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.

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