## Computed torque control method

Suppose that it is desired that the sway motion of the vehicle tracks the preplanned trajectory yd , and that the yaw motion of the vehicle tracks the preplanned trajectory yd . Let the tracking errors be defined by y = y -yd (26)

The control law is given by Eqs. (3) and (4). One can observe that Eq. (3) is obtained by replacing the acceleration term of the equations of motion, q , by the servo law u . Since this process involves the acceleration terms, we take the time derivative of Eqs. (23) and (25), and substitute (21) and (22) into the results. Therefore y = uycosy + (alluv + al2uy + b11u 2Ss + bl2u 2Sb )cosy -vy siny (28)

Next we replace y with the servo law j, and y with the servo law v, and solve these equations for the rudder deflections Ss and Sb to obtain the control law

Ss = {b22(jsecy +vy tany-uy - a11uv - a12ur )-b12(v - a21uv - a22ur )}

Sb = -{b21(jsecy +vy tany-uy - a11uv - a12ur ) - b11 (v-a21uv - a22ur )}

/(b11b22 - b21b12> 2= (31) where the servo laws j and v are given according to Eqs. (4), (5), and (6)

Note that the negative signs in the above equations, are due to the definition of tracking error in Eqs. (26) and (27) which differs in a minus sign from the definition of Eq. (2).

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