## Explicit solution for varying a using piecewise linear functions rotatable thrusters and rudders

An extension of the mp-QP algorithm to marine vessels equipped with azimuthing thrusters and rudders has been given by Johansen et al. (2003). A propeller with a rudder can produce a thrust vector within a range of directions and magnitudes in the horizontal plane for low-speed maneuvering and dynamic positioning. The set of attainable thrust vectors is non-convex because significant lift can be produced by the rudder only with forward thrust. The attainable thrust region can, however, be decomposed into a finite union of convex polyhedral sets. A similar decomposition can be made for azimuthing thrusters including forbidden sectors. Hence, this can be formulated as a mixed-integer-like convex quadratic programming problem and by using arbitrarily number of rudders as well as thrusters and other propulsion devices can be handled. Actuator rate and position constraints are also taken into account. Using a multi-parametric quadratic programming software, an explicit piecewise linear representation of the least-squares optimal control allocation law can be pre-computed. The method is illustrated using a scale model of a supply vessel in a test basin, see Johansen et al. (2003) for details, and using a scale model of a floating platform in a test basin, see Spj0tvold (2008).

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