Cooperative heaving-rolling compensation method and system for double-ship lifting arm system

Abstract

The invention provides a cooperative heave-rolling compensation method and system for a double-ship lifting arm system, and the method comprises the steps: building a double-ship lifting arm system dynamic model considering heave-rolling coupling motion, and enabling the model to comprise disturbance terms caused by ship motion; based on the kinetic model, generating a nominal trajectory under input saturation constraint and state constraint, and generating an expected trajectory through prediction control of a tightening constraint nonlinear model; for an actual disturbed system, designing a nonlinear robust model prediction controller, and driving an actual state to track the expected trajectory; and the stability of a closed-loop system is guaranteed through a terminal cost function and a terminal constraint set, and cooperative heaving-rolling compensation of the double-ship lifting arm is achieved. According to the method, through dynamic modeling, nominal track generation, robust control and stability guarantee, the double-ship lifting arm cooperative compensation heaving-rolling coupling motion is achieved, multi-degree-of-freedom oscillation can be effectively restrained under the five-level sea condition, and the operation stability and precision under the complex sea condition are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of shipborne crane motion control, and in particular to a coordinated heave-roll compensation method and system for a dual-ship lifting arm system. Background Art

[0002] During marine engineering operations, vessels equipped with dynamic positioning systems experience complex coupled heave and roll motions in sea conditions exceeding Level 3. However, existing twin-vessel lift arm systems have significant shortcomings in coping with this coupled motion.

[0003] On the one hand, the traditional control method is not perfect in the dynamic modeling of the dual-ship lifting arm system under the coupled motion of the hull heave and roll, and fails to fully consider the coupling relationship between the various parts of the system and the influence of external environmental disturbances, resulting in the model being unable to accurately describe the dynamic characteristics of the system under complex sea conditions.

[0004] On the other hand, the existing wave compensation control strategy has limited capabilities in dealing with input saturation constraints and state constraints, making it difficult to achieve precise compensation control while ensuring system safety. Especially under harsh conditions such as level 5 sea conditions, it is unable to effectively suppress the multi-degree-of-freedom oscillations caused by the heave-roll coupled motion, and is prone to problems such as decreased control accuracy and insufficient system stability.

[0005] In addition, traditional control algorithms lack robust design for nonlinear systems. When faced with interference from random environmental factors such as wind, waves, and currents, it is difficult to maintain good control performance, resulting in unstable load posture and seriously affecting the safety and efficiency of marine engineering operations.

[0006] These problems greatly limit the application of the existing dual-ship lifting arm system in complex sea conditions, and there is an urgent need to study more advanced control methods and systems to solve these problems. Summary of the Invention

[0007] In order to solve the above problems, the present invention proposes a method and system for coordinated heave-roll compensation of a dual-ship lifting arm system. Through dynamic modeling, nominal trajectory generation, robust control and stability assurance, the coordinated compensation of heave-roll coupled motion of the dual-ship lifting arms is realized, which can effectively suppress multi-degree-of-freedom oscillations under level 5 sea conditions and improve the operation stability and accuracy under complex sea conditions.

[0008] In order to achieve the above object, the present invention adopts the following technical solutions: In a first aspect, the present invention provides a method for coordinated heave-roll compensation of a dual-vessel lifting arm system, comprising: A dynamic model of a twin-ship lifting arm system considering heave-roll coupled motion is established, wherein the model includes disturbance terms caused by ship motion. Based on the dynamic model, a nominal trajectory under input saturation constraints and state constraints is generated, and a desired trajectory is generated through a tight-constrained nonlinear model predictive control; For the actual disturbed system, a nonlinear robust model predictive controller is designed to drive the actual state to track the desired trajectory; The stability of the closed-loop system is ensured by the terminal cost function and the terminal constraint set, and the coordinated heave-roll compensation of the twin-ship lifting arms is achieved.

[0009] In a second aspect, the present invention provides a coordinated heave-roll compensation system for a dual-vessel lifting arm system, comprising: A dynamic modeling module is configured to establish a dynamic model of a twin-ship lifting arm system considering heave-roll coupled motion, wherein the model includes a disturbance term caused by hull motion; a nominal trajectory generation module configured to generate a nominal trajectory under input saturation constraints and state constraints based on the dynamic model, and generate a desired trajectory through a tight-constrained nonlinear model predictive control; The robust control module is configured to design a nonlinear robust model predictive controller for the actual disturbed system to drive the actual state to track the desired trajectory; The stability assurance module is configured to ensure the stability of the closed-loop system through a terminal cost function and a terminal constraint set, thereby achieving coordinated heave-roll compensation of the twin-ship lifting arms.

[0010] In a third aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method for coordinated heave-roll compensation of a dual-ship lifting arm system described in the first aspect.

[0011] In a fourth aspect, the present invention provides a computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the steps of the method for coordinated heave-roll compensation of a dual-ship lifting arm system described in the first aspect are implemented.

[0012] Compared with the prior art, the present invention has the following beneficial effects: Under input saturation constraints and state constraints, the controller based on the nonlinear robust model of the present invention effectively suppresses the multi-degree-of-freedom oscillations caused by the heave-roll coupling motion while maintaining the optimization characteristics of the model predictive control. First, a nonlinear model prediction trajectory generation framework under input saturation constraints and state constraints is designed. The purpose is to generate a reference compensation trajectory that meets the safe operating boundaries such as the boom stroke and load swing angle through terminal constraint set design and Lyapunov function optimization, thereby ensuring the physical feasibility of the control instructions. Furthermore, a nonlinear robust model predictive controller based on tight constraint optimization is proposed, and by introducing the discrete system Lyapunov candidate function and recursive feasibility analysis, the asymptotic stability of the closed-loop system is strictly guaranteed. Finally, a series of simulation and experimental tests are carried out to ensure that the method can still maintain excellent robustness and stability in the presence of various disturbances.

[0013] Advantages of additional aspects of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The accompanying drawings, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their description are used to explain the present invention but do not constitute a limitation of the present invention.

[0015] Figure 1 A main flow chart of a method for coordinated heave-roll compensation of a dual-vessel lifting arm system provided by an embodiment of the present invention; Figure 2 Schematic diagram of a dual-ship lifting arm system considering the coupled motion of ship heave and roll, provided in an embodiment of the present invention; Figure 3 A schematic diagram of the control targets of a dual-ship lifting arm system considering the coupled motion of ship heave and roll, provided in an embodiment of the present invention; Figure 4 A schematic diagram of the heave-roll compensation effect of the transverse lifting arm of a ship 1 provided in an embodiment of the present invention; Figure 5 A schematic diagram of the heave-roll compensation effect of the longitudinal lifting arm of a ship 1 provided in an embodiment of the present invention; Figure 6 A schematic diagram of the heave-roll compensation effect of the transverse lifting arm of a ship 2 provided in an embodiment of the present invention; Figure 7 A schematic diagram of the heave-roll compensation effect of the longitudinal lifting arm of a ship 2 provided in an embodiment of the present invention; Figure 8 A schematic diagram of collaborative compensation load angle changes provided by an embodiment of the present invention; Figure 9A schematic diagram of collaborative compensation for load center point displacement provided by an embodiment of the present invention; Figure 10 A schematic diagram of the verification results of the transverse / longitudinal lifting arm constraints of the ship 1 provided in an embodiment of the present invention; Figure 11 A schematic diagram of the verification results of the transverse / longitudinal lifting arm constraints of the ship 2 provided in an embodiment of the present invention; Figure 12 A schematic diagram of the input saturation constraint verification result provided by an embodiment of the present invention; Figure 13 Schematic diagram of the heave-roll compensation effect of the lateral lifting arm of ship 1 under input saturation and speed constraints provided by an embodiment of the present invention; Figure 14 Schematic diagram of the heave-roll compensation effect of the longitudinal lifting arm of ship 1 under input saturation and speed constraints provided by an embodiment of the present invention; Figure 15 Schematic diagram of the heave-roll compensation effect of the lateral lifting arm of ship 2 under input saturation and speed constraints provided by an embodiment of the present invention; Figure 16 Schematic diagram of the heave-roll compensation effect of the longitudinal lifting arm of ship 2 under input saturation and speed constraints provided by an embodiment of the present invention; Figure 17 The embodiment of the present invention provides a method for collaboratively compensating load angle changes under input saturation and speed constraints. Figure 18 Schematic diagram of collaborative compensation of load center point displacement under input saturation and speed constraint provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0016] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0017] Example 1 like Figure 1 As shown, this embodiment discloses a method for coordinated heave-roll compensation of a dual-vessel lifting arm system, comprising the following steps: S1: A dynamic model of a twin-ship lifting arm system considering heave-roll coupling motion is established, wherein the model includes disturbance terms caused by ship motion; S2: Based on the dynamic model, generate a nominal trajectory under input saturation constraints and state constraints, and generate a desired trajectory through tight-constrained nonlinear model predictive control; S3: Design a nonlinear robust model predictive controller for the actual disturbed system to drive the actual state to track the desired trajectory; S4: The stability of the closed-loop system is ensured by the terminal cost function and the terminal constraint set, and the coordinated heave-roll compensation of the dual-ship lifting arms is achieved.

[0018] Next, combine Figure 1 , a method for coordinated heave-roll compensation of a dual-ship lifting arm system disclosed in this embodiment is described in detail.

[0019] 1. Dynamic modeling of a twin-ship lifting arm system considering the coupled heave-roll motion of the hull The schematic diagram of the double-ship lifting arm system considering the coupled motion of the ship's heave and roll is as follows: Figure 2 As shown, the dual-ship lifting arm system includes ship 1 and ship 2. Table 1 lists in detail the parameters and variables corresponding to the dual-ship lifting arm system in the earth-fixed coordinate system.

[0020] In the inertial coordinate system, the center position of the horizontal lifting arm of ship 1 is , center position of longitudinal lifting arm of ship 1 , load center position , center position of ship 2's horizontal lifting arm , center position of longitudinal lifting arm of ship 2 is defined as: (1) (2) (3) (4) The coordinates of the load near the side support of ship 1 are: (5) The coordinates of the load support point near the ship 2 side are: (6) Table 1 Parameters and variables;

[0021] Based on formulas (5) and (6), the load center coordinates are obtained: (7) By taking the derivative of the load coordinates and the four lifting arm coordinates in the double-ship lifting arm system, the velocity expression is obtained, and then according to the kinetic energy theorem, the total kinetic energy of the double-ship lifting arm system is obtained. The expression is as follows: (8) (9) Total potential energy of the double-ship lifting arm system The expression is as follows: (10) The sum of the virtual work done by the driving force and gravity is: (11) (12) Define the Lagrangian as , and calculate the following Lagrangian dynamic equation: (13) Solving the above equation yields the nonlinear dynamic equation for the dual-ship lift arm system, taking into account roll and heave. It should be understood that this solution process is within the capabilities of those skilled in the art. Arranging the equations based on the state variables yields: (14) in, is a state variable, is the system inertia matrix, is the centripetal-Coriolis force matrix, is the gravity matrix, represents the control input vector, is the external force matrix.

[0022] The specific expressions of each matrix and vector in formula (14) are as follows:

[0023] Among them, the inertia matrix Related to the kinetic energy of the twin-ship lifting arm system. Inertia matrix The construction of needs to extract coefficients from the quadratic form of kinetic energy, respectively deal with the load of the double-ship lifting arm system and the kinetic energy expression of the four lifting arms, and obtain the inertia matrix The detailed expression of each item is as follows:

[0024]

[0025]

[0026] Centripetal-Coriolis force matrix The velocity product term is involved and needs to be extracted from the Lagrange equation. The centripetal-Coriolis force matrix term corresponds to the time derivative of the inertia matrix and the quadratic term of the velocity. The specific expression is as follows:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032]

[0033] Gravity Matrix of a Twin-Vessel Lifting Arm System It is closely related to the mechanical energy (especially potential energy) of the system, which reflects the coupling effect of the gravity field on each degree of freedom of the system. After taking partial derivatives, we get the gravity matrix The detailed expressions of each item are as follows:

[0034] External disturbance matrix of the double-ship lifting arm system It is caused by the hull motion and mainly includes the rolling and heaving motions of the system and the coupling terms of the ship and lifting arm parameters. The specific expression is as follows:

[0035]

[0036]

[0037]

[0038]

[0039] In this embodiment, when constructing the dynamic model of a dual-ship lifting arm system that considers the coupled heave and roll motion of the hull, a centripetal-Coriolis force matrix is ​​introduced. This matrix accurately describes the nonlinear forces generated by the velocity product term in the system's coupled rotational and translational motions, effectively capturing the impact of the Coriolis and centripetal forces induced by roll on the system's dynamic characteristics, and ensuring the model's accurate representation of the mechanical coupling relationships under complex motion states. The external matrix fully considers the coupling effects of the hull's heave and roll motions with the system parameters, quantifying the disturbances induced by the hull motion into a matrix form. This allows the model to fully reflect the external excitation of the hull motion on the dual-ship lifting arm system. This makes the dynamic model more consistent with actual operating conditions, provides a more precise theoretical basis for system control strategy design, stability analysis, and dynamic response prediction, and improves the accuracy of dynamic analysis and engineering application value of the dual-ship lifting arm system in complex sea conditions.

[0040] according to Figure 2 Due to the connection between the lifting arm and the two ships and the various components of the lifting arm, it can be seen intuitively that the system has certain coupling constraints. Reasonable use of system constraints can simplify the order of the system dynamics model and reduce the difficulty of system control. The system has the following constraints: (15) Assumptions: (16) Based on the constraints, we can get the time The derivative and Partial derivatives of : (17) (18) This can be expressed in a more compact form using constraints: (19) The simplified system dynamics model is obtained: (20) in, Represents the system state variables after simplification using constraints. represents the simplified inertia matrix, centripetal-Coriolis force matrix, and system input. represents the simplified gravity matrix, Represents the perturbation external force matrix.

[0041] This embodiment reduces the dimensionality of state variables by coupling constraints on the motion of associated components, effectively lowering the model order, reducing computational effort and complexity. This preserves the system's key mechanical properties while simplifying analysis and control, improving the efficiency and feasibility of system analysis and control.

[0042] 2. Wave compensation control objectives of the twin-ship lifting arm system considering the coupled heave-roll motion of the hull When a ship experiences coupled heave and roll motions due to waves on the sea, the twin-ship lifting arm system controls the four lifting arms to track the desired trajectory, so that the position of the load in the inertial coordinate system remains stable and does not move significantly with the swaying of the ship, thereby achieving wave compensation.

[0043] 1. Movement Patterns and Basic Assumptions When sea conditions rise, the ship's motion pattern becomes more complex. Heave motion no longer exists independently, and roll motion is closely coupled with it to become the ship's main motion form. Even in this complex motion state, the amplitudes of heave and roll still have clear boundaries. Therefore, a reasonable assumption is made: Assume that the ship is not allowed to operate in sea conditions above level 6, so the ship's rolling motion is bounded: (twenty one) 2. Coordinate transformation and desired trajectory solution according to Figure 3, the position of the load in the hull coordinate system can be obtained. After the kinematic coupling analysis of the lifting arm system and conversion to the inertial coordinate system, the positions of the lifting points at both ends of the load are: (twenty two) In a two-dimensional plane, the complex coordinate values ​​obtained from the above formula are , solve the expected trajectory of the state variables of the lifting arm system , through subsequent control, the lifting arm tracks the desired trajectory to achieve the goal of stabilizing the load position.

[0044] 3. Control target decomposition (1) Stable load position Under the coupled motion of hull heave and roll, the four lifting arms keep the load in a stable position in the inertial coordinate system by tracking the desired trajectory, that is: (twenty three) (2) Structural constraints Considering the structural limitations of the dual-ship lifting arm system and the rigid body connection of the load, two constraints need to be met in the inertial coordinate system: one is the distance constraint between the two ships, and the other is to suppress the swing of the load relative to the inertial coordinate system, namely: (twenty four) (3) Constraints on the executive body Consider the input saturation and safety requirements of the actuator: (25) in, represents the maximum actuator input and state constraints that satisfy the saturation requirement.

[0045] 4. Controller design ideas During controller design, the boom travel needed to be constrained within a reasonable safety range. The actuator control input needed to be limited to account for actuator input saturation. For the optimization-based regulator, a model predictive approach was used to generate a desired velocity with corresponding constraints, including upper and lower bounds on the state variables and input saturation constraints. Based on this desired velocity, the twin-vessel boom system could accurately track the reference trajectory kinematically. Finally, a nonlinear robust model predictive controller was designed to achieve optimal control of the twin-vessel boom system.

[0046] 3. Cooperative Heave-Roll Compensation Control Algorithm Based on Nonlinear Robust MPC Based on the nominal system, a nonlinear model prediction trajectory generation method under input saturation and state constraints is designed to generate the desired trajectory. Then, a nonlinear robust model predictive controller is designed again for the actual system with nonlinear characteristics of the actuator, so that the actual state variable trajectory is kept close to the desired trajectory with the state variable trajectory of the nominal system.

[0047] 1. Nonlinear model prediction trajectory generation method under input saturation and state constraints Based on the proposed dynamic model, the expected speed of the nominal model of the double-ship lifting arm system is solved based on model predictive control without considering the disturbance term of the model. Nominal model: (26) In order to facilitate the design of the control law of model predictive control, the double-ship lifting arm system model (26) is firstly discretized by Euler-discretization to obtain: (27) in, , are the discretized system state variables and their derivatives (velocity vectors), is discrete time, is the current sampling moment.

[0048] The system model described in Equation (27) is a nominal discrete model of a dual-ship lifting arm system that considers heave-roll disturbances, and does not consider the influence of disturbances. In the above model, the constraints of state variables and control inputs are expressed as ,in and are all compact sets containing the origin.

[0049] Specifically, considering the impact of increasing disturbances, in order to distinguish the cost function designs of the following speed regulator and nonlinear robust model predictive controller, in the nominal model, the control input is The state variables also have similar format differences. Nominal input required For the nominal system model, a nonlinear model prediction trajectory generation method under input saturation and full state constraints is constructed: (28) in, .

[0050] By adjusting parameters and It can be seen from the value range of the system that the state constraints of the system are and input saturation requirements A certain degree of compression has been performed to ensure that the state constraints and input saturation requirements are met when the disturbance effect is further considered in the future.

[0051] However, how to choose the right and , especially for complex nonlinear systems, is still an open problem. and If the value is small, the feasible domain of the optimization control problem for the nominal system will be smaller, which will bring certain difficulties to the search for feasible solutions and optimal solutions. and If the value is large, the original constraint requirements may be violated when considering the disturbance. and For example, a trial and error method can be used to gradually adjust and , observe the search situation of feasible solutions to the optimization control problem and the satisfaction degree of the constraints, and continuously iterate and test until a suitable parameter combination is found.

[0052] In the cost function (3-3) For time domain prediction, the first term of the cost function is defined as: (29) in, They are diagonal positive definite weight matrices, which can be used to adjust the balance between state tracking accuracy and energy consumption by selecting appropriate values.

[0053] The second term of the cost function is the terminal cost function: (30) in, It is also a diagonal positive definite weight matrix, which can be used to adjust the convergence of the state variables at the terminal moment of the prediction time domain.

[0054] The idea of ​​designing a nominal controller for a nominal system is basically the same as that of traditional nonlinear model predictive control. Under the condition of satisfying the tightening constraint, the nominal system is used to predict the evolution of the system state, and the optimal control law is obtained by minimizing the cost function of formula (28). Specifically, at the current sampling time , assuming that the state variables are measurable, denoted as By solving the optimization control problem, the corresponding optimization control law sequence is recorded as , which is the nominal controller sequence required in this section. According to the operating principle of nonlinear model predictive control, the optimized control sequence The first set of control laws The above process is repeated at each sampling time. For ease of description, the sampling time The state variables and the executed control laws are respectively expressed as That is, in the optimal control law Under the action of , the state evolution of the nominal system conforms to the following relationship: (31) 2. Nonlinear Robust Model Predictive Controller In reality, the structure and dynamic characteristics of a twin-ship crane boom system are extremely complex. During operation, the system is constantly affected by random environmental factors (such as wind, waves, and currents). Therefore, it is unrealistic to obtain a disturbance-free mathematical model of the twin-ship crane boom system. Similarly, the aforementioned nominal model cannot fully and objectively reflect the actual operating characteristics and working environment of the ship crane.

[0055] Based on this, in order to more accurately reflect the actual working state of the ship crane and improve the robustness of the controller, an additional time-varying bounded disturbance is added to the nominal discrete model (27) of the double-ship lifting arm system: , then the double-ship lifting arm system model considering disturbance has the following form: (32) Specifically, compared with the discrete nominal model represented by Equation (27), the discrete model represented by Equation (32) includes the perturbation , which can reflect the influence of various external environmental factors on state variables to a certain extent.

[0056] This section will design a nonlinear robust model predictive controller for model formula (32) to realize the heave-roll coupled wave compensation control of the dual-ship lifting arm system. First, a constraint satisfaction governor based on model predictive control is designed for the nominal system, so that the state variables of the nominal system converge quickly to the expected value in the absence of disturbances and keep the load stable to obtain the expected trajectory. Then, a nonlinear model predictive controller is designed again based on model predictive control for the actual system with disturbances, so that the actual state variables approach the expected trajectory with the state variable trajectory of the nominal system. In this way, although the system may be affected by unpredictable and random disturbances, under all possible disturbances, the actual state variable trajectory will remain in a pipeline centered on the state variable trajectory of the nominal system and bounded by a certain radius. within , thus having better control effect and stronger robustness.

[0057] It can be seen that the design of a new nonlinear robust model predictive controller is to make the actual state variables as consistent as possible with the state variables of the corresponding nonlinear model predictive trajectory generation method. Therefore, this can be achieved by solving the following nonlinear robust model predictive control optimization problem: (33) In the cost function middle To predict the time domain, under the condition of meeting the computing power limit, control the time domain , the first term function of the cost function The specific expression is: (34) in, are diagonal positive definite weight matrices, which can be used to balance the energy consumption of state variable tracking accuracy. The second term of the cost function is Specifically expressed as: (35) in, They are all diagonal positive definite weight matrices, which can be used to adjust the convergence of the state variables at the terminal moment of the prediction time domain.

[0058] The overall goal is to drive the actual state variables as close as possible to the corresponding nominal system state variables by minimizing the cost function (33). The first and second constraints are actually the evolution laws of the dual-ship lifting arm system state under the action of the nominal controller. Since the disturbance is random and cannot be accurately measured or predicted, the changes in the system state can only be deduced with the help of the nominal system model in the prediction domain. Therefore, the third and fourth items are the nominal system models without considering the disturbance. If the prediction domain is selected appropriately, the error caused by this treatment method of ignoring the disturbance is acceptable. The last two constraints are the state constraint and the input saturation requirement.

[0059] By solving the optimization control problem defined in equation (33), the sampling time can be obtained Optimal control law sequence Similarly, the first set of control laws in the control sequence applied to the actual system.

[0060] The design process of the nonlinear model-predictive trajectory generation method and the nonlinear robust model predictive controller (NRMPC) under input saturation and full-state constraints demonstrates that this method can generate ideal state variable trajectories under undisturbed conditions. This ideal trajectory serves as the desired trajectory of the actual state variables. Even under disturbances, the actual state variables, driven by the auxiliary controller, can approach the ideal trajectory as closely as possible and ultimately converge to near the desired values. Although the system may not precisely reach or stabilize at the desired values ​​under the influence of disturbances, the actual state variables will fluctuate slightly around the desired values, stabilizing within a tubular region centered on the ideal trajectory and with a finite radius. This design not only ensures the system's control performance but also significantly enhances its robustness. Furthermore, state constraints and input saturation conditions are fully considered in the controller design, ensuring the system's safety and feasibility.

[0061] 3. Stability analysis This section provides an in-depth theoretical analysis of the stability of the closed-loop heave-roll compensation control for a twin-ship lifting boom system based on nonlinear robust model predictive control (NMPC). By combining Lyapunov stability theory, terminal constraint design, and disturbance robustness analysis, the system's stability and robustness are ensured in complex sea conditions.

[0062] Construction of terminal cost function: (36) in, , is the terminal cost matrix.

[0063] The construction of the terminal cost function needs to meet certain design conditions. First, the cost function must satisfy the positive definiteness: , dissipative: that is, there is a local controller In the terminal area Inside: (37) in is the designed positive definite matrix.

[0064] Determine the matrix through offline optimization , so that it satisfies the geometric characteristics of the nonlinear system: (38) This optimization problem is solved by the sampling-verification method: dense sampling in the state space , verify inequality constraints, stepwise contraction Until the conditions are met.

[0065] Finally, construct the terminal constraint set and define the terminal region as: (39) in By selecting Make Including the balance point area, and then find the largest one through dichotomy Make ,exist Satisfy the dissipative conditions.

[0066] Before using the Lyapunov method to determine the stability of discrete systems, we must first prove the recursive feasibility of the system. Assume that at time , there exists a feasible control sequence: (40) Make . Construct the candidate solution for the next moment: (41) Terminal status , apply the local controller: (42) From the design conditions of the terminal area, we can know that: (43) Therefore , the recursive feasibility is proved.

[0067] Based on the constructed terminal cost function and the recursive feasibility of the system, the Lyapunov candidate function is defined: (45) use Indicates the current The optimal cost function at any moment, when there is no external disturbance in the system, uses the Lyapunov method of discrete systems to determine stability: (46) (47) Substitute the terminal region dissipation condition: (48) The discrete-time Lyapunov stability conditions are satisfied.

[0068] Example 1 In order to verify the effectiveness of this embodiment, the following specific implementation is given.

[0069] The nonlinear robust model predictive control (NRMPC) provided in this embodiment adopts a tight constraint optimization strategy, and designs the prediction time domain to be equal to the control time domain. Step, terminal constraint set parameters , using Sequential Quadratic Programming (SQP) to solve online. The weight matrix is ​​set as:

[0070] The reasonable selection of sampling time is the key to MPC design, and its value directly affects the dynamic balance between real-time computing burden and state estimation accuracy. , prediction time domain The value is 5, which is the same as the control time domain same.

[0071] To ensure statistically significant performance comparisons between nonlinear robust model predictive control (NRMPC) and nonlinear model predictive control (NMPC), this study employed identical parameter configurations in both control structures. Specifically, both control strategies employed the same state weight matrix and control input weight matrix, and the experimental setup used a unified sampling time, prediction horizon, and control horizon, ensuring a fair comparison of the two control strategies within the same optimization framework. This parameter homogeneity effectively avoided performance deviations caused by differences in optimization parameters and provided a standardized experimental benchmark for subsequent robustness comparisons.

[0072] The output feedback control parameters were optimized after multiple experiments, and the following control gains and parameters were used:

[0073] (1) Comparative test with nonlinear MPC and output feedback methods Figures 4 to 9 The following plots show the boom and load position tracking response curves for a dual-vessel boom system under five sea conditions. The red continuous curve represents the desired compensation trajectory generated by the wave spectrum model, the green continuous curve shows the tracking result of the proposed robust nonlinear model predictive control (NRMPC) method, and the yellow and blue dashed lines correspond to the response curves of traditional nonlinear model predictive control (NMPC) and output feedback control, respectively. By calculating the root mean square error (RMSE) of tracking the reference compensation trajectory and the load angle change, the superiority of the nonlinear robust MPC method over the nonlinear MPC and output feedback algorithms is numerically demonstrated. The RMSE results are shown in Table 2.

[0074] Figures 4 to 9 The simulation data of driving state quantity verified the load stability control of the double-ship lifting arm system under the fifth-level sea state. Although the single typical working condition data has intuitively demonstrated the effectiveness of the control method, the reliability and stability of the experiment are still uncertain. Table 2 Performance indicators of different controllers;

[0075] Furthermore, the qualitative results still need to be further verified. To this end, this embodiment carried out three repeated experiments and introduced error statistical analysis. By constructing a root mean square error comparison table, the performance differences of different control strategies were quantified from the data level, striving to conduct a more comprehensive and rigorous analysis to deeply evaluate the actual performance and application value of the control method. The quantitative indicators in Table 2 show that the root mean square error (RMSE) of the boom tracking of the output feedback control is much larger than the root mean square error of the nonlinear robust model predictive control and the nonlinear model predictive control. The RMSE of the load angle change of the output feedback is 0.1214, which is also significantly higher than the 0.0323 of the nonlinear robust model predictive control (NRMPC) and the 0.0765 of the nonlinear model predictive control (NMPC). This confirms the inherent defects of traditional control methods in compensating for the nonlinear coupling dynamics of hull and load. Figures 4 to 7 The boom motion trajectory tracking curve demonstrates the superiority of the NRMPC algorithm for heave-roll compensation in a dual-vessel boom system. Experimental results show that the NRMPC algorithm tracks the reference compensation curve with significantly greater accuracy than traditional NMPC and output feedback control, demonstrating the algorithm's enhanced tracking capabilities under complex dynamic coupling conditions.

[0076] Figure 8-9 The load angle variation and center-point displacement trajectory demonstrate that the nonlinear robust model predictive control (NRMPC) system exhibits excellent load attitude stabilization capabilities under extreme disturbance conditions in level 5 sea states. The NRMPC's load swing angle fluctuations are limited to ±0.05°, and the load motion is strictly confined to a target area of ​​0.003 m × 0.4 m. In contrast, the NMPC's displacement range extends to 0.01 m × 0.04 m, while the output feedback control's maximum displacement deviation reaches 0.07 m × 0.4 m. This result directly demonstrates the NRMPC's ability to precisely decouple multi-degree-of-freedom motion.

[0077] (2) Robustness verification of input saturation and state constraints Figure 10 and Figure 12 The verification experimental results of input saturation constraints and state constraints are shown. The red continuous curve is the expected speed trajectory generated by the wave spectrum model, the green continuous curve is the tracking result of the NRMPC algorithm with constraint processing added, and the blue dotted line represents the physical limit of the actuator speed. Figure 12 The system input response curve under input saturation constraint is shown. The red continuous curve represents the system input of the proposed algorithm under input saturation constraint, and the blue dashed line represents the physical limit of the actuator output. Figures 13 to 18The arm position tracking response curves and load displacement curves under input saturation constraints and velocity constraints are shown. The red line shows the desired compensation trajectory, the green line shows the response curve of the unconstrained NRMPC, and the yellow dashed line shows the response curve of the NRMPC with constraints.

[0078] Input saturation and state constraints were incorporated into the coordinated heave-roll compensation of a dual-ship lift arm system. The NRMPC algorithm was used for both. The NRMPC algorithm with and without these constraints was compared to verify the effectiveness of the designed controller under these constraints. The root mean square error (RMSE) results for tracking the reference compensation trajectory and load angle changes using the nonlinear robust model predictive control with and without constraints are shown in Table 3.

[0079] The quantitative indicators and Figures 13 to 18 Comparative simulation experiments are presented for the NRMPC with input saturation constraints and state constraints and the unconstrained NRMPC. The root mean square error in Table 3 shows that after the introduction of the dual constraints, the dynamic performance of the boom tracking system remains stable, and the tracking accuracy is slightly reduced. The root mean square error of the four boom tracking increases by only 10% compared with the unconstrained NRMPC algorithm, and the load angle is also within the constraint range. The root mean square error of 0.0323 increases by only 9% compared with 0.335 of the unconstrained NRMPC algorithm, avoiding the input amplitude saturation phenomenon.

[0080] Table 3 Performance indicators of NRMPC algorithm with different constraints;

[0081] Figures 13 to 16 The boom tracking curve and Figure 17 The coordinated compensation load angle variation curve shows the robustness of the NMPC algorithm in heave-roll compensation of the twin-ship lifting arm system, demonstrating the superiority of the algorithm. Figure 18 The displacement trajectory of the collaborative compensation center point shows that under level 5 sea conditions, after adding state constraints and input saturation constraints, the load swing angle fluctuation range of NRMPC is also maintained within the range of ±0.1°, and the load movement is strictly limited to the target area of ​​0.01m×0.4m. This result intuitively reflects that NRMPC ensures the stability of the control effect while processing constraints, showing the robustness of the NRMPC algorithm.

[0082] Existing technologies suffer from imperfect dynamic modeling, insufficient constraint processing capabilities, and poor robustness in sea conditions above level 3. This specific embodiment establishes a coupled motion model with disturbance terms to accurately describe the system's dynamic characteristics; designs a nonlinear MPC trajectory generator with tight constraints, balancing control accuracy and physical feasibility; and introduces a robust MPC controller that ensures closed-loop stability through terminal constraint sets and Lyapunov functions. This allows the system to maintain trajectory tracking accuracy under input saturation and random perturbations, reducing the root mean square error by over 60% compared to traditional methods, breaking through the limitations of operations in complex sea conditions.

[0083] Example 2 This embodiment provides a coordinated heave-roll compensation system for a dual-vessel lifting arm system, comprising: A dynamic modeling module is configured to establish a dynamic model of a twin-ship lifting arm system considering heave-roll coupled motion, wherein the model includes a disturbance term caused by hull motion; a nominal trajectory generation module configured to generate a nominal trajectory under input saturation constraints and state constraints based on the dynamic model, and generate a desired trajectory through a tight-constrained nonlinear model predictive control; The robust control module is configured to design a nonlinear robust model predictive controller for the actual disturbed system to drive the actual state to track the desired trajectory; The stability assurance module is configured to ensure the stability of the closed-loop system through a terminal cost function and a terminal constraint set, thereby achieving coordinated heave-roll compensation of the twin-ship lifting arms.

[0084] Example 3 This embodiment provides a computer-readable storage medium having a computer program stored thereon. When the program is executed by a processor, the steps of the method for coordinated heave-roll compensation of a dual-vessel lifting arm system as described in the first embodiment above are implemented.

[0085] Example 4 This embodiment provides a computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the steps of the method for coordinated heave and roll compensation of a dual-vessel lifting arm system as described in the first embodiment above are implemented.

[0086] The steps or modules involved in Examples 2 to 4 above correspond to those in Example 1. For detailed implementations, please refer to the relevant description of Example 1. The term "computer-readable storage medium" should be understood to mean a single medium or multiple media that includes one or more instruction sets; it should also be understood to include any medium that can store, encode, or carry an instruction set for execution by a processor and cause the processor to perform any method of the present invention.

[0087] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention are intended to be within the scope of protection of the present invention.

Claims

1. A method for coordinated heave-roll compensation of a dual-ship lifting arm system, characterized in that: include: A dynamic model of a twin-ship lifting arm system considering heave-roll coupled motion is established, wherein the model includes disturbance terms caused by ship motion. Based on the dynamic model, a nominal trajectory under input saturation constraints and state constraints is generated, and a desired trajectory is generated through a tight-constrained nonlinear model predictive control; For the actual disturbed system, a nonlinear robust model predictive controller is designed to drive the actual state to track the desired trajectory; The stability of the closed-loop system is ensured by the terminal cost function and the terminal constraint set, and the coordinated heave-roll compensation of the twin-ship lifting arms is achieved.

2. The method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to claim 1, characterized in that: The process of establishing the dynamic model of the twin-ship lifting arm system considering the heave-roll coupled motion specifically includes: Obtain the center positions of the transverse and longitudinal lifting arms of the two ships, and calculate the coordinates of the contact points between the two ships and the load; Calculating the load center coordinates based on the coordinates of the contact points; The coordinates of the four lifting arms and the load center are derived, and the total kinetic energy of the double-ship lifting arm system is obtained according to the kinetic energy theorem. The dynamic model is obtained based on the total kinetic energy and total potential energy of the dual-ship lifting arm system according to Lagrange's theorem; wherein the disturbance term of the model is the coupling term of the roll and heave motion of the dual-ship lifting arm system and the ship and lifting arm parameters.

3. The method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to claim 1, characterized in that: The generating of a nominal trajectory under input saturation constraints and state constraints based on the dynamic model and generating a desired trajectory through a tight-constrained nonlinear model predictive control specifically includes: Discretizing the dynamic model to construct a nominal discrete model; Under the conditions of input saturation and state constraints, a tight-constrained nonlinear model predictive control is used to generate the desired trajectory. This includes compressing the original constraint bounds and designing an optimization objective function that includes a state tracking term and a terminal cost term. The optimization objective function is solved to obtain a nominal control sequence that satisfies the tight constraints and generate a desired trajectory of the twin-ship lifting arm system.

4. The method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to claim 1, wherein: The nonlinear robust model predictive controller is designed for the actual disturbed system to drive the actual state to track the desired trajectory, specifically including: Introducing bounded disturbance terms into the nominal discrete model to construct the actual disturbed system model; Taking the expected trajectory generated by the nominal system as the tracking target, a robust optimization objective function is designed; By minimizing the deviation between the actual state and the desired trajectory, the control instructions that satisfy the input saturation constraint and the state constraint are solved, and the actual state is driven to converge to a pipeline area centered on the desired trajectory and bounded by a finite distance.

5. The method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to claim 1, characterized in that: The method of ensuring the stability of the closed-loop system through the terminal cost function and the terminal constraint set specifically includes: Design a terminal cost function that satisfies the positivity and dissipativeness conditions; Construct a terminal constraint set including a terminal cost function and a local controller; The closure of the state within the constraint set is verified by recursive feasibility analysis, and the asymptotic stability of the closed-loop system is proved by combining the Lyapunov function.

6. The method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to claim 1, characterized in that: The method of realizing coordinated heave-roll compensation of the dual-ship lifting arms specifically includes: The four sets of lifting arms coordinately track the desired trajectory, so that the position of the load in the inertial coordinate system remains stable; And suppress the load's attitude angle change relative to the inertial coordinate system, so that the load center displacement and swing angle meet the preset safety boundaries.

7. A coordinated heave-roll compensation system for a dual-ship lifting arm system, characterized in that: include: A dynamic modeling module is configured to establish a dynamic model of a twin-ship lifting arm system considering heave-roll coupled motion, wherein the model includes a disturbance term caused by hull motion; a nominal trajectory generation module configured to generate a nominal trajectory under input saturation constraints and state constraints based on the dynamic model, and generate a desired trajectory through a tight-constrained nonlinear model predictive control; The robust control module is configured to design a nonlinear robust model predictive controller for the actual disturbed system to drive the actual state to track the desired trajectory; The stability assurance module is configured to ensure the stability of the closed-loop system through a terminal cost function and a terminal constraint set, thereby achieving coordinated heave-roll compensation of the twin-ship lifting arms.

8. The coordinated heave-roll compensation system for a dual-vessel lifting arm system according to claim 7, characterized in that: The process of establishing the dynamic model of the twin-ship lifting arm system considering the heave-roll coupled motion specifically includes: Obtain the center positions of the transverse and longitudinal lifting arms of the two ships, and calculate the coordinates of the contact points between the two ships and the load; Calculating the load center coordinates based on the coordinates of the contact points; The coordinates of the four lifting arms and the load center are derived, and the total kinetic energy of the double-ship lifting arm system is obtained according to the kinetic energy theorem. The dynamic model is obtained based on the total kinetic energy and total potential energy of the dual-ship lifting arm system according to Lagrange's theorem; wherein the disturbance term of the model is the coupling term of the roll and heave motion of the dual-ship lifting arm system and the ship and lifting arm parameters.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to any one of claims 1 to 6 are implemented.

10. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method for coordinated heave-roll compensation of a dual-vessel lifting arm system according to any one of claims 1 to 6 are implemented.

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