## 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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This first volume will guide you through the basics of Photoshop. Well start at the beginning and slowly be working our way through to the more advanced stuff but dont worry its all aimed at the total newbie.

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