# Experimental Validation of Ellipsoidal Techniques for State Estimation in Marine Applications
## 2 Autonomous DDboats
### Tracking Control by Means of an Artificial Potential Field Method  
The aforementioned tracking control procedure for both DDboats makes use of an artificial potential field method [18; 19; 20] to minimize the deviations between the desired trajectories \(\mathbf{y}_{\text{d}}^{(i)}(t)\) and the actual robot positions denoted by the vector \(\left[x_{1}^{(i)}(t)\quad x_{2}^{(i)}(t)\right]^{T}\). For that purpose, the squared distances between the desired and actual positions are employed as energy-like functions. Computing the temporal derivatives under the assumption of a quasi-stationary boat position and performing a superposition with the desired robot velocities yields the vector-valued input signals  
\[\mathbf{w}^{(i)}(t)=\gamma\cdot\left\|\hat{\mathbf{y}}_{\text{d}}^{(i)}(t) \right\|\cdot\left(\mathbf{y}_{\text{d}}^{(i)}(t)- \begin{bmatrix}x_{1}^{(i)}(t )\\ x_{2}^{(i)}(t)\end{bmatrix} \right)+\hat{\mathbf{y}}_{\text{d}}^{(i)}(t) \tag{4}\]  
for both DDboats \(i\in\{\text{A},\text{B}\}\). Here, the heuristically tuned parameter \(\gamma>0\) is employed to specify the speed of convergence towards the desired trajectories. The computed vectors \(\mathbf{w}^{(i)}(t)\) according to (4) determine the desired orientations  
\[\phi_{\mathbf{w}}^{(i)}(t)=\arg\!\left(\mathbf{w}^{(i)}(t)\right) \tag{5}\]  
of the robots' force input vectors and the corresponding amplitudes, computed as the Euclidean vector norms  
\[\omega^{(i)}(t)=\left\|\mathbf{w}^{(i)}(t)\right\|\, \tag{6}\]  
to specify their required velocities. Using this information, a proportional-differentiating control law can be established for the propulsion systems of both robots, which consist of a pair of propellers with the rotational speeds \(u_{1}^{(i)}(t)\) and \(u_{n}^{(i)}(t)\) at the stern of each boat, where the subscripts 1 and \(\mathbf{r}\) denote the left and right actuators, respectively. In such a way, the thrusts of both actuators \(j\in\{\text{l},\mathbf{r}\}\) are given by the saturated input signals  
\[\tau_{j}^{(i)}(t)= \begin{cases}0&\text{if}\quad\tau_{j}^{(i)}\leq 0\\ \tau_{\text{max}}&\text{if}\quad\tau_{j}^{(i)}\geq\tau_{\text{max}}\\ \tau_{j}^{(i)}(t)&\text{else}\end{cases}  \tag{7}\]  
Figure 2: Reference trajectories of both DDboats and video snapshot of the experimental data acquisition. with  
\[\tilde{\tau}_{j}^{(i)}(t)=\left(K_{0}-K_{\mathrm{P},j}\cdot\left(\phi_{\mathbf{w}}^ {(i)}(t)-\phi^{(i)}(t)\right)+K_{\mathrm{D},j}\cdot\phi^{(i)}(t)\right)\cdot \left(\omega^{(i)}(t)\right)^{2},\;\;j\in\left\{\mathrm{l},\mathrm{r}\right\}\;, \tag{8}\]  
where \(\phi^{(i)}(t)\) and \(\phi^{(i)}(t)\) denote the current headings and angular velocities in a body-fixed coordinate frame of each DDboat. The experimentally tuned gains in (8) satisfy the relations \(K_{0}>0\), \(K_{\mathrm{P},\mathrm{l}}=-K_{\mathrm{P},\mathrm{r}}\) and \(K_{\mathrm{D},\mathrm{l}}=-K_{\mathrm{D},\mathrm{r}}\), respectively. Finally, the rotational speeds of the actuators are computed as  
\[u_{1}^{(i)}(t)=\sqrt{\tau_{1}^{(i)}(t)}\quad\text{and}\quad u_{\mathrm{r}}^{( i)}(t)=\sqrt{\tau_{\mathrm{r}}^{(i)}(t)}\;\;. \tag{9}\]  
Despite the fact that this simple control approach does not account for specific measures aiming at a reduction of the influence of external disturbance forces (such as robustifying the controller by means of sliding mode techniques with variable-structure gains chosen so that the worst-case effect of all possible matched uncertainty can be compensated for, or by enhancing the dynamics by means of an observer-based disturbance reduction), Figure 3 shows a quite accurate tracking of the desired trajectories by each of the DDboats. In addition to the position information, GPS- and compass-based heading measurements are also compared with their corresponding desired trajectories in Figure 3. Note that the aim of this control design is not the minimization of tracking errors but rather the implementation of a simple technique to provide experimental data as the input for the following estimation algorithm.