## 3 Evaluating the Accuracy of Determining the Position Coordinates Using Dead Reckoning  
If the impact of current, drift, and waving of sea on the BAUV movement can be avoided, the only sources of errors are the devices used to dead reckoning. In the case of the BAUV, these are the Inertial Navigation System, INS \"VN-200\", which determines course over ground (COG) [21] and the electromagnetic log \"ALIZE\", which determines speed through water (STW). In this case, the STW is equal to the speed over ground (SOG) [22]. By measuring using these devices COG \(\left(k\right)\), SOG \(\left(k\right)\) at the time of \(\left(k\right)\), vector \(\mathbf{x_{h}}\) of horizontal coordinates and their accuracy can be calculated for the time \(\left(k+1\right)\). Functions \(\mathbf{f}\left(\mathbf{x_{h}}\left(k\right),\mathbf{u_{h}}\left(k\right), \mathbf{w_{h}}\left(k\right)\right)\) can be used for this purpose describing the non-linear BAUV movement model:  
\[\mathbf{x_{h}}\left(k+1\right)\ =\left[ \begin{array}{c}f_{x}\left(k+1\right)\\ f_{y}\left(k+1\right)\end{array} \right]=\left[ \begin{array}{c}x\left(k \right)\\ y\left(k\right)\end{array} \right]+\left[ \begin{array}{c}\Delta t\left(k \right)\cdot\text{SOG}\left(k\right)\cdot\text{sinCOG}\left(k\right)\\ \Delta t\left(k\right)\cdot\text{SOG}\left(k\right)\cdot\text{cosCOG}\left(k \right)\end{array} \right]+\left[ \begin{array}{c}w_{\text{x}}\left(k\right) \\ w_{y}\left(k\right)\end{array} \right] \tag{8}\]  
where \(x\left(k\right),y\left(k\right)\)--dead reckoned horizontal coordinates of BAUV position at the time of \(k\), COG \(\left(k\right)\)--the BAUV course over ground at the time of \(k\); SOG \(\left(k\right)\)--the BAUV speed over ground at the time of \(k\); \(\Delta t\left(k\right)\)--duration between the time of \(k\) and \(k+1\); \(w_{x}\left(k\right),w_{y}\left(k\right)\)--the so-called intentional interference in determining the coordinates at the time of \(k\) (expressed as a zero mean normal distribution N[0, 1]); and calculated on the basis of Equation (8) the covariance matrix  
\[\mathbf{P}\left(k+1\right)=\mathbf{F_{x}}\left(k+1\right)\mathbf{P}\left(k \right)\mathbf{F_{x}}\left(k+1\right)^{\mathbf{T}}+\mathbf{Q}\left(k\right) \tag{9}\]  
here: \(\mathbf{F_{x}}\left(k+1\right)=\left[ \begin{array}{cc}\frac{\partial f_{x}} {\partial x}&\frac{\partial f_{x}}{\partial y}\\ \frac{\partial f_{y}}{\partial x}&\frac{\partial f_{y}}{\partial y}\end{array}  \right]=\left[ \begin{array}{cc}1&0\\ 0&1\end{array} \right]\)--the so-called matrix system is calculated as Jacobian matrix from the function \(\mathbf{f}\left(\mathbf{x}\left(k\right),\mathbf{u}\left(k\right),0\right)\); \(\mathbf{P}\left(k\right)\)--the covariance matrix, determined at the time of \(k\). \[\mathbf{Q}\left(k\right)=\left[ \begin{array}{cc}\frac{\partial f_{x}}{ \partial COG}&\frac{\partial f_{x}}{\partial SOG}\\ \frac{\partial f_{y}}{\partial COG}&\frac{\partial f_{y}}{\partial SOG}\end{array}  \right]\left[ \begin{array}{cc}\sigma_{COG}{}^{2}&0\\ 0&\sigma_{SOG}{}^{2}\end{array} \right]\left[ \begin{array}{cc}\frac{\partial f _{x}}{\partial COG}&\frac{\partial f_{x}}{\partial SOG}\\ \frac{\partial f_{y}}{\partial SOG}\end{array} \right]^{T}\]  
\(\sigma_{COG}\)--the course over ground mean measurement error; \(\sigma_{SOG}\)--the speed over ground mean measurement error. Knowing the covariance matrix of initial (the previous) position coordinates \(\mathbf{P}(k)\) and the covariance matrix of the vector of growth coordinates \(\mathbf{P}(k+1/1)\), taking into account the errors resulting from the operation of dead reckoning navigation, the covariance matrix of the coordinates of the current BAUV position can be determined  
\[\mathbf{P}\left(k+1\right)=\mathbf{P}\left(k\right)+\mathbf{P}\left(k+1/1\right) \tag{10}\]  
where  
\[\mathbf{P}\left(k+1\right)=\left[ \begin{array}{cc}\sigma_{x}{}^{2}&\sigma_{ xy}\\ \sigma_{yx}&\sigma_{y}^{2}\end{array}