# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 4 Modeling of Uncertainty
### Quantification of Measurement Uncertainty: Bounded Measurement Noise  
To identify a set-valued uncertainty model for the compass sensor, as well as for the GPS-based position measurements, independently for each of the DDboats, assume additive, time-varying interval uncertainties in the model for the compass sensor  
\[\phi(t_{k})\in\phi_{\mathrm{m}}(t_{k})+\left[\Delta\phi(t_{k})\right] \tag{61}\]  
as well as time-invariant uncertainty bounds for the GPS-based position measurement according to  
\[ \begin{bmatrix}x_{1}(t_{k})\\ x_{2}(t_{k})\end{bmatrix} \in \begin{bmatrix}x_{1,\mathrm{m}}(t_{k})\\ x_{2,\mathrm{m}}(t_{k})\end{bmatrix} + \begin{bmatrix}\left[\Delta x_{1}\right] \\ \left[\Delta x_{2}\right]\end{bmatrix} \enspace. \tag{62}\]  
A corresponding uncertainty representation of the velocity vectors of the DDboats is obtained by an explicit Euler method-based approximation in the form  
\[ \begin{bmatrix}v_{1}(t_{k})\\ v_{2}(t_{k})\end{bmatrix} \in\frac{1}{T_{\mathrm{s}}}\cdot\left( \begin{bmatrix} x_{1,\mathrm{m}}(t_{k+1})\\ x_{2,\mathrm{m}}(t_{k+1})\end{bmatrix} - \begin{bmatrix}x_{1,\mathrm{m}}(t_{k}) \\ x_{2,\mathrm{m}}(t_{k})\end{bmatrix} + \begin{bmatrix}\left[\Delta\hat{x}_{1} \right]\\ \left[\Delta\hat{x}_{2}\right]\end{bmatrix} \right)\enspace, \tag{63}\]  
where, under the assumption of symmetric intervals \(\left[ \begin{array}{cc}\Delta x_{1}&\left[\Delta x_{2}\right]\end{array}  \right]^{T}\
i\boldsymbol{0}\), the error bounds  
\[ \begin{bmatrix}\left[\Delta\hat{x}_{1}\right]\\ \left[\Delta\hat{x}_{2}\right]\end{bmatrix} =2\cdot \begin{bmatrix}\left[ \Delta x_{1}\right]\\ \left[\Delta x_{2}\right]\end{bmatrix}  \tag{64}\]  
of the velocity model (63) are obtained. A contractor-based identification of the interval bounds \(\left[\Delta\phi(t_{k})\right]\), \(\left[\Delta x_{1}\right]\), and \(\left[\Delta x_{2}\right]\) has to make sure that, for a reasonable initial search space for the measurement uncertainties, the compatibility constraint  
\[\tan(\phi_{\mathrm{m}}(t_{k})+\left[\Delta\phi(t_{k})\right])\cap\left(\frac {x_{2,\mathrm{m}}(t_{k+1})-x_{2,\mathrm{m}}(t_{k})+\left[\Delta\hat{x}_{2} \right]}{x_{1,\mathrm{m}}(t_{k+1})-x_{1,\mathrm{m}}(t_{k})+\left[\Delta\hat{x} _{1}\right]}\right)\
eq\emptyset \tag{65}\]  
is satisfied at each measurement instant \(t_{k}\), where--in this equation--we assume for ease of notation that the denominator term in brackets does not include the value zero. A comparison of the point-valued measurements according to Figure 3c,d motivates the initialization of \([\Delta\phi(t_{k})]=0.6\cdot[-1\;;\;1]\) rad (lower and upper interval bounds are separated by semi-colons to distinguish them from vectors, which are also typeset by using square brackets). These graphs correspond equally to the measurement models (61) as well as to \(\texttt{atan2}(v_{2},v_{1})\) with the velocity components defined in (63), in which, all uncertainty intervals would be zero in the noise-free setting. Moreover, the worst-case GPS uncertainty is set to the independent intervals \([\Delta x_{1}]=[-2\;;\;2]\) m and \([\Delta x_{2}]=[-2\;;\;2]\) m. Using this initialization, Algorithm 1 is used to contract the error bounds so that the constraint (65) is satisfied.