 Coordinate systems

The final navigation output that most users need to receive includes the coordinates of a point called latitude and longitude as well as their altitude and accuracy (Noureldin et al., Reference Noureldin, Karamat and Georgy2012). Susceptible measurements are performed by an inertial navigation system. For instance, there are three components perpendicular to body rotation axes and three accelerometers, each of which is not directly related to any coordinate system (linear or curvature). These measurements must be integrated analytically and eventually. Therefore, they are converted to the elliptical coordinate system (Wrigley, Reference Wrigley1977; Noureldin et al., Reference Noureldin, Karamat and Georgy2012). As a result, all coordinate systems must be well defined in the measurement transformations and integral outputs. The definition of different coordinate systems associated with an inertial navigation system is given in the next section.

2.2 Inertial coordinate system

A graphic sample for an inertial coordinate system is displayed in Figure 1. This is a summarised definition of an inertial coordinate system for computational purposes. The directions of the inertial coordinate system axes are demonstrated in Figure 1.

Figure 1.

Inertial and Earth coordinate systems

An inertial coordinate system is defined as follows:

• Centre: in the centre of the Earth

• X_I axis: towards the average vernal equinox at time T ₀

• Y_I axis: a complete right-turn inertial coordinate system

• Z_I axis: towards the north celestial pole at time T ₀

2.3 Earth-Centred-Earth-Fixed (ECEF)

Earth-Centred-Earth-Fixed (ECEF) is a coordinate system that contains the mapping system coordinates. It is not an inertial navigation system and orbits around the Sun at the rate of 7.292115 × 10⁻⁵ rad/s (Kayton and Fried, Reference Kayton and Fried1997; Noureldin et al., Reference Noureldin, Karamat and Georgy2012; Ramesh et al., Reference Ramesh, Jyothi, Vedachalam, Ramadass and Atmanand2016).

The directions of the ECEF axes are demonstrated in Figure 1.

ECEF is defined as follows:

• Centre: in the center of Earth's mass

• X^e axis: towards the Greenwich meridian on the equator plate

• Y^e axis: 90° east of the Greenwich meridian on the equator plate

• Z^e axis: the rotation axis of the basis elliptic

The coordinates in ECEF can be converted with an inertial navigation system by a negative rotation around the Z axis. Essential parameters in the Earth ellipsoid model are shown in Figure 2. The equations related to these parameters will be as follows:

(1)\begin{gather}\begin{array}{l} R = 6378137\,\textrm{m}\\ r = R({1 - f} )= 6,356,752.3142 \end{array}\end{gather}

(2)\begin{gather}f = 1 - ({r/R} )= \frac{1}{{298.257223563}}\;\textrm{flattening}\end{gather}

(3)\begin{gather}e = {[{1 - {{({r/R} )}^2}} ]^{0.5}} = 0.0818191908426\;\textrm{eccentricity}\end{gather}

Figure 2.

Essential parameters in the Earth ellipsoid model (Noureldin et al. Reference Noureldin, Karamat and Georgy2012)

In the above equations, ‘f’ is the flattening coefficient of the Earth and ‘R’ is the Earth's equatorial radius. Also, Radios East-West ‘R _EW’ and Radios North-South ‘R _NS’ can be defined as (Kayton and Fried, Reference Kayton and Fried1997; Noureldin et al., Reference Noureldin, Karamat and Georgy2012; Ramesh et al., Reference Ramesh, Jyothi, Vedachalam, Ramadass and Atmanand2016)

(4)\begin{gather}{R_{\textrm{NS}}} = \frac{{R({1 - {e^2}} )}}{{{{({1 - {e^2}{{\sin }^2}\Lambda } )}^{3/2}}}}\end{gather}

(5)\begin{gather}{R_{\textrm{EW}}} = \frac{R}{{{{({1 - {e^2}{{\sin }^2}\Lambda } )}^{1/2}}}}\end{gather}

where ‘Λ’ is latitude and ‘λ’ is longitude, ‘e’ is the eccentricity of the Earth, ‘R _EW’ is Radios East-West, and ‘R _NS’ is Radios North-South.

2.4 Local horizon system

The local horizon system is a type of coordinate system known to surveyors as Earth's coordinate system (Kayton and Fried, Reference Kayton and Fried1997; Noureldin et al., Reference Noureldin, Karamat and Georgy2012). If a change is made in ‘X’ and ‘Y’ directions, the velocity of the inertial mapping system is usually defined as components along the axes of the local horizontal coordinate system (Noureldin et al., Reference Noureldin, Karamat and Georgy2012). The directions of the local horizontal coordinate system axes are demonstrated in Figures 3 and 4.

Figure 3.

Local horizon system (NEU)

Figure 4.

Local horizon system (NED)

The definition of a local horizon system is as follows:

• Centre: upper centre

• X axis: east of the elliptical curve (also displayed as E axis)

• Y axis: north of the elliptical curve (also shown as N axis)

• Z axis: upwards along the upright ellipsoid (also expressed as U axis, or Z axis downward displayed as  D axis in some Russian books and references)

The conversion matrix between the local horizon and ECEF coordinate systems is as follows:

(6)\begin{equation}\left[ {\begin{array}{*{20}{c}} {{X_E}}\\ {{Y_E}}\\ {{Z_E}} \end{array}} \right] = \underbrace{{\left[ {\begin{array}{*{20}{c}} { - \sin \lambda }& { - \textrm{snin}\Lambda \cos \lambda }& {\cos \Lambda \cos \lambda }\\ {\cos \lambda }& { - \sin \Lambda \sin \lambda }& {\cos \Lambda \sin \lambda }\\ 0& {\cos \Lambda }& {\sin \Lambda } \end{array}} \right]}}_{{C_{\textrm{LL}}^e}}\left[ {\begin{array}{*{20}{c}} E\\ N\\ {UP} \end{array}} \right]\end{equation}

The angular velocity of the local horizontal coordinate system relative to the Earth can be displayed as follows:

(7)\begin{gather}{\omega _E} = \frac{{{V_N}}}{{R + h}}\end{gather}

(8)\begin{gather}{\omega _N} = \frac{{{V_E}}}{{R + h}} + U\cos \Lambda \end{gather}

(9)\begin{gather}{\omega _T} = \frac{{{V_E}}}{{R + h}}\tan \Lambda + U\sin \Lambda \end{gather}

• U: Earth's rotation rate

• R: Earth's radius

• V _N and V _E: relative velocities of the local horizontal coordinate system with respect to the Earth

• h: Earth's ellipsoid height

3.1 Dead reckoning navigation equations

Dead reckoning navigation method equations of an AUV are the equations whose solution leads to an estimate of the current position of the vehicle (Kayton and Fried, Reference Kayton and Fried1997). Another method is inertial navigation, which estimates the current position by taking the initial position of the moving object as well as the accelerations of the object over time (Grewal et al., Reference Grewal, Weill and Andrews2007; Noureldin et al., Reference Noureldin, Karamat and Georgy2012; Gharib et al., Reference Gharib, Heydari and Salehi Kolahi2024). These methods are based on the famous kinematic relations between acceleration, speed and position. Based on these relations, the acceleration integral produces speed, and the speed integral generates position (Leonard and Bahr, Reference Leonard, Bahr and Xiros2016). One of the advantages of DR and inertial navigation methods is that they do not need external data for mobile vehicles from the outside. However, the accuracy of the estimation will increase by correcting navigation errors (Carlson et al., Reference Carlson, Gerdes and Powell2004). Most of the navigation methods can be considered as the processes in which an integral operator leads to provide the outputs (Leonard and Bahr, Reference Leonard, Bahr and Xiros2016). Table 1 displays this issue. In GPS-based position stabilisation navigation systems, integration is unnecessary to convert the sensor output to the position. However, GPS sensitivity can be affected by a number of factors, including the following.

1. 1. Signal blockage: GPS signals can be blocked or weakened by tall buildings, mountains, trees and other obstacles.

2. 2. Atmospheric conditions: GPS signals can be affected by atmospheric conditions such as solar flares, ionospheric disturbances and weather conditions like heavy rain or snow.

3. 3. Satellite position: The accuracy of GPS signals can be influenced by the position of the satellites in the sky.

4. 4. Multi-path errors: GPS signals can be reflected off surfaces like buildings, water and mountains, causing the receiver to receive multiple signals that can lead to errors.

5. 5. Receiver errors: GPS receivers can introduce errors due to factors like clock drift, signal processing and antenna placement.

Table 1.

Integral process in navigation systems

Integrating once in DR and twice in inertial navigation converts the sensor output to the position. These methods to find the position of an object have an essential problem. In these methods, the precision of identifying the vehicle's ultimate position decreases as time passes during navigation.

The practical procedure for solving DR equations involves several steps, which will be described later.

To better understand the concepts in DR, it is necessary to briefly review the relations between the coordinate systems used in this navigation first.

The main equation of DR navigation is the velocity vector equation, which can be displayed as $\overrightarrow p (t )= \int_0^t {\overrightarrow v (\tau )d\tau }$. Its input is the velocity vector ‘$\overrightarrow v (t )$’ whose components are the velocity components north, south and down. The output of the navigation equation is the position vector ‘$\overrightarrow p$’, which in NED coordinates is the equation $\overrightarrow v = \left[ {\begin{array}{*{20}{c}} {{V_N}}& {{V_E}}& {{V_D}} \end{array}} \right]$ and the result after integration will be $\overrightarrow p = \left[ {\begin{array}{*{20}{c}} {{p_n}}& {{p_e}}& h \end{array}} \right]$, where the components of this vector are the coordinates of north, east and height, respectively. The velocity vector in the body coordinate system has to be converted to the velocity vector in the geographical coordinate system. This is done with the use of direction cosine matrices. Suppose that the velocity vector in the body coordinate systems $({{{\vec{V}}_{\textrm{Body}}}} )$ and in the geographical coordinate systems $({{{\vec{V}}_{\textrm{NED}}}} )$ are as follows:

(10)\begin{gather}{\vec{V}_{\textrm{Body}}} = [{{V_x},{V_y},{V_z}} ]\end{gather}

(11)\begin{gather}{\vec{V}_{\textrm{NED}}} = [{{V_N},{V_E},{V_D}} ]\end{gather}

In these relations, velocity ‘V_N’ is towards the geographical north and velocity ‘V_E’ is towards the east.

For converting the velocity vector in the body coordinate system to the velocity vector in the geographical coordinate system, we use the following relation:

(12)\begin{gather}\vec{V}_{\textrm{NED}}^T = T \cdot \vec{V}_{\textrm{Body}}^T\end{gather}

(13)\begin{gather}\boldsymbol{T} = \left[ {\begin{array}{*{20}{c}} {\cos (\psi )\; \cos (\theta )}& {\; - \cos (\varphi )\; \sin (\psi ) + \sin (\varphi )\; \sin (\theta )\; \cos ()}& {\sin (\psi )\sin (\varphi )+ \sin (\theta )\; \cos ()\cos (\varphi )}\\ {\sin (\psi )\cos (\theta )}& {\cos (\psi )\cos (\varphi )+ \sin (\psi )\sin (\varphi )\; \sin (\theta )}& {\sin (\psi )\cos (\varphi )\sin (\theta )- \sin (\varphi )\cos (\psi )}\\ {\; - \sin (\theta )}& {\cos (\theta )\sin (\varphi )}& {\cos (\theta )\cos (\varphi )} \end{array}} \right]\end{gather}

The following relations will be achieved by using the navigation formulae:

(14)\begin{equation}\left\{ {\begin{array}{@{}l} {\dot{\Lambda } = \dfrac{{{V_N}}}{{{R_{\textrm{NS}}} - h}}}\\ {\dot{\lambda } = \dfrac{{{V_E}}}{{({{R_{\textrm{EW}}} - h} )\cos \Lambda }}}\\ {\dot{h} ={-} {V_D}} \end{array}} \right.\end{equation}

where ‘h’ represents the height, which is negative for altitude above sea level and positive for below sea level.

Using these components, latitude ‘Λ’ and longitude ‘λ’ at any time can be calculated:

(15)\begin{gather}V = \sqrt {{v_x}^2 + {v_y}^2 + {v_z}^2} \end{gather}

(16)\begin{gather}\begin{array}{*{20}{l}} {\Lambda (t )= \int_{{t_0}}^t {\dfrac{{{V_N}}}{{({{R_{\textrm{EW}}} - h} )}}dt + \Lambda ({{t_0}} )} }\\ {\; \lambda (t )= \int_{{t_0}}^t {\dfrac{{{V_E}}}{{({{R_{\textrm{EW}}} - h} )\cos \Lambda }}dt + \lambda ({{t_0}} )} }\\ {h(t )={-} \int_{{t_0}}^t {{V_D}dt + h({{t_0}} )} } \end{array} \end{gather}

3.2 Sensor deviations in dead reckoning navigation

DR points show approximate underwater vehicle positions. During the movement of the vehicle from the origin point to the destination point, some factors affect movement and cause the DR navigation points to not exactly match the designated points along the way, so the final point is not obtained precisely (Donovan, Reference Donovan2012; Filaretov et al., Reference Filaretov, Zhirabok, Zyev, Protsenko, Tuphanov and Scherbatyuk2015). Generally, dead reckoning navigation errors can be divided into the two following categories: errors of the measurement instruments and errors due to sea current (Filaretov et al., Reference Filaretov, Zhirabok, Zyev, Protsenko, Tuphanov and Scherbatyuk2015).

The general state of an underwater vehicle's possible situations is shown in Figure 6, considering the errors of the speedometer and gyro navigation sensors. If its sensors are not faulty, the vehicle starts moving from point A and arrives at point B over time. Suppose errors in the range of ‘±ΔΨ’ and ‘±ΔV’ in gyro and speedometer sensors have occurred. In that case, the vehicle will be at a point in the domain of CDEF instead of point B. Navigation values (velocity and north angle) are needed to achieve accurate and momentary navigation, as displayed in Figure 6. We need the actual values of the speedometer and heading angle sensors, as well as their error rates, as illustrated in Figure 6.

Figure 6.

Determining the probable domain

3.3 Error analysis for conventional dead reckoning navigation

The accuracy of dead reckoning navigation depends on the sensors’ precision in measuring the vehicle's heading and speed. We can use the error propagation theory to analyse the navigation error caused by the sensor errors. This theory states that the error in the estimated position is proportional to the errors in the measured parameters and their derivatives. Generally, the navigation error caused by sensor errors can be reduced using sensor fusion techniques, such as Kalman filtering or particle filtering, which combine the measurements from multiple sensors to estimate the vehicle's position more accurately. These techniques can also take into account the correlation between the sensor errors and the vehicle's motion dynamics, which can improve the accuracy of the estimated position.

While sensor fusion techniques can improve the accuracy of dead reckoning navigation, there are some potential problems and negative points that should be considered.

1. 1. Sensor limitations: The accuracy of sensor measurements can be limited by factors such as environmental conditions, sensor drift and hardware limitations. If the sensors used in the sensor fusion system are not accurate, or if they are not properly calibrated, then the accuracy of the estimated position will be reduced.

2. 2. High computational complexity: Sensor fusion techniques are computationally intensive and require significant processing power. This can be an issue in systems with limited computing resources, such as embedded systems or mobile devices.

3. 3. Modeling errors: The accuracy of the estimated position depends on the accuracy of the models used to represent the system dynamics and the sensor errors. If these models are not accurate, then the estimated position will be less accurate and the navigation error may increase.

4. 4. System complexity: Sensor fusion systems can be complex and may require significant expertise to design, implement and maintain. This can increase the cost and complexity of the navigation system.

5. 5. Integration with other navigation systems: Sensor fusion techniques are often used in conjunction with other navigation systems, such as GPS or inertial navigation systems. The integration of these systems can be challenging and may require careful calibration and coordination to ensure that the estimated position is accurate.

6. 6. Limited applicability: Sensor fusion techniques may not be applicable in all navigation scenarios. For example, dead reckoning navigation may be the only option in environments with limited or no access to external sensors, such as underground tunnels or underwater environments. In such scenarios, the accuracy of dead reckoning navigation can be limited by the sensor's accuracy and sensor fusion techniques may not be effective.

7. 7. Cost: Implementing a sensor fusion system can be expensive, particularly if high-quality sensors are needed or if custom hardware and software are required.

8. 8. Maintenance: Sensor fusion systems require regular maintenance and calibration to continue operating accurately. This can be time-consuming and costly.

9. 9. Security: Sensor fusion systems can be vulnerable to cyber attacks, compromising the estimated position's accuracy and reliability.

Overall, while sensor fusion techniques can improve the accuracy of dead reckoning navigation, they are not without their challenges and limitations. These considerations should be carefully weighed when deciding whether to use sensor fusion techniques for a given navigation application.

Considering the problems mentioned in the preceding section in obtaining accurate navigation for a vehicle (marine), we will address this problem by employing a novel and practical method that considers the uncertainties in the data collection from sensors.

4.1 Induced frequency difference

In this section, we investigate solving induced frequency differences between input data of velocity and angle sensors. Gyro and speedometer sensors have different frequencies between input and output data; however, they do not have much of an impact due to the underwater vehicle system, which is inertial (Noureldin et al., Reference Noureldin, Karamat and Georgy2012). Furthermore, the minimum rate of sensor data retrieval can be chosen for more accuracy and optimal use of navigation sensors, and the remaining data obtained from other sensors can be assumed to be fixed. Additionally, the interpolation methods (linear or nonlinear) were applied to obtain more accuracy. Considering the short time interval of sensor data retrieval in the navigation, using the Bisection numerical method is suggested to achieve the sensor's data when needed.

4.2 Flowing water error

This section explores the errors caused by flowing water and how to correct them. Deviation of the movement of water is the volume of deviation that happens in the direction of the float while traveling. The water flow on the sea maps is calculated by the angle of the real water flow direction. The intensity of the current in miles per hour is the magnitude of the current in inches per hour. If the difference between the two deviation points is greater than one hour (or less than one hour), the flow intensity should be measured on a scale of one hour; for example, the flow intensity should be multiplied by 1.5 if it is one and a half hours. Based on Figure 7, assume that the float moves with the course of ‘C ₀’ and the velocity of ‘V’. The water movement often moves with the course of ‘W ₀’ and the velocity of ‘V’. The purpose is to measure the vector ‘R’ (the velocity vector of the vessel influenced by the water flow) and the angle ‘R ₀’ (angle between ‘V’ and ‘R’). Calculations of the vector and the laws of the triangle are used to measure ‘R ₀’. So, the vector ‘R’ can be determined first, and then the angle ‘R ₀’ can be calculated using the sine rules and the values ‘R’ and ‘α’.

(26)\begin{gather}\frac{W}{{\textrm{sin}\;{R_0}\; }}\; = \frac{R}{{\textrm{sin}\; \propto }}\end{gather}

(27)\begin{gather}\textrm{sin}\;{R_0} = \frac{{W\; \times \,\textrm{sin}\; ({\propto} )}}{R}\end{gather}

(28)\begin{gather}{R_0} = {\sin ^{ - 1}}\left( {\frac{{W \times \; \sin \; ({\propto} )}}{R}} \right)\end{gather}

Figure 7.

Correcting water flow errors in the DR method

If W ₀ > C ₀, the R ₀ value should be subtracted from C ₀ to fix the volume of water flow variance R ₀. This implies that the vessel can travel down the course: C ₀₁ = C ₀ − R ₀ to be influenced by the flowing water at point B. If C ₀ > W ₀, the amount of R ₀ is to be applied to C ₀, i.e. C ₀₁ = C ₀ + R ₀; if W ₀ = C ₀, it is C ₀₁ = C ₀.

Table 2 shows the various scenarios to address the issue of water flow in the dead reckoning method.

Table 2.

Various scenarios to address the issue of water flow in the dead reckoning method

4.3 Software simulation of underwater vehicle's dead reckoning navigation

As an illustration, we are going to simulate software with experimental data to determine the dead reckoning navigation of a submarine under the following conditions.

In our simulation, the following assumptions are considered.

• • The underwater vehicle maintains a constant speed and heading throughout the journey.

• • The underwater vehicle does not encounter any winds that affect its movement.

• • The underwater vehicle does not change its course or speed to avoid obstacles or perform manoeuvres.

• • The Earth is assumed to be an ellipsoid model, and the underwater vehicle's movement is calculated using ellipsoid trigonometry by using Equations (1)–(5).

  (29)\begin{gather}\begin{array}{l} {v_x} = 3,{v_y} = 4,{v_z} = 0.1,\;{V_N} = 4.33\,\textrm{m/s,}\;{V_E} = 2.5\;\textrm{m/s},\\ \mathrm{\Lambda } = 27^\circ ,\lambda = 54^\circ ,h = 20\;\textrm{m},\theta = 5^\circ ,\varphi = 5^\circ ,\;\mathrm{\Psi } = 30^\circ ,\mathrm{\delta \Psi } = 0.7\;\textrm{seclat}\\ \delta \Psi = 0.7\;\textrm{seclat} = 0.786^\circ{=} {0.01372^{\textrm{rad}}} \end{array}\end{gather}
  (30)\begin{gather}\begin{array}{l} \delta {V_N} = \cos (\psi )\delta V - V\sin (\psi )\delta \psi ={-} 0.0145\,\textrm{m/s}\\ \delta {V_E}\sin (\psi )\delta V + V\cos (\psi )\delta \psi = 0.0708\,\;\textrm{m/s}\end{array}\end{gather}
  (31)\begin{gather}\left\{ {\begin{array}{@{}l} {\dot{\Lambda } = \dfrac{{{V_N}}}{{{R_{\textrm{NS}}} - h}} = 3.92\; \times {{10}^{ - 7}}}\\ {\dot{\lambda } = \dfrac{{{V_E}}}{{({R_{\textrm{EW}}} - h)\cos \Lambda }} = 4.4\; \times {{10}^{ - 7}}}\\ {\dot{h} ={-} {v_D}} \end{array}} \right.\end{gather}

Now if 10 h is considered for the navigation time, we are going to specify the longitude and latitude of the desired endpoint by using real data from sensors.

(32)\begin{gather}\begin{array}{*{20}{l}} {\dot{\Lambda } = \dfrac{{{V_N}}}{{{R_{\textrm{NS}}} - h}}}\\ {\dot{\lambda } = \dfrac{{{V_E}}}{{({{R_{\textrm{EW}}} - h} )\cos \Lambda }}\; } \end{array}\end{gather}

(33)\begin{gather}\begin{array}{l} \displaystyle\Lambda = \int_{{t_1}}^{{t_2}} {\frac{{{V_N}}}{{{R_{\textrm{NS}}} - h}}dt = 27.0304\;\textrm{degree}}\\ \displaystyle\lambda = \int_{{t_1}}^{{t_2}} {\frac{{{V_E}}}{{({{R_{\textrm{EW}}} - h} )\cos \Lambda }}dt = 54.0205\,\textrm{degree}}\end{array} \end{gather}

Knowing these components, we can calculate latitude ‘Λ’ and longitude ‘λ’ at any time. The essential issue in numbering is to consider this: h = −20 m.

(34)\begin{equation}\left\{ {\begin{array}{*{20}{c}} {\dot{\Lambda } = \dfrac{{{V_N}}}{{{R_{\textrm{NS}}} - h}}\mathop \to \limits^{\textrm{yields}} \; \Lambda = \int_{{t_1}}^{{t_2}} {\dfrac{{{V_N}}}{{{R_{NS}} - h}}dt} \; }\\ {\dot{\lambda } = \dfrac{{{V_E}}}{{({{R_{\textrm{EW}}} - h} )\cos \Lambda }}\mathop \to \limits^{\textrm{yields}} \; \; \lambda = \int_{{t_1}}^{{t_2}} {\dfrac{{{V_E}}}{{({{R_{EW}} - h} )\cos \Lambda }}dt} \; }\\ {\dot{h} ={-} {v_D}\mathop \to \limits^{\textrm{yields}} h ={-} \int_{{t_1}}^{{t_2}} {{V_D}dt} \; } \end{array}} \right.\end{equation}

To solve the above integral by using experimental data, discretisation should be used:

(35)\begin{align}\delta ({{\Lambda _n}} )\; & = \delta {\Lambda _0} + \; \mathop \sum \limits_{k = 1}^n \left( \frac{1}{{({6,378,137 + 20} )}}\; \cos ({{\Psi _k}} )\delta {V_k} - \frac{1}{{({6,378,137 + 20} )}}{V_k}\sin ({\Psi _k})\right.\nonumber\\ & \left.\quad\times \; 0.7\;\textrm{seclat}\; + \frac{{{V_k}\cos ({{\Psi _k}} )}}{{{{({6,378,137 + 20} )}^2}}}\; 0.21 \right)\Delta {t_k}\end{align}

\begin{align*} \delta ({{\lambda_n}} )& = \delta {\lambda _0} + \; \mathop \sum \limits_{k = 1}^n \left( \frac{1}{{({ + 20} )\cos {\Lambda _k}}}\; \sin ({\Psi _k})\delta {V_{k\; \; }} + \frac{1}{{({6,378,137 + 20} )\cos {\Lambda _k}}}{V_k}\cos ({\Psi _k})\right.\nonumber\\ & \left.\quad \times\, 0.7\;\textrm{seclat}\; + \frac{{{V_k}\sin ({{\Psi _k}} )}}{{{{({6,378,137 + 20} )}^2}\cos {\Lambda _k}}}\; 0.21 + \; \frac{{{V_k}\sin ({{\Psi _k}} )\; \tan ({\Lambda _k})}}{{({6,378,137 + 20} )\cos ({{\Lambda _k}} )}}\delta {\Lambda _k}\right)\Delta {t_k}\end{align*}

According to calculation, the following is the tolerance of error for identifying a location's latitude and longitude of the underwater vehicle during 10 h of traveling:

(36)\begin{gather}\delta (\Lambda )={-} 3.2 \times {10^{ - 5}}\Delta t ={-} 0.0066\;\textrm{degree}\end{gather}

(37)\begin{gather}\delta (\lambda )= 1.55 \times {10^{ - 8}}\Delta t = 0.032\;\textrm{degree}\end{gather}