## Adaptive computed torque control method

In this method, the control law estimates the unknown parameters. As stated before, all the parameters ajJ and btJ comprise hydrodynamic uncertainties which must be estimated. On the other hand, the vehicle's forward velocity u is assumed to be constant, but subjected to changes from environment, and ocean currents. Thus all terms including u must also be estimated. But instead of estimating all the atJ and btJ terms, we define parameter functions, pi , in a linear parameterization process. This process does not reveal a unique parameterization and the results depend on the way one defines pi s. One can show that a possible parameterization of Eqs. (30) and (31) is given by

cos\

(b11b22 - b21b12)u 2 b12a22u - b22a12u - b22u

(b11b22 - b21b12)u 2 b21

(b11b22 - b21b12)u 2 b21a21u - b11a22u + b21u

(b11b22 - b21b12)u 2

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