## Nonlinear model representation of the dynamic equations of motion

The motion of the UUV in space needs to be defined with respect to some certain coordinate frames. One of the coordinate frames can be chosen to be fixed to the vehicle and is called the body-fixed reference frame. The advantage of defining the motion of the UUV in terms of the linear and angular motion components about the orthogonal body axes leads to define the body-fixed frame, xyz. The body-fixed frame is chosen so as to coincide with the center of buoyancy (CB), which is the volumetric center of the fluid displaced. It implies that the CB vector is zero vector, r= [xb,yb,Zb]T=[0,0,0]T. Another orthogonal coordinate system is defined to describe the motion of the moving body-fixed frame relative to an inertial frame.

The earth-fixed reference frame, xoyozo, is assumed as fixed in Earth and accepted as inertial. These right-handed frames are shown in Fig.1.

The rotation of a rigid-body can be represented in many ways. The well-known and mostly used representation is Euler angles. This representation is practical, popular and has intuitive physical meaning. However, Euler angle parameterization causes some singularities (Ang & Tourassis, 1987) and inaccuracies in calculations, e.g. discontinuous changes may occur in the attitude when the rotation is changed incrementally. A singularity-free and well-suited quaternion parameterization is preferred for accurate calculations.

The transformation between the body-fixed frame and the earth-fixed frame is given by (Fossen, 1994):

where E is the transformation matrix and T]E=[x,y,z/Ei, e2l £3/]]T. Here, e=[si, e2l s3/tf] is a unit quaternion vector. The unit quaternion vector represents the rotation with respect to an axis of rotation, s=[si, s2, £3, ]]] T, and a rotation of angle, ], about that axis. The linear and angular velocities of the vehicle are described as v = [U/V/W/P/q/r]T.

The dynamic equations of motion for the UUV can be written as follows (Fossen, 1994):

where V is the time derivative of the velocity vector, and t is propulsion forces and moments vector. The simplified inertia matrix, M, is written as

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