N

0 0 Nr+Nrlrllr

The reason why the coupled model is limited to the horizontal plane is the simplicity -underwater vehicles can be trimmed in such a way that the heave motion is not affected by the motions in other DOFs. It can be seen from equation (4) that in general case, yaw motion depends on v + ur, u — vr, uv and N; surge motion depends on r, u, r2, rv and X; and sway motion depends on r, u, r2 and ur. It is not possible to make general conclusions on which of the parameters can be further neglected - identification has to be performed for each vehicle separately in order to determine the dominant coefficients.

An example of deciding on the model parameters for the coupled model is shown on results for the VideoRay ROV. The correlation between all possible coupling parameters for the three DOFs has been determined and is shown in Fig. 4. The data were obtained using a vision-based method described in Section 2. The terms with the greatest percentage are the dominant ones.

Fig. 4. Correlation coefficients for a) yaw, b) surge and c) sway motion, see Ljung (1999)

These figures let us conclude that parameter X„ from the added mass matrix is negligible (i.e. the added mass matrix is diagonal) and that the centre of buoyancy is practically equivalent to the centre of gravity (xG = yG = 0) in the horizontal plane. Both of these statements are true for micro-ROVs, Miskovic et al. (2007a). Details on the identified coupled model can be found in Section 4. Uncoupled model

Additional simplifications can be introduced if the vehicle is trimmed in such a way that roll and pitch are negligible while the vehicle is moving in other controllable degrees of freedom. In order to neglect the coupling due to the Coriolis' forces, it can be assumed that the vehicle is moving at low speed. Using these two assumptions, the coupling can be completely omitted leaving the equation (5) for surge (where q(t) is surge speed u) and yaw (q(t) is yaw rate r) degree of freedom and equation (6) for heave degree of freedom (where q(t) is heave speed w).

In both models, parameter r(t) is the excitation force (e.g. surge force X, yaw moment N). Parameter S can either be external disturbance (in the case of yaw model) or a vehicle physical parameter such as difference between weight and buoyancy (in the case of heave model). This model gives an uncoupled dynamic model of underwater vehicles. While describing marine vehicle dynamics, usually two models are used based on the drag: the linear one, which has a constant drag coefficient , and the nonlinear one, whose drag coefficient is linear pNlq(t)l, Fossen (1994), Caccia et al. (2000), Ridao et al (2004). Linear model is usually used at low speeds, where higher order drag terms can be neglected. This is usually the case when the vehicle is being dynamically positioned. The nonlinear model is suitable for applications where the vehicle is moving at higher speed, i.e. in the cases when the vehicle is in motion.

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