## Numerical model and controller design 41 Numerical model

The three-dimensional non-linear dynamic equation of submersible motion has been described by Gertler and Hagen (1967), Feldman (1979) and Fossen (1994). Ignoring the effect of current, we can describe the dynamic equation of ISiMI as the form in the study of Feldman (1979) or Gertler and Hagen (1967). Considering the body coordinate system in Fig. 9 and classifying the terms in the dynamic equation, we get the following equation for ISiMI:

wherein "V = {u, v, w, p, q, r}T is the linear and angular velocity vector with respect to the body coordinate frame, M is the inertial term including the added mass, FCC is the coriolis and centrifugal force term for a rigid body, Fvh is the velocity-dependent hydrodynamic force acting on the body, Frest is the restoring force, Fthrust is the thrust force, and Ffn is the lift and drag force on the fins. Each element term in (3) is listed in (Jun et al., 2008). The hydrodynamic coefficients in the model were estimated using the Nernstein and Smith method and the Prestero method. The developed coefficients were non-dimensionalized with the length of the vehicle and are listed in (Jun et al., 2008). The roll damping coefficient was analogized from that of similar vehicles.

The velocity V in (3) can be written with respect to the earth-fixed coordinate frame with the transformation matrices J s as follows:

wherein Vj = {U, v, w}T and V2 = {p, q, r}T are the linear and angular velocities with respect to the body coordinate frame, respectively; = {XY,Z}T and ^2 ={$>&> y} are the position and attitude vectors with respect to the earth-fixed frame, respectively; and Jj, J 2 are the linear and angular velocity transformation matrices referred to in (Jun et al., 2008). Based on the non-linear dynamics of ISiMI, we developed a simulation environment for it using MatLab and Simulink. All the simulation results presented in this chapter were derived from the simulation environment.

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