## Extended thrust configuration matrix for rotatable actuators

When solving the control allocation optimization problem an alternative representation to (1.10) is attractive to use. Equation (1.11) is nonlinear in the controls a and u. This implies that a nonlinear optimization problem must be solved. In order to avoid this, the rotatable thrusters can be treated as two forces.

Consider a rotatable thruster in the horizontal plane (the same methodology can be used for thrusters that can be rotated in the vertical plane),

Next, we define an extended force vector according to fe = K eue (1 16)

where Te and Ke are the extended thrust configuration and thrust coefficient matrices, respectively and ue is a vector of extended control inputs where the azimuth controls are modelled as uix = u cos «i

The following examples show how this model can be established for an underwater vehicle equipped with two main propellers and two azimuth thrusters in the horizontal plane. Example 1: Thrust configuration matrices for an ROV/AUV with rotatable thrusters

The horizontal plane forces X and Y in surge and sway, respectively and the yaw moment N satisfy (see Figure 2), t = T(« )Ku

 " X " 