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Mv (■X]) i) + -Mr, (rj) f] + Dv{v, 77)17 + g„ (77) - T„.

From this expression, one can recognize the generalized thrust force t^ and the so-called generalized matrix of the Coriolis and centripetal force given by

On the other side, the most appropriate form of the equation of motion is related to the body-fixed frame. In order for (11) to be expressed in this frame, let us consider the frame coordinate relation

where J is the rotation matrix (see Fossen, 1994) depending on the Euler angles 0 and p. The matrix J is not singular as long as the pitch angle fulfills | 0 |< n/2. So, with one obtains

With (14) in (16) and comparing this result with (11), one can identify the matrices of the new dynamics description

with M, C, D, g and Tt being the generalized matrices and vectors accounting for the inertia matrix, the Coriolis and centripetal matrix, the drag matrix, the buoyancy and the thrust force, respectively, all them with respect to the body-fixed frame. Replacing (17) of Mn in (10) of Mn and this result in (12) one accomplishes

Now from (22) in (18) one obtains the final expression for the generalized matrix of Coriolis and centripetal force with respect to the body-fixed frame

It is noticing from (23) that the supposed time variance of the dynamics at the beginning of the section, leads to the appearance of the new term ^ M in comparison with the time invariant model in Fossen, 1994.

Finally, the dynamics of the time-varying dynamics of the underwater vehicle is described by two ordinary differential equations (ODEs)

where the time dependence in the systemmatrices is explicitly declared in the notation. Moreover, it is assumed the existence of ikf1(i) uniformly in time, and Cc is the part of C in (23) that does not contain the term ^ M .

2.2 System matrices description

In the following, a detailed description of expressions for the system matrices M, C, D and g will be given from a physical point of view.

The inertia matrix M can be decomposed into the so-called body inertia matrix Mb and the added mass matrix Ma which accounts for the surrounded fluid mass. So, it is valid

The body inertia matrix is

 m(t) 