## Mathematical models of underwater vehicles

In order to define the full mathematical model of an underwater vehicle (UV) we will use the terminology adopted from Fossen (1994). Vector of positions and angles of an underwater vehicle q = [x y z ^ 9 is defined in the Earth-fixed coordinate frame {E} and vector of linear and angular velocities v = [u v w P Q r] (surge, sway, heave, roll, pitch and yaw velocity, respectively) is defined in a body-fixed {B} coordinate frame, see Fig. 1. Vector T = [X YZ P Q /?] represent the external forces that act on the vehicle, vector Tl are commanded thrusts for each actuator and nl are commanded inputs for the actuators themselves. Here we make an assumption that the vehicle is actuated by thruster force, even though other actuator types possible and appear in practice. Using this notation, the complete mathematical model can be represented with Fig. 2. In the following sections, all parts of the model will be described. Fig. 1. Body-fixed and Earth-fixed coordinate frames (taken from Omerdic (2004))

actuators

 t uv dynamics 