## 5 Simulations and Comparison
### Vehicle Models and Test Conditions
#### 5.1.3 Comparison with Another Controller  
The trajectory tracking algorithm proposed in [42] was chosen for comparative simulation studies. It is suitable for marine vehicles in horizontal motion, for which the inertia matrix in the mathematical model is symmetric. The reason for choosing this algorithm is that it, too, is based on a transformation of variables, except that instead of a velocity transformation (as in the proposed control scheme), it uses a transformation of coordinates describing the vehicle's position. Moreover, this control scheme has the advantage of having been tested in both simulation studies and a sea trial. In both tests it proved to be effective. In this regard, its effectiveness was compared with that of the proposed controller by simulation. This is not equivalent to an experiment, of course, but it indirectly provides information on the importance of the presented control approach. Simulation tests were conducted according to the method described in detail in [42]. In this work, the tests performed for this controller are designated CL (classical controller). Verification of the selected algorithm could be performed only for constant velocity disturbances as these are the requirements of its applicability. The vehicles and trajectories used for tracking were as in the tests of the previously proposed control algorithm. The transformed equations of motion described in [42] are as follows (\(z_{1}=\psi,z_{2}=r\)):  
\[\dot{z}_{1}=z_{2},\] (71) \[\dot{z}_{2}=F_{z_{2}}(z_{1},\xi_{3},\xi_{4})+\tau_{r},\] (72) \[\left[ \begin{array}{c}\xi_{1}\\ \xi_{2}\end{array} \right]=\left[ \begin{array}{c}\xi_{1}\\ \xi_{2}\end{array} \right]+\left[ \begin{array}{c}V_{x}\\ V_{y}\end{array} \right],\] (73) \[\left[ \begin{array}{c}\xi_{3}\\ \xi_{4}\end{array} \right]=\left[\begin{array}{c}F_{\xi_{3}}(z_{1},\xi_{3}, \xi_{4})\\ F_{\xi_{4}}(z_{1},\xi_{3},\xi_{4})\  
Next, the simulation test of the controller for Kambara, using the same velocity disturbances and for both linear and cycloid trajectory, was performed. The following set of parameters was applied:  
\[k_{vx}=20,\ \ k_{xy}=20,\ \ k_{px}=2.5,\ \ k_{py}=2.5,\ \ k_{I_{x}}=0.1,\ \ k_{I_{y}}=0.1, \tag{79}\]  
and \(l=1.2\) (to avoid oscillations). As it shown in Figure 11a, the desired linear trajectory is also not tracked exactly, which is confirmed in Figure 11b here it is seen that the position errors do not tend to zero. On the contrary, all variables \(\Delta_{1}^{x},\Delta\xi_{2},\Delta\xi_{3},\Delta\xi_{4}\) go to zero in the considered time as is presented in Figure 11c,d. Moreover, velocity \(v\) given in Figure 11e exceeds the allowed value assumed for the test, whereas the force and torque have small values (Figure 11f). Figure 11: Simulation results for Kambara, CL controller and linear trajectory: (**a**) desired and realized trajectory; (**b**) position errors; (**c**) position error states; (**d**) velocity error states; (**e**) velocities; (**f**) applied force and torque. Figure 10: Simulation results for SIRENE, CL controller and linear trajectory: (**a**) desired and realized trajectory; (**b**) position errors; (**c**) position error states; (**d**) velocity error states; (**e**) velocities; (**f**) applied force and torque. For the cycloid trajectory, the results are shown in Figure 12. From Figure 12a,b, it can be observed that the trajectory position tracking errors are significant and thus the tracking task is not performed satisfactorily. Meanwhile, the time history of the variables \(\Delta\xi_{1},\Delta\xi_{2},\Delta\xi_{3}^{*},\Delta\xi_{4}\), as is shown in Figure 12c,d, indicates that they converge to zero. The velocity \(r\) (Figure 12e) has big value (although this is an acceptable value). In practice, however, this means that the vehicle will perform an oscillating motion about a vertical axis. The applied force and torque have very small values (Figure 12f). The performance of the algorithm is collected in Table 4. For SIRENE values of indexes are grater than for the proposed control algorithm (with exception of MIAC).