## Iterative solutions using linear programming

Linear approximations to the thrust allocation problem have been discussed by Webster and Sousa (1999) and Lindfors (1993). In Linfors (1993) the azimuth thrust constraints f =V(f cos a )2 + (f sin a )2 < f — (1.46)

are represented as circles in the (f cos a, ft sin a) -plane. The nonlinear program is transformed to a linear programming (LP) problem by approximating the azimuth thrust constraints by straight lines forming a polygon. If 8 lines are used to approximate the circles (octagons), the worst case errors will be less than ± 4.0%. The criterion to be minimized is a linear combination of | f |, that is magnitude of force in the x- and y-directions, weighted against the magnitudes

representing azimuth thrust. Hence, singularities and azimuth rate limitations are not weighted in the cost function. If these are important, the QP formulation should be used.

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