# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 5 Estimation Results
### Localization without Heading Measurement  
Figure 7 gives a comparison of the localization results after the ellipsoidal state prediction step, as well as after the subsequent set-based innovation step. It can be seen that  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **Constant Bounds \(\left\langle\mathbf{A}\right\rangle\)** & **Time-Varying \(\left\langle\mathbf{A}\right\rangle\)** & **Constant Bounds \(\left\langle\mathbf{B}\right\rangle\)** & **Time-Varying \(\left\langle\mathbf{B}\right\rangle\)** \\ \hline \(\mu_{\phi,\text{GPS}}\) in rad & 0.6077 & 0.2883 & 0.6054 & 0.3638 \\ \(\mu_{\phi,\text{GPS}}\) in rad & 0.2000 & 0.1872 & 0.1593 & 0.1564 \\ \(\sup\left(\left[\Delta x_{1}\right]^{\left|\phi\right\rangle}\right)\) in m & 0.0474 & 0.0350 & 0.0500 & 0.0456 \\ \(\sup\left(\left[\Delta x_{2}\right]^{\left|\phi\right\rangle}\right)\) in m & **0.1447** & **0.1445** & **0.0955** & **0.0977** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Measurement uncertainty (mean of the interval radius of the compass-based heading measurement \(\phi_{\text{m,c}}\) and the GPS-based heading reconstruction \(\phi_{\text{m,GPS}}\), as well as interval boxes \(\left[\Delta x_{1}\right]^{\left|\phi\right\rangle}\), \(\left[\Delta x_{2}\right]^{\left|\phi\right\rangle}\).  
Figure 6: Combination of the information obtained by the compass- and GPS-based heading intervals for the DDboat A. (**a**) Intersection of compass- and GPS-based measurement intervals for DDboat A (constant error bounds). (**b**) Intersection of compass- and GPS-based measurement intervals for DDboat A (time-varying error bounds). (**c**) Interval diameters corresponding to Figure 6a. (**d**) Interval diameters corresponding to Figure 6b.  
the additional use of velocity information, represented by the interval data described in Section 4.2, only slightly improves the estimation accuracy in comparison with a pure position measurement.  
It has to be pointed out that the heading angle included as the fourth state variable in the model (55) is unobservable if only the coordinates \(x_{1}\) and \(x_{2}\) are measured by the available GPS. For that reason, a countermeasure against an unbounded growth of the uncertainty in the heading angle information needs to be included as a _virtual_ measurement in the ellipsoidal innovation stage. This restriction has been set to the interval \([-\pi\;;\;\pi]\) rad in both subgraphs of Figure 7, while the velocity was further restricted in Figure 7a to an interval \([-2\;;2]\frac{\mathrm{m}}{\mathrm{s}}\), centered around the midpoint of the current state prediction.  
**Remark 5**.: _In future work, further quasi-linear state-space representations (also ensuring observability of the heading for non-vanishing boat velocities based on position measurements) can be included in the estimation procedure. These reformulated system models may also serve as virtual measurement generators._  
_The corresponding analytical reformulations of the system model (55) split up the term \(\sin(\phi(t_{k}))\cdot v(t_{k})\) into the form_  
\[\sin(\phi(t_{k}))\cdot v(t_{k})=\begin{bmatrix}0&0&\alpha\cdot\sin(\phi(t_{k} ))&(1-\alpha)\cdot\frac{\sin(\phi(t_{k}))}{\phi(t_{k})}\cdot v(t_{k})\end{bmatrix} \cdot\mathbf{x}(t_{k}) \tag{67}\]  
_with \(\alpha\in\mathbb{R}\) as an optimization degree of freedom. The optimal choice of this parameter \(\alpha\) is currently being investigated in a parallel research activity aiming at the use of an ellipsoidal calculus for analyzing the stability of discrete-time dynamic systems, as well as for extensions of the control procedure presented in [31]._