Table 1. Example of actuators and control variables.

Table 1. Example of actuators and control variables.

For underwater vehicles the most common actuators are:

• Main propellers/longitudinal thrusters are mounted aft of the hull usually in conjunction with rudders. They produce the necessary force in the x-direction needed for transit.

• Transverse thrusters are sometime going through the hull of the vessel (tunnel thrusters). The propeller unit is then mounted inside a transverse tube and it produces a force in the y -direction. Tunnel thrusters are only effective at low speed which limits their use to low-speed maneuvering and DP.

• Rotatable (azimuth) thrusters in the horizontal and vertical planes are thruster units that can be rotated an angle a about the z-axis or y-axis to produce two force components in the horizontal or vertical planes, respectively. Azimuth thrusters are attractive in low-speed maneuvering and DP systems since they can produce forces in different directions leading to an over-actuated control problem that can be optimized with respect to power and possible failure situations.

• Aft rudders are the primary steering device for conventional vessels. They are located aft of the vessel and the rudder force Fy will be a function of the rudder deflection (the drag force in the x-direction is usually neglected in the control analysis). A rudder force in the y-direction will produce a yaw moment which can be used for steering control.

• Stabilizing fins are used for damping of vertical vibrations and roll motions. They produce a force Fz in the z-directions which is a function of the fin deflection. For small angles this relationship is linear. Fin stabilizers can be retractable allowing for selective use in bad weather. The lift forces are small at low speed so the most effective operating condition is in transit.

• Control surfaces can be mounted at different locations to produce lift and drag forces. For underwater vehicles these could be fins for diving, rolling, and pitching, rudders for steering, etc.

Table 1 implies that the forces and moments in 6 DOF due to the force vector f = [Fx, Fy, Fz can be written f r x f

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