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This paper deals with a nonlinear model predictive control (NMPC) scheme for a winch servo motor to overcome the sudden peak tension in the lifting wire caused by a lumped-mass payload at the beginning of a lifting off or a lowering operation. The crane-wire-payload system is modeled in 3 degrees of freedom with the Newton-Euler approach. Direct multiple shooting and real-time iteration (RTI) scheme are employed to provide feedback control input to the winch servo. Simulations are implemented with MATLAB and CaSADi toolkit. By well tuning the weighting matrices, the NMPC controller can reduce the snatch loads in the lifting wire and the winch loads simultaneously. A comparative study with a PID controller is conducted to verify its performance. INTRODUCTION Due to the oil crisis in 1973-74 and anti-nuclear sentiment, wind farm industry was motivated and started to grow in the late 1970s. At the beginning stage, investment subsidy, tax-free policy, CO2 bonus, and long-term policy goals stimulated the industry, while it is meanwhile blocked by local protests, siting difficulties, and the inner resistance in the power company [1]. In recent years, there is a significant development of offshore wind farms with an increasing rated power per turbine. Large amounts of newly built offshore wind farms, e.g. in United Kingdom, Denmark, and Germany, have sparked an increasing requirement for safe and cost-and-time efficient installation schemes. Lifting is one of the most common operations during wind turbine installation. Due to the great mass of the payload, sudden wire rope tension at the initial stage of the lifting off and the lowering operation is crucial to the marine installation safety [2]. As largely depending on pragmatic experiences, marine operations ordinarily follow the criteria from guidance documents and confidential data from decades of experience. For example, alpha factor, which is a correction factor on the operational limit in terms of significant wave height, is used to modify the operational criteria and contributes to the decision making process, e.g., when to start a specific operation based on the weather forecasts. To reduce the environmental influence and to provide a stable platform for lifting crane operations, a jackup is the most widely applied scheme. Such methods can merely reduce the risk; however, they are insufficient to satisfy strict deadlines and limitations in operational time windows. The vessel may wait for quite a long period until a calm and long enough weather window for positioning of the jackup rigs. Such operations can only be conducted during 1 Copyright c © 2017 by ASME the summer time in the North Sea. Cranes are normally modeled in the framework of Lagrange mechanism by disregarding the axial wire rope elongation. Large amounts of crane models have been developed, for example [3–7]. Hence, a series of basic properties of Lagrange and Hamiltonian system can be applied during the controller design phase [8]. However, the model complexity is unmanageable for those with high degrees of freedom (DOF). In such situation, Newton-Euler modeling method is more straightforward and simpler in the form. For an overhead crane or a ship-mounted onshore situations, the payload is lumped-mass or distributedmass, e.g., [9–11]. To achieve an overall goal of heave compensation through the wave zone during moonpool operations, a number of researches have modeled such process in 1DOF and treat the servo motor as the control input, e.g., [12–15]. However, the model is limited in 1DOF. The main drawback of NewtonEuler model is the difficulty of the model nonlinearities. Model predictive control (MPC) is widely applied in industry to large multi-variable constrained control problem by solving an online constrained optimization problem repeatedly. Nonlinear model predictive control (NMPC) is an optimal feedback control approach which provides real-time feedback to a nonlinear system through real-time optimizations. An NMPC controller solves a nonlinear dynamic optimization problem at each sampling time. The direct approach performs a discretization for a continuous system on a time grid and solves the programming. It can be categorized into the sequential approach and the simultaneous approach. Direct single shooting method is one of the sequential approaches, and it is unsuitable for longer horizon. Compared to single shooting, the main disadvantages of the simultaneous approach lay on the fact of unnecessary strong nonlinearity of the optimization problem and poor convergence behavior to the desired reference trajectory which may be caused by poor initial guess, especially for unstable system [16]. Due to the limited computational capacity of hardware, online and realtime NMPC is demanding. Real-time iteration (RTI) scheme is an approximation technique for optimal feedback control [17]. The most obvious benefit of RTI scheme is that it performs only one Newton-type iteration per NMPC sample [18]. Automatic code generation is a popular topic due to the requirement for fast solver by the embedded platforms. There exists a group of popular toolboxs with user-friendly interfaces, e.g., CaSADi, ACODO, and CVXGen [19–21]. A number of quadratic programming (QP) solvers which can automatically generate C++ code for the embedded systems have been published, such as FORCES, CVXGen, qpOASES, and FiOrdOs [22, 23]. This paper proposes an NMPC control scheme for a winch servo to overcome the sudden tension caused by a lumped-mass payload at the beginning stage of a lifting off or a lowering operation. The paper is structured as follows. In Section 2, the control plant model is proposed. Basic concepts and theories FIGURE 1. FREEBODY DIAGRAM. about direct multiple shooting approach is introduced in Section 3. Formulation of the optimization problem is shown in Section 4. Simulation studies and discussions are presented in the Section 5. Finally, a conclusion summarizes the paper. Notation: | · | and ‖x‖Q, respectively, stand for the Euclidean norm and the weighted Euclidean norm, i.e., |x|2 = x>x and ‖x‖A = x>Ax. The overlines and underlines, · and ·, are the lower and upper limits for the variable (·). Normal lowercase letters represent scalars. Vectors and matrices are, respectively, expressed with bold lowercase and capital letters . SYSTEM MODELING A jackup vessel is considered in this paper, and it is assumed to be rigidly fixed on the seafloor. A ship-mounted boom crane on it is responsible for lifting a payload through a wire rope; see Fig. 1. In this paper, we applied a reduced model where the servo motor is the only control input. Hereafter, we consider an infinitely stiff crane, whose position is not influenced by the payload nor the motor. The mass of the payload is mt , which is significantly greater than the mass of the rope. Furthermore, we assume a light-weight wire rope, that is, its mass is disregarded. The rope connects to the crane through a pulley at the crane tip. We assume the position of the wire tangency point coincides with the center of the pulley, and its relative position to the boom is fixed. Therefore, the rope and the hook are modeled as a tensile spring. The length of the unstretched wire rope between the pulley and the payload is l̄1. The mass of the hook is combined into the payload, which does not interact with the water surface. Only the wind load is included. Inspired by the finite element method model, a lifting model with elastic wire rope and controllable winch is deduced in the perspective of the Newton-Euler method. The system is modeled 2 Copyright c © 2017 by ASME in the 3DOF (NED) frame, and it has a relatively simple form, when comparing with the Euler-Lagrange approach. The positive directions of the coordinate point are to the north, the east, and downward, respectively. Now, define the following vectors: the position of the payload: rt = [xt ,yt ,zt ]>, the position of the pulley: r p = [xp,yp,zp], the velocity of the payload: ut = [ẋt , ẏt , żt ]>, the velocity of the pulley: up = [ẋp, ẏp, żp], where the subscripts t and p represent the payload and the pulley, respectively. In this paper, up = [0,0,0]> as we assume both the jackup and the crane are rigidly fixed. We define two additional vectors to simplify the following expressions. Position vector from the pulley to payload and its time derivative are defined as ∆1 := rt − r p, (1a)

Modeling and Control of Crane Overload Protection During Marine Lifting Operation Based on Model Predictive Control