## Fl F l

^ yx ^ x^y where r = [lx, ly, lz are the moment arms. For azimuth thrusters in the horizontal plane the control force F will be a function of the rotation angle. Consequently, an azimuth thruster will have two force components Fx = Fcosa and Fy = Fsina, while the main propeller aft of the vehicle only produces a longitudinal force Fx = F, see Table 1.

2.1 Thrust configuration matrix for non-rotatable actuators

The control forces and moments for the fixed thruster case (no rotatable thrusters) can be written t = Tf where T e R"xr is the thrust configuration matrix. The control forces satisfies, f = Ku, with control inputs u = [u1,..., ur ]r. The force coefficient matrix K e Rrxr is diagonal,

The actuator configuration matrix is defined in terms of a set of column vectors t; e Rn according to

If we consider 6 DOF motions, the columns vectors can be derived from (1.5) and (1.9) according to

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