Abstract: The course of an unmanned surface vessel (USV) is easily affected by external disturbances such as wind, waves, and currents during navigation. Frequent rudder adjustments to correct the course not only accelerate the wear of the servo motors but also increase the vessel's power consumption. To address this issue, a model-predictive interval control algorithm is proposed. This method uses the course reference trajectory and rudder increment as variables in the online optimization solution. The reference trajectory automatically tracks the course, thereby reducing the number and magnitude of rudder adjustments and ultimately lowering the vessel's power consumption. Finally, a simulation of the course interval control problem of a specific ship is conducted to verify the effectiveness of the algorithm.
Keywords: low power consumption; heading section control; predictive control
Chinese Library Classification Number: TP27 Document Identification Code: A
1 Introduction
When unmanned surface vessels (USVs) navigate in the ocean, they are easily disturbed by the ocean environment, such as wind, waves and currents. If they rely on frequent ruddering to resist the interference of wind, waves and currents to maintain their course, it will not only accelerate the wear of the rudder motor but also greatly increase the power consumption of the ship. How to reduce the power consumption of USVs is a problem worth studying. In terms of ship motion control, Jawhar Ghommam et al. transformed the system into a pure serial structure and then used a discontinuous backstepping controller to make the whole system achieve global asymptotic stability. The effectiveness of the algorithm was verified by simulation [1]. KDDo designed a controller by combining Lyapunov direct method, backstepping and parameter estimation, which can make the path tracking error of underwater vehicles arbitrarily small [2]. Thor I. Fossen proposed an integral adaptive LOS navigation algorithm, which identifies the drift angle as a fixed parameter through an adaptive algorithm and verified the feasibility of the algorithm by simulation [3]. MortezaMoradi et al. designed a sliding mode controller based on a known upper limit of disturbance during the ship's roll motion, and adopted a second-order adaptive sliding surface to reduce the output chattering phenomenon. The effectiveness of the algorithm was verified by simulation [4]. WeiMeng et al. proposed a controller based on two sliding surfaces for ship trajectory tracking. One first-order sliding surface was used to track longitudinal error, and the other second-order sliding surface was used to track lateral error. The effectiveness of the algorithm was verified by simulation [5]. In terms of reducing ship energy consumption, Liu Yong et al. proposed an S/KS hybrid sensitivity H∞ robust control algorithm. It aims to reduce the number of steering actions while maintaining the course. The effectiveness of the algorithm was verified by simulation [6]. Chen Xin proposed a method to minimize the overall fuel consumption of the entire voyage by optimizing the ship's forward speed in each segment of the entire voyage and minimizing the fuel consumption in each segment [7]. Ren Hongying established a mathematical model of ship, engine, propeller, and wing, and proposed relevant measures to improve the performance of the ship's main propulsion system and a new speed control strategy for a wind-wing-diesel hybrid ship [8]. Li Sheng observed the pattern of changes in the relevant parameters of the main engine with the rotational speed. He also gave specific measures to be taken when implementing ship speed reduction during navigation [9].
This paper starts from the actual needs of unmanned surface vessel (USV) navigation, and designs a controller based on the constraints of rudder angle and rudder speed to reduce the rudder amplitude and number of rudder actions during USV cruise to achieve course range control. An experimental platform is built using Simulink in MATLAB, and the effectiveness of the method is verified through simulation.
2 Mathematical Model of Horizontal Motion of Unmanned Surface Vessel
Figure 1. Horizontal coordinate system of the unmanned surface.
Fig.1The coordinateframeofshiphorizontalmovement
As shown in Figure 1, the motion of the unmanned surface vessel on the horizontal plane is divided into a geodetic coordinate system and a motion coordinate system. The meanings of the parameters in the coordinate systems are as follows:
Eξη and Oxy represent the geodetic coordinate system and the motion coordinate system, respectively; u, v, and U represent the longitudinal velocity, lateral velocity, and resultant velocity of the unmanned surface vessel, respectively.
β , φ , and γ represent the drift angle, heading angle, and the angle between the unmanned surface vessel's velocity and the horizontal axis of the geodetic coordinate system, respectively.
r represents the bow angular velocity of the unmanned surface vessel;
δr represents the rudder angle of the unmanned surface vessel;
The general equations for the horizontal motion model of an unmanned surface vessel are shown below:
To facilitate controller design, the nonlinear motion equations of the unmanned surface vessel (USV) shown above need to be simplified. First, consider the linearization of the following two hydrodynamic functions.
Assuming the ship's mid-section remains symmetrical and its fore and aft are approximately symmetrical,
The longitudinal velocity remains approximately linear during micro-maneuvers, and its increment is negligible. Therefore, Y <sub>0</sub> = N<sub> O</sub> = 0, u <sub> 0</sub> = 0. Substituting equations (5) and (6) into equation (1), we obtain the linear equation for the horizontal maneuvering of the unmanned surface vessel. Rewriting this equation in matrix form yields...
in
3. Heading Controller Design
Predictive control is an optimization control algorithm developed from industrial applications. The principle of predictive control can be summarized as follows: at a certain sampling time, the object model is used to predict the future state of the object under the action of a certain control variable. Based on this, the controller is solved according to the given constraints. At this sampling time, the prediction of the future state of the system is corrected by detecting the real-time state of the system. In summary, it consists of three steps: prediction model, rolling optimization, and feedback correction. Since predictive controllers can handle the solution of constrained problems well, this paper will design a controller based on predictive control to handle the heading interval control problem of unmanned surface vessels under rudder angle and rudder speed constraints.
3.1 Model Prediction
Multi-step prediction is performed based on the linear equation (7) of the unmanned surface vessel's horizontal motion from Chapter 2. Considering the actual modeling errors and noise present in the control process, a state-space equation based on the augmented state is used for prediction, as shown below.
Using a prediction step size ny and a control step size nu, the horizontal heading of the unmanned surface vessel is predicted. The prediction equation is as follows:
in
The prediction equation shown below is obtained.
3.2 Constraint Handling
Due to physical limitations, the rudder of unmanned surface vessels (USVs) is subject to physical constraints on rudder angle and rudder speed. In order to reduce the magnitude and number of rudder movements of USVs in the event of interference, a control range will be set for the heading control.
The constraint on rudder speed is also the constraint on the increment of the control quantity. The upper and lower limits of the control quantity increment are expressed as matrix inequalities.
The upper and lower limits of the rudder angle are denoted as and , respectively. The rudder angle constraint can be written in the form of a matrix inequality.
To achieve heading interval control, an optimized reference trajectory is introduced, where the reference trajectory α is also treated as an output variable for solution and constraint. The constraint interval is the heading control interval . The constraints of the reference trajectory can be expressed in matrix inequality form as follows:
3.3 Solving the control law
Under the constraints of rudder angle, rudder speed, and heading range, the control law is solved using the following cost function.
Substitute equation (10) into the cost function to solve.
The constant term in the cost function has no effect on the solution of the control law. After removing the constant term from the cost function, the result can be written in the following form.
From equations (14) and (15), it can be seen that solving the control law is a standard quadratic programming (QP) problem. Using the quadratic programming method, the optimal solution O* under constraints can be obtained, and then the output at the current time and the optimal reference trajectory at the current time can be obtained.
4. Simulation Analysis
To verify the effectiveness of the interval control algorithm proposed above, a simulation is performed using the heading control problem of an unmanned surface vessel as an example.
The hydrodynamic coefficients of the unmanned surface vessel are shown in the table below.
Table 1 Hydrodynamic coefficients of unmanned surface vessels
Tab.1tableofunmannedsurfacevesselhydrodynamiccoefficients
The linear equation of motion for the unmanned surface vessel (USV) on the horizontal plane at a speed of 9 knots is:
Assuming the system's initial state is zero, and both initial input and output are zero, the simulation parameters are set as shown in the table below.
Table 2 Simulation Parameters
Tab.2 Simulation parameter
Using the parameters shown in the table above, interval predictive control simulation was performed in MATLAB with a control cycle of 0.2s. The simulation curves are shown in the figure below.
Figure 2 Heading Curve
Fig.2simulationresultsofheading
In Figure 2, the solid line represents the heading curve, and the dashed line represents the upper and lower boundaries of the defined interval. Figure 2 shows that even with bow acceleration disturbances, the heading is well controlled within the given interval. Figure 3 shows the curves of the reference trajectory and the heading. The dashed line represents the reference trajectory, and the solid line represents the actual heading. It can be seen from the figure that in the initial stage, the heading is outside the constraint interval, and the reference trajectory represents the minimum value (27°) of the heading constraint interval. Once the heading enters the constraint interval, by treating the reference trajectory as a constraint variable for online optimization, the reference trajectory can automatically follow the heading. Figure 4 shows the actual rudder angle curve. It can be seen from the figure that both the rudder angle and rudder speed are controlled within the given interval. When the heading changes within the given interval, the rudder hardly moves, significantly reducing the number and magnitude of rudder inputs. This simulation verifies the effectiveness of the heading interval predictive control algorithm proposed in this paper.
5. Conclusion
This paper first analyzes the motion characteristics of an unmanned surface vessel (USV) under disturbances such as wind, waves, and currents, and proposes a control method that significantly reduces the number and magnitude of rudder inputs through course interval control. This method uses the course reference trajectory and rudder increment output as constraint variables in online optimization. When the course is outside the constraint interval, the reference trajectory serves as the boundary value of the constraint interval, and the controller guides the USV's course towards the reference interval. When the course is within the control interval, the reference trajectory, obtained through online optimization, automatically tracks the course, minimizing the cost function and thus reducing the number and magnitude of rudder inputs, thereby lowering the overall power consumption of the USV. Finally, this method is applied to a simulation of a USV course interval control problem, achieving good control results.
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