Optimal control of a rudder based on a predictive horizon and an adaptive dynamic model for trajectory tracking of an unmanned surface vehicle with rudder rate constraints

Abstract

The application proposes a rudder speed limited unmanned surface vehicle trajectory tracking method based on a preview optimal rudder controller and an adaptive dynamics model, which realizes unmanned surface vehicle trajectory tracking through control of the rudder angle and propeller speed, and realizes rudder angle instruction optimization by a preview optimal rudder controller; the preview optimal rudder controller calculates the predicted turning trajectory and the expected turning trajectory of the unmanned surface vehicle in the future preview time domain under the condition of meeting the rudder rate constraint based on the ship kinematics model, the guidance law and the adaptive rudder dynamics model in each control period, and determines the optimal rudder angle control quantity by minimizing the deviation between the predicted turning trajectory and the expected turning trajectory. The application embeds the rudder turning rate constraint in the optimization algorithm itself through the preview optimal rudder controller, so as to reduce overshoot and delay caused by inertia.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering and autonomous control systems, specifically relating to a method for tracking the trajectory of unmanned surface vessels with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model. Background Technology

[0002] With the rapid development of unmanned aerial and ground systems, unmanned surface vessels (USVs) have become a key area of ​​robotics research. Their versatility enables them to perform a wide range of marine applications, such as cargo transport, environmental monitoring, scientific research, and commercial activities, reflecting their enormous future development potential. Therefore, achieving precise trajectory tracking control (ATTC) is considered a core problem in the fields of marine engineering and autonomous control systems. However, underactuated unmanned underwater vehicles (UUVs) have fewer independent actuators than degrees of freedom, leading to more challenging control problems compared to fully actuated UUVs. The dynamic models of these systems exhibit highly nonlinear and strongly coupled characteristics, making them extremely sensitive to environmental disturbances such as waves, wind, and currents, which can severely affect tracking accuracy. Therefore, designing robust and efficient control frameworks to cope with uncertain and changing marine environments and achieve reliable trajectory tracking remains a key and unresolved challenge in the field of autonomous vessel control.

[0003] To achieve accurate trajectory tracking control (ATTC) for unmanned surface vessels (USVs), various control strategies have been developed, covering feedback linearization, backstepping control, nonlinear model predictive control, sliding mode control, neural networks, fuzzy logic, and adaptive control systems. Further progress is reflected in the design of guidance laws: typically, the guidance module calculates the desired heading angle and forward speed, and then the controller tracks the direction. The integration of artificial intelligence has also had a significant impact on the control methods of unmanned surface vessels.

[0004] Recent research has made significant progress in the field of unmanned surface vessel (USV) trajectory tracking and control, covering both model-driven and data-driven methods. However, practical deployment under rudder speed constraints still presents challenges. Most methods still neglect key implementation issues when turning to physical platforms, such as unmodeled dynamics, actuator response delays during maneuvers, and especially the coupling effect of rate saturation and large inertia in rudder operation: due to mechanical structure and safety limitations, the actual USV rudder system has strict rudder steering rate constraints. These constraints are coupled with the ship's large inertia and inherent steering delay, which can easily lead to overshoot and trajectory instability.

[0005] In view of this, the present invention proposes a framework that integrates a predictive optimization rudder controller and an adaptive dynamics model to bridge the above gap. This framework explicitly incorporates rate constraints and mechanical response characteristics, thereby improving the reliability of simulated physical object migration in unmanned surface vessel trajectory tracking. Summary of the Invention

[0006] The purpose of this invention is to propose a trajectory tracking method for unmanned surface vessels with constrained rudder speed based on a predictive optimized rudder controller and an adaptive dynamics model. This method integrates a predictive optimized rudder controller and an adaptive dynamics model. The predictive optimized rudder controller uses a predictive mechanism to predict trajectory requirements in future time periods and actively optimizes rudder commands under rudder speed constraints, thereby significantly suppressing overshoot while maintaining high tracking accuracy. The performance of the predictive optimized rudder controller is enhanced by an adaptive dynamics model, which can be calibrated online using limited measured data to ensure the reliability of predictions and facilitate a smooth transition from simulation environments to physical platforms.

[0007] To achieve the above objectives, the technical solution of this invention is: a trajectory tracking method for unmanned surface vessels (USVs) with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamic model. This method achieves trajectory tracking by controlling the rudder angle and propeller speed, and optimizes the rudder angle command using a predictive optimization rudder controller.

[0008] The predictive optimization rudder controller calculates the predicted and desired steering trajectories of the unmanned surface vessel in the future preview time domain under the condition of satisfying the rudder speed constraint, based on the ship's kinematics model, guidance law, and adaptive rudder dynamics model in each control cycle. The optimal rudder angle control quantity is determined by minimizing the deviation between the predicted and desired steering trajectories.

[0009] Preferably, the servo dynamics model is used to predict yaw acceleration based on propeller speed, rudder angle, and yaw rate, and the model parameters are adaptively calibrated using historical operating data.

[0010] Preferably, under near-constant speed operating conditions, an incremental PID controller is used to adjust the propeller speed control.

[0011]

[0012] in: The expected oscillation speed at the current time step t With oscillation speed The difference, This is the optimal propeller speed control value output in the current control cycle. , These are the proportional coefficient and the integral coefficient, respectively. This represents the propeller speed control value at the previous time step t.

[0013] Preferably, the servo motor dynamics model is as follows:

[0014] ,

[0015] in: Indicates yaw rate, Indicates yaw acceleration. Indicates the propeller speed. represents the rudder angle, This represents the yaw acceleration function, which is determined by the propeller speed, rudder angle, and yaw rate. The density of water (taken as ) ), Indicates the length of the unmanned surface vessel. Indicates the mass of the unmanned surface vessel. Indicates that it follows a normal distribution Environmental noise torque, The standard deviation of the distribution Indicates the moment of inertia. and This represents the parameters of the servo motor dynamics model.

[0016] Preferably, adaptive calibration of the servo motor dynamics model parameters is performed using historical operating data, as follows:

[0017] Transform the servo dynamics model into a linear regression form:

[0018]

[0019] in: Represents the target variable containing noise. Indicates the independent variable;

[0020] Collect yaw acceleration data of N groups of unmanned surface vessels during operation. propeller speed and rudder angle And calculate the independent variable corresponding to each set of data. and target variable :

[0021]

[0022] Determining parameters using the least squares method and :

[0023]

[0024] in: This represents the target variable calculated using the operational data of the i-th unmanned surface vessel. and Let represent the independent variable calculated using the operational data of the i-th unmanned surface vessel.

[0025] Preferably, the optimal rudder angle control quantity is determined by minimizing the deviation between the predicted steering trajectory and the desired steering trajectory, and the specific expression is as follows:

[0026]

[0027] in: This is the optimal rudder angle control value output in the current control cycle. This represents the predicted rudder angle at the j-th future time step. This indicates that the predicted rudder angle at the j-th future time step satisfies the servo steering rate constraint. , This represents the predicted yaw rate at the j-th future time step. Let represent the expected yaw rate at the j-th future time step. This represents the maximum time step in the future preview time domain.

[0028] Preferably, the calculation of the predicted yaw rate is based on a servo dynamics model under the assumption of constant speed:

[0029]

[0030] in: Let be the predicted oscillation velocity at the j-th future time step; Let be the predicted sway velocity at the j-th future time step; Let be the predicted yaw rate at the j-th future time step;

[0031] When j=1: This indicates the propeller speed at the current time step. , Indicates the rudder angle at the current time step. , This represents the yaw rate at the current time step; and These represent the sway velocity and oscillation velocity at the current time step, respectively.

[0032] When j=2,...,N p hour: Let represent the predicted propeller speed at the (j-1)th future time step, and ; This represents the predicted rudder angle at the (j-1)th future time step. The value is based on the rudder angle at the current time step. and servo steering rate constraints ,and The values ​​are equal; This represents the predicted yaw rate at the (j-1)th future time step;

[0033] Then the predicted yaw rate at the j-th future time step The calculation is as follows:

[0034]

[0035] in: As a discount factor, .

[0036] Preferably, the calculation of the desired yaw rate is as follows:

[0037]

[0038]

[0039]

[0040]

[0041] in, Let be the expected heading angle at the j-th future time step; Let yaw angle be the predicted yaw angle for the j-th future time step; For the sampling time of the control system; Let be the expected oscillation velocity at the j-th future time step; This is the guidance law function, used to determine the position coordinates. With speed state Calculate the desired heading angle and desired sway velocity under the corresponding working conditions; , These are the predicted x-coordinate and predicted y-coordinate for the j-th future time step, respectively. This is a model of ship kinematics.

[0042] Preferably, the calculation of the desired heading angle and desired sway velocity is as follows:

[0043]

[0044] in: Let the tangent angle of the task path be the j-th future time step. Let $\frac{j}{j}$ be the predicted transverse track tracking error at the $j$-th future time step. To adapt to the forward sight distance, For hydrodynamic drift angle, Let the expected summation velocity be at the j-th future time step. Let the total speed of the task path be the speed at the j-th future time step. , Let be the lateral velocity of the task path at the j-th future time step. Let the longitudinal velocity of the task path be the velocity at the j-th future time step. To control the gain, Let $\frac{j}{j}$ be the predicted tracking error along the path at the $j$-th future time step. Forward sight distance, All of these are adjustable hyperparameters for adjusting adaptive forward gaze distance. is a natural constant and the base of the natural exponential function. The weighting coefficients are used to characterize the trade-off between tracking performance and heading smoothness.

[0045] Preferably, the calculation of the predicted trail tracking error and the predicted transverse trail tracking error is as follows:

[0046]

[0047] in: , The x and y coordinates of the task path point at the j-th future time step.

[0048] Compared with the prior art, the present invention has the following beneficial effects:

[0049] (1) Adaptive dynamic model: The adaptive dynamic model proposed in this invention can autonomously complete the calibration using limited running data, thereby significantly improving the prediction accuracy, effectively bridging the difference between simulation and physical deployment, and eliminating the need for a large amount of manual parameter tuning.

[0050] (2) Predictive optimization of rudder controller: This invention proposes a novel control framework that explicitly considers rudder speed constraints (e.g., ±6° / s) during trajectory tracking optimization. Through a rudder control strategy that deeply integrates prediction and guidance laws, the overshoot and delay problems caused by large inertia are effectively alleviated. Attached Figure Description

[0051] Figure 1 This is a schematic diagram of the motion coordinate system of the unmanned surface vessel of the present invention;

[0052] Figure 2 This is a diagram of the overall architecture of the present invention;

[0053] Figure 3 This is a visual schematic diagram of the control process of the present invention;

[0054] Figure 4 This is a diagram showing the results of validating the adaptive dynamics model of this invention.

[0055] Figure 5 This is a preview diagram of the influence of time domain length on the present invention;

[0056] Figure 6 This is a physical image of the unmanned surface vessel platform used in the sea trial of this invention.

[0057] Figure 7This is a diagram illustrating the sea trial linear trajectory tracking performance of the present invention.

[0058] Figure 8 This is a diagram showing the circular trajectory tracking performance of the present invention during sea trials;

[0059] Figure 9 This is a diagram illustrating the performance of the invention in tracking complex trajectories during sea trials. Detailed Implementation

[0060] The following is in conjunction with the appendix Figure 1-9 The technical solution of the present invention will be described in detail below.

[0061] This invention addresses the key challenge of trajectory tracking control for unmanned surface vessels (USVs) under rudder speed constraints (e.g., ±6° / s) caused by mechanical limitations of the rudder mechanism. Due to the large inertia and steering delay of ships, rudder speed constraints often lead to significant overshoot and trajectory instability. To overcome these shortcomings, this invention proposes a novel control framework that integrates a predictive-optimized rudder controller with an adaptive dynamics model. The proposed method employs a predictive mechanism to forecast trajectory requirements in future time periods and proactively optimizes rudder commands under rudder speed constraints, thereby significantly suppressing overshoot while maintaining high tracking accuracy. The performance of the rudder controller is enhanced by an adaptive dynamics model, which can be calibrated online using limited measured data to ensure the reliability of the predictions and facilitate a smooth transition from simulation environments to physical platforms. Sea trials covering challenging scenarios such as high inertial loads and complex paths have confirmed the effectiveness of the proposed method, demonstrating its excellent robustness and tracking accuracy under stringent rudder speed constraints.

[0062] 1. Unmanned Surface Vessel Modeling and Coordinate System Construction:

[0063] like Figure 1 This invention employs two reference coordinate systems: an inertial coordinate system and an inertial coordinate system. (That (The plane is horizontal) and the body coordinate system fixed to the hull. (The origin is located amidships). The body coordinate system's x-axis points towards the bow, and the y-axis points towards starboard. The hull's heading is determined by the yaw angle. It indicates that its measurement benchmark is Axis to The rotation angle of the shaft. The sway and roll velocities are denoted as u and v, respectively, and the yaw rate is r. The control input is the rudder angle. The hydrodynamic drift angle is defined as... The combined velocity is The center of gravity G is located in the body coordinate system. Its corresponding lateral velocity is expressed as

[0064] The kinematic model is described by equation (1):

[0065] (1)

[0066] 2. High-stability guidance law:

[0067] Tracking error Horizontal tracking error and desired heading angle and expected oscillation speed Defined by equations (2) and (3) respectively:

[0068] (2)

[0069] (3)

[0070] in, Indicates the tangent angle of the task path. , Let x and y represent the first derivatives of the x and y coordinates of the task path point at the current time step t with respect to time, respectively; the resultant velocity of the task path is defined as... . The desired velocity. For adaptive forward sight distance; Forward sight distance, >0; k is the control gain, k > 0; All of these are adjustable hyperparameters for adjusting adaptive forward gaze distance. >0, >0 and >0. is a natural constant and the base of the natural exponential function. The weighting coefficients represent the trade-off between tracking performance and heading smoothness. The mission path referred to in this invention is the global planning result of the unmanned surface vessel (USV) completing a specific task; the expected path is the specific trajectory generated based on the mission path and real-time environmental information; and the actual path is the trajectory of the USV during execution.

[0071] Due to mechanical structure and safety limitations, practical unmanned surface vessel (USV) steering systems face strict rate constraints. These constraints, coupled with the ship's large inertia and inherent steering delay, can easily lead to overshoot and trajectory instability. For example... Figure 2As shown, the rudder speed limit in the light blue area (left side) is ±6° / s. Controllers lacking explicit rate compensation exhibit significant oscillations and overshoot (red dashed trajectory). This invention assumes that the speed only undergoes minor adjustments or remains constant, employing an incremental PID controller for propeller speed regulation, with a primary focus on rudder control optimization. This invention proposes a predictive optimization rudder controller that uses a guidance law to predict future steering needs in the preview time domain, optimizing rudder angle commands to achieve trajectory tracking while satisfying physical rate constraints. Figure 2 In the white area on the left, state t The upper and lower semicircles at the point represent the expected and actual heading changes, respectively. To accurately describe the servo motor dynamics, a self-tuning dynamic model is established. This model can automatically calibrate parameters using limited historical operating data, effectively promoting the transfer of simulation to the actual physical model. Figure 2 As shown in the light blue area on the right, the controller outputs the rudder angle in each control cycle. With propeller speed The command instructs the submarine to acquire new state variables via the strapdown inertial navigation system (SINS) after one propulsion cycle and initiate the next control loop. This mechanism generates a smooth, high-precision trajectory (illustrated by the blue dashed line). Finally, extensive sea trials demonstrated the robustness and superiority of the proposed method. Detailed designs for the rudder controller, dynamic model, and experimental evaluation are provided in Sections 3, 4, and 6, respectively.

[0072] 3. Adaptive dynamic model:

[0073] The proposed model is based on Euler's rotation theorem and achieves a compact representation of rudder dynamics through parameter integration:

[0074] (4)

[0075] in, , and These represent the propulsion torque, damping torque, and ambient noise torque, respectively. Moment of inertia. Inclusion and quality Related uncertainty factors :

[0076] (5)

[0077] in Represents the set of positive real numbers, i.e. ;

[0078] propulsion torque It integrates control input and hydrodynamic effects:

[0079] (6)

[0080] in, The propulsion coefficient is represented; the damping effect is expressed using a factor incorporating hydrodynamic damping. The model is based on the secondary resistance model:

[0081] (7)

[0082] in, The density of water (taken as ) ), Indicates the length of the unmanned surface vessel;

[0083] Environmental disturbances are modeled as additive noise terms:

[0084] (8)

[0085] The dynamic model can integrate the unknown coefficients into two key parameters. and ( Perform parameterized characterization:

[0086] (9)

[0087] in: This represents the yaw acceleration function, which is determined by the propeller speed, rudder angle, and yaw rate. This represents the normalized environmental noise torque.

[0088] Therefore, the rudder dynamics model can be transformed into a linear regression form:

[0089] (10)

[0090] in: Represents the target variable containing noise. Indicates the independent variable; This represents the second-normalized environmental noise torque.

[0091] During the online parameter estimation process, the yaw angle acceleration of the USV during operation is collected in real time. propeller speed and rudder angle Measured values. According to the definition in equation (10), these measured values ​​can be directly calculated. and Combining the simplified linear model shown in equation (11), the parameters can be identified using the least squares method described in equation (12). and .

[0092] (11)

[0093] (12)

[0094] in: This represents the target variable calculated using the operational data of the i-th unmanned surface vessel. and Let represent the independent variable calculated using the operational data of the i-th unmanned surface vessel.

[0095] Among them, noise item As a residual term, it functions similarly to a regularization term in machine learning, effectively preventing overfitting and improving generalization ability during the optimization process of equation (12). This mechanism ensures that the estimated parameters... and It has strong robustness and can achieve [the desired result] under most operating conditions. Stable predictions.

[0096] 4. Predict and optimize the rudder controller:

[0097] The primary objective of a ship's heading control system is to track and maintain the desired course through rudder angle adjustments. However, due to mechanical constraints and safety requirements, the steering rate of the rudder is strictly limited. Furthermore, ships exhibit significant inertia and time lag, causing the effects of control actions to persist for a considerable period. To suppress rudder angle overcompensation and improve tracking performance, this invention proposes a predictive optimization rudder controller. Figure 3 As shown in the gray area, the system predicts the ship's position under feasible rudder speed constraints at each time step. ) next future The turning behavior of the step; at the same time, such as Figure 3 The light yellow area shows the calculation of the desired steering trajectory. The optimal rudder angle command is selected by minimizing the difference between the predicted and desired trajectories. This process is formalized as the following optimization problem:

[0098] (13)

[0099] In the optimization framework, This is the optimal rudder angle control value output in the current control cycle. This represents the predicted rudder angle at the j-th future time step. This indicates that the predicted rudder angle at the j-th future time step satisfies the servo steering rate constraint. , This represents the predicted yaw rate at the j-th future time step. This is the expected yaw rate at the j-th future time step. This represents the maximum time step in the future preview time domain. The calculation methods for both will be explained in detail below. The prediction is based on the rudder dynamics model under the constant speed assumption. Relying on equation (9), the system dynamics evolve in the following discretized form:

[0100] (14)

[0101] in, Let be the predicted oscillation velocity at the j-th future time step; Let be the predicted sway velocity at the j-th future time step; Let be the predicted yaw rate at the j-th future time step;

[0102] When j=1: This indicates the propeller speed at the current time step. , Indicates the rudder angle at the current time step. , This represents the yaw rate at the current time step; and These represent the sway velocity and oscillation velocity at the current time step, respectively.

[0103] When j=2,...,N p hour: Let represent the predicted propeller speed at the (j-1)th future time step, and ; This represents the predicted rudder angle at the (j-1)th future time step. The value is based on the rudder angle at the current time step. and servo steering rate constraints ,and The values ​​are equal; This represents the predicted yaw rate at the (j-1)th future time step;

[0104] Based on the ship kinematics and dynamics relationship established by equations (1) and (14), the motion parameters during the uniformly accelerated turning maneuver evolve according to the following discrete-time kinematic equations:

[0105] (15)

[0106] (16)

[0107] in, Let j be the predicted yaw angle for the j-th future time step. For ship kinematics model, For the sampling time of the control system, , These are the predicted x-coordinate and predicted y-coordinate for the j-th future time step, respectively.

[0108] To mitigate the inherent cumulative error in long-term ship yaw rate prediction, an exponentially weighted loss term is introduced. This formula prioritizes meeting near-term steering control requirements while gradually attenuating the impact of long-term predictions.

[0109] (17)

[0110] in: As a discount factor, By integrating equations (3), (14), and (16), a highly stable guidance law can be used to predict the desired trajectory of the ship. As shown in equation (18), this method calculates the desired heading angle at the j-th future time step based on the ship's kinematics and dynamics. With expected oscillation speed This ensures robust trajectory tracking and effective disturbance suppression. The corresponding expected yaw rate for each preview step is then derived from equation (19):

[0111] (18)

[0112] (19)

[0113] in, This is the guidance law function, used to determine the position coordinates. With speed state Calculate the desired heading angle and desired sway velocity under the corresponding working conditions;

[0114] The proposed predictive optimization framework can generate optimal rudder commands that simultaneously achieve accurate trajectory tracking, effective disturbance suppression, and mitigation of overshoot caused by rate constraints in large-inertia marine systems. By explicitly incorporating ship dynamics constraints, this control strategy achieves... Maximize navigation performance within the preview time domain.

[0115] 5. Incremental PID controller:

[0116] (20)

[0117] in: The expected oscillation speed at the current time step t With oscillation speed The difference, This is the optimal propeller speed control value output in the current control cycle. , These are the proportional coefficient and the integral coefficient, respectively. This represents the propeller speed control value at the previous time step t.

[0118] 6. Experimental verification:

[0119] This section evaluates the effectiveness, robustness, and engineering applicability of the proposed method through a systematically designed experiment. The experiment consists of two core parts: an ablation experiment analyzing the contributions of each key module, and a sea trial in open water.

[0120] 6.1. Ablation experiment:

[0121] To evaluate the independent contributions of the adaptive dynamics model and the predictive optimization rudder controller, ablation experiments were conducted. By comparing the performance under different hyperparameter values, the mechanism by which each module improves tracking accuracy and suppresses overshoot under rudder speed constraints was verified.

[0122] 6.1.1. Adaptive dynamic model verification:

[0123] Experimental verification, through systematic data collection and analysis, confirmed the effectiveness of the proposed adaptive dynamics model. Figure 4 As shown, a comprehensive simulation was performed within the full operating range of propeller speed ([0 RPS, 50 RPS]) and the full range of rudder angle ([-30°, 30°]). Parameter identification adopted a structured test scheme: based on discrete speed samples [10, 20, 30, 40, 50] RPS, the rudder angle was sequentially switched every 2 seconds according to the pattern [0, -6, -12,..., -24, -30, -24,..., 24, 30, 24,..., 12, 6]. The generated trajectory data ( Figure 4 (a) Supports efficient hyperparameter estimation based on Equation (12) and The 2-second short sampling interval design ensures the feasibility of real-world ship applications.

[0124] Verification results show a significant performance improvement. At a constant speed of 40 RPS ( Figure 4 In (b), the adaptive model achieves high-precision trajectory prediction, significantly outperforming the untuned baseline model. Quantitative analysis further demonstrates that ( Figure 4 (c) The model consistently maintains excellent accuracy: the cumulative position error is less than 0.1 meters within 30 seconds and less than 0.5 meters within 60 seconds. Although there are minor errors due to transient dynamics during the initial positioning and heading conversion phases, the overall accuracy fully meets the operational requirements of the predictive optimization algorithm.

[0125] Figure 5 shows the preview time domain length (in Equation (13)). The results of the ablation study on the impact of rudder speed on trajectory tracking performance. Under the constraint of ±6° / s rudder speed, a distinct behavioral pattern was observed: shorter preview time (1≤ ≤4) Significant overshoot and oscillation phenomena occur because the system cannot adequately compensate for the ship's large inertia. As shown in Table 1, when the preview time domain is extended to... The system achieves stable tracking when the time domain is ≥ 5. While a longer time domain slightly increases the computational load, it generates smoother trajectories while keeping the error within acceptable limits. Based on this, the system is selected... = 5 is the optimal compromise between tracking accuracy and computational efficiency, achieving an effective balance between predictive control requirements and practical engineering constraints.

[0126] Table 1. Error analysis of the effect of time domain length in preview (after 50 seconds)

[0127]

[0128] 6.2. Sea Trial Results and Analysis:

[0129] like Figure 6 As shown, the sea trial used an experimental vessel 4.5 meters long and 2 meters wide, with a displacement of approximately 2 tons. The vessel is equipped with twin propulsion systems, achieving a maximum speed of approximately 5 knots at a maximum speed of 800 RPM. Due to mechanical limitations, the rudder angle is limited to ±30° and the maximum rudder speed is ±6° / s. These stringent actuator constraints provide a real-world platform for verifying the effectiveness and practicality of the proposed algorithm.

[0130] Tracking performance was evaluated for three types of trajectories: straight, circular, and complex, under challenging environmental conditions including unknown ocean currents and wind speeds up to 15 m / s. All reference trajectories used a constant desired speed of 1.5 m / s. Initial navigation data were acquired using the same sampling protocol as the ablation experiment (Section 6.1.1) to support parameter self-tuning of the rudder control system. After initialization, comprehensive trajectory tracking experiments were conducted under different route scenarios to evaluate robustness and accuracy under real disturbances.

[0131] 6.2.1. Results of the straight-line trajectory sea trial:

[0132] In the straight-line trajectory tracking experiment, the ship started from a near-stationary initial state with a certain offset distance from the reference path, and the entire test lasted 310 seconds. Figure 7 As shown, the algorithm maintained strong tracking performance during sea trials. Specifically, Figure 7 (a) shows that the actual trajectory is smooth and closely follows the desired path; Figure 7 Figure (b) shows that the heading angle remains stable over time, and the rudder angle is smoothly adjusted within the ±6° / s rate constraint, providing continuous and effective control even under time-varying environmental disturbances.

[0133] like Figure 7As shown in (c), both the tracking error and the transverse tracking error converged rapidly and smoothly. The controller's strong anti-disturbance capability ensured that the ship maintained high-precision tracking throughout the experiment. According to the data in Table 2, after 50 seconds, the average tracking error dropped to 0.513 meters, and the average rudder angle remained at a low level of 1.2°. These results confirm that the proposed strategy not only minimizes mechanical wear by smoothing control actions, but also achieves reliable and accurate trajectory tracking performance under actual sea conditions.

[0134] Table 2. Analysis of tracking error during sea trials (after 50 seconds)

[0135]

[0136] 6.2.2. Results of the circular trajectory sea trial:

[0137] In the circular trajectory tracking scenario, the ship begins its journey from a near-stationary state with a significant initial offset relative to the tangent of the reference circle. This initial condition causes a significant heading error, and the 307-second experiment effectively verifies the algorithm's convergence capability. The circular path requires the ship to maintain continuous steering motion under environmental disturbances, posing a unique challenge. Figure 8 As shown, the proposed algorithm demonstrates superior performance in this challenging scenario: Figure 8 (a) shows that the actual trajectory precisely coincides with the reference circle, forming a smooth and continuous path without significant overshoot or lag; Figure 8 Figure (b) reveals the internal dynamics of the maneuver—the bow angle changes continuously and smoothly at a near-constant rate, while the rudder angle alternates smoothly and periodically within a rate constraint of ±6 ° / s to maintain uniform circular motion. This controlled actuation effectively compensates for the influence of time-varying environmental factors during the turning process.

[0138] like Figure 8 As shown in (c), the tracking and lateral tracking errors converge rapidly and remain stable during circular navigation. According to the data in Table 2, despite continuous steering requirements and environmental disturbances, the algorithm still achieves an average tracking error of 0.512 meters after 50 seconds, with an average rudder angle of only 1.4°. The proposed strategy not only provides accurate trajectory tracking in continuous steering scenarios but also maintains smooth and efficient control under demanding steering conditions, as fully demonstrated by the experimental results.

[0139] 6.2.3. Results of complex trajectory sea trials:

[0140] The complex trajectory tracking experiment lasted 481 seconds, with the ship starting from a near-stationary state and deviating somewhat from the reference path. This scenario was used to evaluate the algorithm's ability to handle complex maneuvering challenges, including straight sections, curved sections, and course changes. Figure 9 As shown, the proposed method maintains excellent tracking performance throughout the complex sea trial: Figure 9(a) shows that the actual trajectory closely follows the complex reference path, accurately tracks curve segments and straight line segments, and achieves smooth transitions between different path geometries; Figure 9 (b) illustrates the state response during this challenging operation—the bow angle adjusts in real time according to changes in path curvature, while the rudder angle is smoothly adjusted within a rate constraint of ±6 ° / s, providing stable and continuous control under varying environmental conditions and complex path requirements.

[0141] like Figure 9 As shown in (c), the along-track and lateral-track errors remained bounded and stable throughout the experiment, exhibiting rapid convergence and sustained accuracy despite changes in path characteristics. The controller effectively managed the transition between different path segments while compensating for external disturbances. According to the data in Table 2, the algorithm achieved an average tracking error of 0.536 meters after 50 seconds, with an average rudder angle of 1.2°. These results validate that the proposed strategy provides robust and accurate trajectory tracking performance even under the most demanding maritime conditions involving complex trajectories and environmental disturbances.

[0142] 7. Conclusion:

[0143] This invention proposes a predictive optimization rudder control framework integrating an adaptive dynamics model to address the core challenge of trajectory tracking for unmanned surface vessels (USVs) under strict rudder speed constraints. By integrating three key technologies: a predictive optimization control strategy that actively compensates for steering limitations, an adaptive dynamics model that can reduce the gap between simulation and reality using only a small amount of measured data, and an optimization mechanism with embedded guidance laws, the proposed method effectively suppresses overshoot and instability caused by large inertia and execution delays in ships. Sea trials on straight, circular, and complex trajectories demonstrate that this method significantly improves tracking accuracy, reduces mechanical wear, and maintains robust performance under environmental disturbances and rudder speed constraints, providing a practical and efficient solution for the actual deployment of USVs.

[0144] The above are preferred embodiments of the present invention. Any changes made to the technical solution of the present invention that do not exceed the scope of the technical solution of the present invention shall fall within the protection scope of the present invention.

Claims

1. A trajectory tracking method for unmanned surface vessels with constrained rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model, characterized in that, Unmanned surface vessel (USV) trajectory tracking is achieved by controlling the rudder angle and propeller speed, and the rudder angle command is optimized by the predictive optimization rudder controller. The predictive optimization rudder controller calculates the predicted and desired steering trajectories of the unmanned surface vessel in the future preview time domain under the condition of satisfying the rudder speed constraint, based on the ship kinematics model, guidance law and adaptive rudder dynamics model in each control cycle. The optimal rudder angle control quantity is determined by minimizing the deviation between the predicted and desired steering trajectories. The servo dynamics model is used to predict yaw acceleration based on propeller speed, rudder angle, and yaw rate, and the model parameters are adaptively calibrated using historical operating data. The specific dynamic model of the servo motor is as follows: ， in: Indicates yaw rate, Indicates yaw acceleration. Indicates the propeller speed. represents the rudder angle, This represents the yaw acceleration function, which is determined by the propeller speed, rudder angle, and yaw rate. This indicates the density of water. Indicates the length of the unmanned surface vessel. Indicates the mass of the unmanned surface vessel. Indicates that it follows a normal distribution Environmental noise torque, The standard deviation of the distribution Represents the moment of inertia. and Indicates the parameters of the servo motor dynamics model; The optimal rudder angle control quantity is determined by minimizing the deviation between the predicted steering trajectory and the desired steering trajectory, and the specific expression is as follows: in: This is the optimal rudder angle control value output in the current control cycle. This represents the predicted rudder angle at the j-th future time step. This indicates that the predicted rudder angle at the j-th future time step satisfies the servo steering rate constraint. , This represents the predicted yaw rate at the j-th future time step. Let represent the expected yaw rate at the j-th future time step. This indicates the maximum time step in the future preview time domain.

2. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 1, characterized in that, Under near-constant speed conditions, an incremental PID controller is used to regulate the propeller speed control. in: The expected oscillation speed at the current time step t With oscillation speed The difference, This is the optimal propeller speed control value output in the current control cycle. , These are the proportional coefficient and the integral coefficient, respectively. This represents the propeller speed control value at the previous time step t.

3. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 1, characterized in that, Adaptive calibration of the servo motor dynamics model parameters is performed using historical operating data, as detailed below: Transform the servo dynamics model into a linear regression form: in: Represents the target variable containing noise. Indicates the independent variable; Collect yaw acceleration data of N groups of unmanned surface vessels during operation. propeller speed and rudder angle And calculate the independent variable corresponding to each set of data. and target variable : Determining parameters using the least squares method and : in: This represents the target variable calculated using the operational data of the i-th unmanned surface vessel. and Let represent the independent variable calculated using the operational data of the i-th unmanned surface vessel.

4. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 1, characterized in that, The calculation of predicted yaw rate is based on a servo dynamics model under the assumption of constant speed: in: Let be the predicted oscillation velocity at the j-th future time step; Let be the predicted sway velocity at the j-th future time step; Let be the predicted yaw rate at the j-th future time step; When j=1: This indicates the propeller speed at the current time step. , Indicates the rudder angle at the current time step. , This represents the yaw rate at the current time step; and These represent the sway velocity and oscillation velocity at the current time step, respectively. When j=2,...,N p hour: This represents the predicted propeller speed at the (j-1)th future time step; This represents the predicted rudder angle at the (j-1)th future time step. The value is based on the rudder angle at the current time step. and servo steering rate constraints ,and The values ​​are equal; This represents the predicted yaw rate at the (j-1)th future time step; Then the predicted yaw rate at the j-th future time step The calculation is as follows: in: As a discount factor, .

5. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 4, characterized in that, The calculation of the desired yaw rate is as follows: in, Let be the expected heading angle at the j-th future time step; Let yaw angle be the predicted yaw angle for the j-th future time step; For the sampling time of the control system; Let be the expected oscillation velocity at the j-th future time step; This is the guidance law function, used to calculate the desired heading angle and desired sway velocity under the corresponding working conditions based on the position coordinates and velocity state; , These are the predicted x-coordinate and predicted y-coordinate for the j-th future time step, respectively. This is a model of ship kinematics.

6. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 5, is characterized in that, The calculations for the desired heading angle and desired sway velocity are as follows: in: Let the tangent angle of the task path be the j-th future time step. Let $\frac{j}{j}$ be the predicted transverse track tracking error at the $j$-th future time step. To adapt to the forward sight distance, For hydrodynamic drift angle, Let the expected summation velocity be at the j-th future time step. Let the total speed of the task path be the speed at the j-th future time step. , Let be the lateral velocity of the task path at the j-th future time step. Let the longitudinal velocity of the task path be the velocity at the j-th future time step. To control the gain, Let $\frac{j}{j}$ be the predicted tracking error along the path at the $j$-th future time step. Forward sight distance, All of these are adjustable hyperparameters for adjusting adaptive forward gaze distance. is the base of the natural exponential function. The weighting coefficients are used to characterize the trade-off between tracking performance and heading smoothness.

7. The method for tracking the trajectory of an unmanned surface vessel with limited rudder speed based on a predictive optimization rudder controller and an adaptive dynamics model according to claim 6, characterized in that, The calculation of the predicted trail tracking error and the predicted transverse trail tracking error is as follows: in: , The x and y coordinates of the task path point at the j-th future time step.

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