**2. UUV dynamics**

Let *η*= [*x*, *y*, *z*, *ϕ*, *θ*, *ψ*] *T* be the generalized position vector of the UUV referred to an earth-fixed coordinate system termed *O*′ , with displacements *x*, *y*, *z*, and rotation angles *ϕ*, *θ*, *ψ* about these directions, respectively. The motions associated to the elements of *η* are referred to as surge, sway, heave, roll, pitch and yaw, respectively.

Additionally let *v*= [*u*, *v*, *w*, *p*, *q*,*r*] *T* be the generalized rate vector referred on a vehicle-fixed coordinate system termed *O*, oriented according to its main axes with translation rates *u*, *v*, *w* and angular rates *p*, *q*,*r* about these directions, respectively.

The vehicle dynamics with a time delay in the communication system, is described by the ODE (*cf*. Jordán and Bustamante, 2009a; *cf*. Fossen, 1994)

$$\dot{\boldsymbol{\sigma}} = \boldsymbol{M}^{-1} \left( -C[\boldsymbol{\sigma}] \boldsymbol{\sigma} - D[|\boldsymbol{\sigma}|] \boldsymbol{\sigma} - \mathbf{g}[\eta] + \boldsymbol{\tau}\_{\complement} + \boldsymbol{\tau}(t - T\_d) \right) \tag{1}$$

$$
\dot{\eta} = f[\eta](\upsilon + \upsilon\_c). \tag{2}
$$

Here *M*, *C* and *D* are the inertia, the Coriolis-centripetal and the drag matrices, respectively and *J* is the matrix expressing the transformation from the inertial frame to the vehicle-fixed frame. Moreover, *g* is the restoration force due to buoyancy and weight, *τ* is the generalized propulsion force whose action is delayed *Td* seconds, *τ<sup>c</sup>* is a generalized perturbation force (for instance due to cable tugs in ROV's) and *vc* is a velocity perturbation (for instance the fluid current in ROV's/AUV's), all of them applied to *O*.

From now on, brackets are employed to indicate functional dependence and parenthesis to denote common factor. Besides vectors are indicated in bold, variables in italics and matrices in capital letters.

Notice from (1) the nonlinear dependence of *C*, *D* and *g* with the states *v* and *η.*

Moreover, we will concentrate henceforth on disturbed measures *η<sup>δ</sup>* and *vδ*, and not on exogenous perturbations *τ<sup>c</sup>* and *vc*, so we have set *τc*=*vc*=0 throughout the Chapter. For more explanations about the influence of *τc* and *vc* on adaptive guidance systems see (Jordán and Bustamante, 2008; Jordán and Bustamante 2007), respectively.
