Recursive nullspace-based control allocation with strict prioritization for marine craft ⋆ Roger Skjetne ∗ Øivind K. Kjerstad ∗ ∗ Department of Marine Technology, Norwegian University of Science and Technology, Trondheim, NO-7491 Norway (e-mails: Roger.Skjetne@ntnu.no & oivind.k.kjerstad@ntnu.no) Abstract: In control allocation it is often important to prioritize among the directions to produce control effort, especially in cases of limited capacity. Motivated by a requirement for strict prioritization among the control directions, a recursive nullspace-based allocation design based on direct use of the generalized pseudoinverses and nullspace matrices is proposed. The allocation design divides the overall problem into r subproblems according to a prioritization sequence, and a recursive method is proposed to solve the allocation subproblem in each step. By hiding the allocated controls from subsequent steps in the nullspace corresponding to the present step, the inﬂuence of lower priority control actions onto higher priority directions are nulliﬁed to achieve the speciﬁed prioritization. The method is veriﬁed by analyzing the thruster capacity of an Arctic intervention vessel based on experimental ice towing data. Keywords: Control allocation; thrust allocation; motion control; dynamic positioning; Arctic operations. 1. INTRODUCTION For dynamic positioning (DP) of offshore vessels, the limiting capacity of stationkeeping in heavy environmental conditions is the thruster conﬁguration and the maximum resultant forces and moment that can be produced for surge, sway, and yaw motions (Sørensen, 2005). The DP control law calculates a commanded net force/moment to be produced by the propul- sion system to compensate the environmental loads. A thrust allocation algorithms distributes this net force/moment in an efﬁcient manner to a commanded force and direction for each individual thruster based on their rated power, locations in the hull, thruster types, and constraints. The yaw moment is typi- cally prioritized before surge and sway for operational safety. h i = 0.3 m h i =0.5 m h i =0.8 m Ice thickness: Fig. 1. DP-Ice Capability Plot of the CIVArctic vessel (left) with magniﬁcation (right), parametrized by ice drift speed entering at the respective angles. The 3 plots indicate 3 different ice thicknesses. (Courtesy: Su et al., 2013) In Arctic DP operations of offshore vessels in ice (NTNU, 2010-2014), the ice loads on the vessel are generally larger and more aggressive than open water environmental forces. More- over, vessels that operate in ice have typically been designed ⋆ Funded by the Research Council of Norway project 199567 “Arctic DP” and the consortium partners Kongsberg Maritime, Statoil, and Det Norske Veritas. with an icebreaking bow or stern to minimize the loads and make stationkeeping feasible. This is clearly seen from the DP- Ice Capability Plot in Figure 1, presented by Su et al. (2013) for the customized design of an Arctic intervention vessel to operate partly in ice (Berg et al., 2011). This shows that if the ice enters outside the narrow longitudinal sectors from ahead or astern, then the icebreaking ability – and thus the station- keeping capability – is signiﬁcantly deteriorated. Consequently, prioritizing the yaw moment to keep the vessel heading (bow or stern) pointed against the ice drift becomes an even stricter requirement for DP in ice compared to open water (Moran et al., 2006). A good heading control will as a consequence minimize the sway loads, which therefore implies that the vessel must secondly prioritize surge forces – to push the vessel towards the incoming ice – before lastly the sway forces are given priority. In general, having a commanded vector τ ∈ R m of net or resultant control effort related to the individual control effectors u ∈ R n through the state (x)- and time (t)-dependent map u → h(u, x, t)= τ , then control allocation, being the generalized notion of thrust allocation for marine vessels, is to calculate the individual distribution of control efforts in u. Having typically an overactuated setup, with n>m, there will be inﬁnitely many solutions, and the objective is to ﬁnd the “most optimal” u ∈ U where U is a constraint set for the control effectors. Control allocation is a mature research ﬁeld in aerospace appli- cations (Oppenheimer et al., 2010) and for marine applications (Sørensen, 2005; Fossen, 2011), and is emerging also in other ﬁelds such as in process control and electrical power production systems. Solutions range from simple unconstrained pseudoin- verse methods (Horn and Johnson, 1985; Fossen, 2011; Virnig and Bodden, 1994) to advanced constrained optimization-based methods such as linear programming, quadratic programming, nonlinear programming, and mixed-integer programming. The survey article by Johansen and Fossen (2013) gives an overview of such methods and their references.