## Ii

Fig. 2. Scheme of a complete mathematical model actuators

### 2.1 Actuators

In addition to thrusters as UV actuators, rudders, fins etc. appear in practice also. Here we will limit our discussion on propulsors. According to Fossen (1994), force T that is exerted by a thruster can be described using a bilinear model, T = ¿1|n|n — b2[n\v, where and b2 are positive constants. This model has revolution rate n as input and vehicle's forward speed v as an additional variable. A simpler model which appears in literature is a model that neglects forward speed, and is given in a form T = a|n|n + bn. This model is more applicable in practice especially at low speeds. Further simplification gives that linear part of the model can also be neglected, i.e. b = 0.

However, the force exerted by thrusters is rarely the same when the propulsor is rotating in both directions. This is why a more complex model (1) should be used where sub indices f and b denote 'forward' and 'backward', and super index i stands for a specific thruster.

af\nl\nl + bfU1, nl > 0 ab \nl\nl + bbnl, nl <0

Determining the static characteristic of a thruster, i.e. the relation between the exerted thrust and the thruster control signal is called thruster mapping. The procedure consists in exciting the vehicle causing vehicle motion in such a way that the pull-force of the vehicle can be recorder by a dynamometer, as shown in Fig. 3a). An example of thruster mapping results is shown in Fig. 3b) where a VideoRay ROV (two horizontal thrusters and one vertical) is used as a case study. In Fig. 3. dots represent measured values and the full line gives the approximated curve. 