M1x u1y u2x u2y u3

Notice that Te is constant while T(a) depends on a. This means that the extended control input vector ue can be solved directly from (1.21) by using a pseudo-inverse. This is not the case for (1.20) which represents a nonlinear optimization problem. The azimuth controls can then be derived from the extended control vector ueby mapping the pairs (u1x, u1y) and (u2x, u2 y) using the relations,

u1 ulx + u1y , a1 = atan2(u1y,uu), u2 uL + u2y , a2 = atan2(u2y,u2x).

The last two controls u3 and u4 are elements in ue. □

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