A Path Tracking Method for a Turtle-Inspired Amphibious Robot Based on Model Predictive Control

Abstract

This invention discloses a path tracking method for a turtle-inspired amphibious robot based on model predictive control, comprising the following steps: First, a simplified multi-rigid-body dynamics model of the turtle-inspired amphibious robot with hydrodynamic coefficients is established, and the control dimension is simplified by combining biological motion characteristics; hydrodynamic parameters are quantified through computational fluid dynamics simulation, combined with steady-state towing tests, rotating arm tests, and analysis of unsteady planar motion mechanisms; a robot model and underwater environment are constructed in a dynamics simulator; a model predictive control-fuzzy logic cascaded controller is designed, where the upper layer generates reference forces and torques through rolling time-domain optimization, and the lower layer maps them to actuator commands based on fuzzy rules; finally, the method is verified in a simulation environment to evaluate the robustness and tracking accuracy of the controller. This method effectively copes with nonlinear hydrodynamic effects, parameter uncertainties, and interference from unmodeled environments. Validated through U-shaped and figure-eight path tests, it demonstrates high control accuracy and strong robustness.

Description

Technical Field

[0001] This invention relates to amphibious robot technology, and in particular to a path tracking method for a turtle-inspired amphibious robot based on model predictive control. Background Technology

[0002] In aquatic environments, turtle species demonstrate remarkable complex path-tracking abilities thanks to their unique hydrodynamic morphology and flexible three-dimensional locomotion mechanisms. This survival strategy not only helps them evade predators and forage efficiently, but also enables long-distance migration. This biological intelligence has inspired the development of biomimetic robotic systems, with high-precision path-tracking control technology becoming a key challenge for achieving autonomous underwater operations.

[0003] Precise control relies on establishing accurate hydrodynamic models. Even small deviations in hydrodynamic coefficients can lead to the accumulation of nonlinear errors in the kinematic equations, significantly weakening the robustness of path-following control. While traditional modeling methods using empirical formulas offer advantages in computational efficiency, the unique carapace curvature of the sea turtle robot and the resulting complex vortex structure deviate from existing hydrodynamic coefficient databases, making them unsuitable for its modeling requirements. Experimental fluid dynamics methods, such as towed tank experiments, can obtain high-fidelity data, but their practical application is limited by equipment dependence and high costs. In contrast, computational fluid dynamics (CFD) methods, through numerical solutions to the Navier-Stokes equations, enable parametric analysis of hydrodynamic responses under different operating conditions, thereby facilitating the establishment of high-fidelity hydrodynamic models.

[0004] The scientific community has invested significant effort in developing robust control strategies to address complex fluid dynamics and parameter uncertainties in aquatic environments. Widely used PID controllers suffer from parameter calibration challenges, making it difficult to guarantee accuracy. Fuzzy logic controllers (FLCs) perform better in nonlinear systems by handling uncertainty, but their effectiveness still relies on experience. Sliding mode controllers, while insensitive to environmental disturbances, are limited by oscillatory effects when switching inputs. In contrast, the Model Predictive Control (MPC) framework, through backsight optimization, explicitly integrates kinematic constraints, effectively addressing issues such as parameter uncertainty, operator inexperience, and control output fluctuations.

[0005] It should be noted that the information disclosed in the background section above is only for understanding the background of the present invention, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0006] The main objective of this invention is to overcome the deficiencies in the aforementioned background technology and provide a path tracking method for a turtle-like amphibious robot based on model predictive control.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A path tracking method for a turtle-inspired amphibious robot based on model predictive control includes the following steps:

[0009] S1. Establish a multi-rigid-body dynamics model of the turtle-like amphibious robot, derive simplified dynamic equations including hydrodynamic coefficients, and simplify the control dimension based on biological motion characteristics.

[0010] S2. Through computational fluid dynamics simulation, combined with steady-state towing test, rotating arm test and analysis of unsteady planar motion mechanism, the hydrodynamic parameters are quantified.

[0011] S3. Construct robot models and underwater environments in a dynamics simulator;

[0012] S4. Design Model Predictive Control - Fuzzy Logic Cascade Controller:

[0013] The upper-level model predictive controller generates reference forces and torques through rolling time-domain optimization;

[0014] The lower-level fuzzy logic controller adaptively maps reference forces and torques into actuator commands based on fuzzy rules;

[0015] S5. Conduct path tracking control verification in a simulation environment to evaluate the robustness of the controller and the tracking accuracy.

[0016] Furthermore, the establishment of the simplified dynamic model in step S1 specifically includes:

[0017] By utilizing the robot's dual symmetry about the longitudinal-lateral plane and the longitudinal-vertical plane, the coupling effect of longitudinal, lateral and yaw motions is ignored;

[0018] Based on biomimetic motion characteristics, the active control dimension is limited to longitudinal propulsion and yaw moment;

[0019] The additional mass effect, linear damping term, and nonlinear fluid resistance term are integrated into the dynamic equation.

[0020] Furthermore, the quantification of hydrodynamic parameters in step S2 specifically includes:

[0021] The velocity-related damping factor was identified by dragging tests, and the drag data under different flow velocities were correlated and polynomial coefficients were fitted.

[0022] Rotational motion resistance was identified through a rotating arm test, and the relationship between yaw torque and angular velocity was fitted.

[0023] Inertial effects were identified through analysis of unsteady planar motion mechanisms, and acceleration-related hydrodynamic coefficients were extracted using Fourier analysis.

[0024] The transient flow field is analyzed using superimposed mesh technology, and data transmission between meshes is achieved through multi-domain computation.

[0025] Furthermore, step S3 specifically includes:

[0026] The robot hardware architecture is designed using a biomimetic propulsion system, which includes a central waterproof cabin and multiple three-degree-of-freedom fin-like limbs.

[0027] Importing a simplified robot model into the dynamics simulator and constructing an underwater environment reduces distortion in parametric simulations.

[0028] Furthermore, the input generation for the model prediction controller in step S4 specifically includes:

[0029] The desired position and heading angle are obtained using the line-of-sight method;

[0030] Decouple the path tracking problem into a thrust channel and a torque channel:

[0031] The thrust channel controls the longitudinal velocity by adjusting the oscillation frequency of the flippers;

[0032] The torque channel achieves directional adjustment by coordinating the amplitude difference of the fins and the rudder angle of the tail fin.

[0033] Furthermore, the optimization process of the model prediction controller in step S4 specifically includes:

[0034] Solve the finite-time optimal control problem at discrete time points and generate the optimal control sequence within the prediction step size;

[0035] A moving view strategy is adopted, and the first element of the sequence is used as the current control action output.

[0036] Furthermore, the design of the fuzzy logic controller in step S4 specifically includes:

[0037] Construct a fuzzy rule table to define the mapping relationship between reference force / torque error and actuator command;

[0038] By using asymmetric response surfaces to handle nonlinear and transient conditions, dynamic response and stability are balanced.

[0039] Furthermore, the cooperative mechanism of the model predictive control-fuzzy logic cascade controller specifically includes:

[0040] The upper-level model predictive controller explicitly integrates kinematic constraints and optimizes reference forces / torques to address parameter uncertainties;

[0041] The lower-level fuzzy logic controller compensates for unmodeled fluid dynamic coupling and environmental disturbances through adaptive mapping.

[0042] Furthermore, the path tracking verification in step S5 specifically includes:

[0043] Design a U-shaped path containing straight segments and arcs, and evaluate straight-line tracking and steering capabilities;

[0044] Design a figure-eight path with bidirectional curvature variation to evaluate lateral error suppression and transient stability;

[0045] A random hydrodynamic disturbance model was applied to test the controller's robustness against disturbances.

[0046] A computer program product includes a computer program that, when executed by a processor, implements the model predictive control-based path tracking method for a turtle-like amphibious robot.

[0047] The present invention has the following beneficial effects:

[0048] This invention proposes a path tracking method for a turtle-inspired amphibious robot based on model predictive control. A high-fidelity hydrodynamic model is established through parameterization driven by computational fluid dynamics (CFD), and a hierarchical control framework integrating model predictive control (MPC) and fuzzy logic controller (FLC) is proposed. This method combines the stability of the high-fidelity hydrodynamic model obtained by CFD numerical calculation with the MPC-FLC cascade control, effectively solving problems such as nonlinear hydrodynamic interactions, coefficient uncertainties, and unmodeled environmental disturbances.

[0049] The turtle-inspired amphibious robot platform established based on this invention possesses reliable characteristics. As a biomimetic prototype, the turtle's propulsion and steering can be decoupled during movement, providing a stable foundation for the robot's path tracking algorithm. Simultaneously, a simplified dynamic model based on a multi-rigid-body dynamics model parameterizes the influence of the aquatic environment on the robot as hydrodynamic coefficients, and simplifies the control dimension through mechanical structure analysis and biomimetic prototype analysis. Furthermore, by integrating steady-state towing tests, rotating arm simulations, and PMM analysis, CFD simulations can be performed using STAR-CCM+ software to obtain the corresponding hydrodynamic coefficients, establishing a high-fidelity hydrodynamic model. A hydrodynamic environment model suitable for the turtle-inspired robot was also established in the dynamics simulator Webots, reducing model distortion problems in ordinary parametric simulation processes and enabling a more systematic evaluation of the operational performance of the constructed path tracking framework.

[0050] The model predictive control framework of this invention explicitly integrates kinematic constraints through back-look optimization, which is more effective than traditional PID control strategies and sliding mode controllers in addressing issues such as parameter uncertainty, operator inexperience, and control output oscillations. Furthermore, it proposes a two-layer control strategy combining model predictive control and fuzzy logic control. The upper-layer MPC optimizes reference forces and torques using a rolling time-domain optimization method, while the lower-layer FLC adaptively maps these reference signals into actuator commands based on fuzzy rules, effectively addressing nonlinear hydrodynamic interactions, parameter uncertainties, and unmodeled environmental disturbances. In addition, the proposed control framework has undergone simulation comparisons in U-shaped and figure-eight path tracking, achieving higher control accuracy and more agile robot movement.

[0051] Other beneficial effects of the embodiments of the present invention will be further described below. Attached Figure Description

[0052] Figure 1 This is the coordinate system configuration of the amphibious robot system in this embodiment of the invention.

[0053] Figure 2 This is the dynamic model and joint configuration of the amphibious robot system in this embodiment of the invention.

[0054] Figure 3 The following is a fluid dynamics characteristic analysis (based on CFD simulation) of the biomimetic turtle robot in this embodiment of the invention: (a) velocity distribution of the flow field under an inflow velocity of 0.6 m / s in the oscillating direction; (b) velocity distribution around the biomimetic turtle at a computational domain radius of 3.3 m under an inflow velocity of 0.5 m / s; (c) velocity distribution of the flow field during pure pitching motion at 1 Hz; (d) relationship between pitching drag and flow velocity; (e) relationship between yaw rate and yaw torque; (f) time-dependent drag and Fourier fitting results during pure pitching motion at 1 Hz.

[0055] Figure 4 This is a snapshot of a turtle-inspired robot in the Webots dynamics simulator of this invention.

[0056] Figure 5 This is a block diagram of a turtle-inspired robot path tracking control in an embodiment of the present invention.

[0057] Figure 6 The input and output rules of the fuzzy inference system in this embodiment of the invention are: (a) Input and With output (a) Relationships between them; (b) Input and With output The relationship between them.

[0058] Figure 7 The performance evaluation of the path tracking controller in this embodiment of the invention includes: (a) visualization of the comparison between the U-shaped path and the actual motion; and (b) the time evolution of the lateral path tracking deviation.

[0059] Figure 8 This is the signal output of the MPC-FLC control strategy in the U-shaped trajectory tracking of this invention embodiment.

[0060] Figure 9 The performance evaluation of the path tracking controller in this embodiment of the invention includes: (a) a visual comparison of the figure-eight path and the actual motion; and (b) the time evolution of the lateral path tracking deviation.

[0061] Figure 10 This refers to the signal output of the MPC-FLC control strategy in figure-eight trajectory tracking in this embodiment of the invention.

[0062] Figure 11 This is a flowchart of the path tracking method for a turtle-like amphibious robot based on model predictive control according to the present invention. Detailed Implementation

[0063] The embodiments of the present invention will be described in detail below. It should be emphasized that the following description is merely exemplary and not intended to limit the scope and application of the present invention.

[0064] This invention aims to address the problems of nonlinear hydrodynamic interactions, coefficient uncertainties, and unmodeled environmental interference in path tracking of turtle-inspired amphibious robots. It proposes a path tracking method for turtle-inspired amphibious robots based on model predictive control (MPC). A high-fidelity hydrodynamic model is established through CFD-driven parameterization, and a hierarchical control framework integrating MPC and fuzzy logic control (FLC) is constructed. This method relies on a reliable platform for decoupling propulsion and steering in turtle-inspired robots. Combined with a simplified dynamic model, accurate hydrodynamic parameter acquisition, and the Webots simulation environment, and validated through U-shaped and figure-eight path tests, it achieves higher control accuracy and motion agility, effectively addressing issues such as parameter uncertainties.

[0065] See Figure 11 This invention provides a path tracking method for a turtle-inspired amphibious robot based on model predictive control, comprising the following steps:

[0066] Step S1: Establish a multi-rigid-body dynamics model of the turtle-inspired amphibious robot, derive simplified dynamic equations including hydrodynamic coefficients, and simplify the control dimension based on biological motion characteristics.

[0067] In some embodiments, the establishment of the simplified dynamic model in step S1 specifically includes: utilizing the robot's dual symmetry about the longitudinal-lateral plane and the longitudinal-vertical plane, ignoring the coupling effect of longitudinal, lateral and yaw motions; based on biomimetic motion characteristics, limiting the active control dimension to longitudinal propulsion and yaw torque; and integrating the additional mass effect, linear damping term and nonlinear fluid resistance term into the dynamic equation.

[0068] Step S2: Quantify hydrodynamic parameters by computational fluid dynamics simulation, combined with steady-state towing test, rotating arm test and analysis of unsteady planar motion mechanism.

[0069] In some embodiments, the hydrodynamic parameter quantification in step S2 specifically includes: identifying velocity-related damping factors through towing tests, correlating resistance data under different flow velocities and fitting polynomial coefficients; identifying rotational motion resistance through rotating arm tests and fitting the relationship between yaw torque and angular velocity; identifying inertial effects through analysis of unsteady planar motion mechanisms and extracting acceleration-related hydrodynamic coefficients using Fourier analysis; analyzing the transient flow field using superimposed mesh technology and realizing inter-mesh data transmission through multi-domain computation.

[0070] Step S3: Construct the robot model and underwater environment in the dynamics simulator.

[0071] In some embodiments, step S3 specifically includes: designing a robot hardware architecture using a biomimetic propulsion system, including a central waterproof hull and multiple three-degree-of-freedom fins; importing a simplified robot model into a dynamics simulator and constructing an underwater environment to reduce parametric simulation distortion.

[0072] Step S4: Design Model Predictive Control-Fuzzy Logic Cascade Controller: The upper-level model predictive controller generates reference force and torque through rolling time-domain optimization; the lower-level fuzzy logic controller adaptively maps the reference force and torque into actuator instructions based on fuzzy rules.

[0073] In some embodiments, the input generation of the model prediction controller in step S4 specifically includes: obtaining the desired position and heading angle through the line-of-sight method; decoupling the path tracking problem into a thrust channel and a torque channel: the thrust channel controls the longitudinal velocity by adjusting the oscillation frequency of the flippers; the torque channel achieves directional adjustment by coordinating the amplitude difference of the flippers and the tail fin rudder angle.

[0074] In some embodiments, the optimization process of the model predictive controller in step S4 specifically includes: solving the finite-time optimal control problem at discrete time points to generate the optimal control sequence within the prediction step size; and using a moving horizon strategy to output the first element of the sequence as the current control action.

[0075] In some embodiments, the design of the fuzzy logic controller in step S4 specifically includes: constructing a fuzzy rule table, defining the mapping relationship between reference force / torque error and actuator command; and handling nonlinear and transient conditions through asymmetric response surfaces to balance dynamic response and stability.

[0076] In some embodiments, the cooperative mechanism of the model predictive control-fuzzy logic cascaded controller specifically includes: the upper-level model predictive controller explicitly integrates kinematic constraints and optimizes reference forces / torques to cope with parameter uncertainties; the lower-level fuzzy logic controller compensates for unmodeled fluid dynamic coupling and environmental disturbances through adaptive mapping.

[0077] Step S5: Conduct path tracking control verification in a simulation environment to evaluate the robustness of the controller and the tracking accuracy.

[0078] In some embodiments, the path tracking verification in step S5 specifically includes: designing a U-shaped path containing straight segments and arcs to evaluate straight-line tracking and steering capabilities; designing a figure-eight path with bidirectional curvature changes to evaluate lateral error suppression and transient stability; and applying a random hydrodynamic disturbance model to test the controller's anti-interference robustness.

[0079] The main technical advantages of this invention are as follows: It fully utilizes the biological characteristics of sea turtles—propulsion and steering decoupling during movement—through biomimetic design, providing a stable foundation for path tracking; it combines high-precision modeling methods from computational fluid dynamics (CFD), integrating steady-state towing experiments, rotating arm experiments, and analysis of unsteady planar motion mechanisms, systematically quantifying hydrodynamic parameters, significantly overcoming the limitations of traditional empirical formulas due to eddy current modeling bias caused by carapace curvature and the high cost of experimental fluid dynamics; it innovatively adopts a cascaded architecture of model predictive control (MPC) and fuzzy logic control (FLC), where the upper-layer MPC explicitly processes kinematic constraints and generates reference forces / torques through rolling time-domain optimization, effectively addressing parameter uncertainties and control oscillations; the lower-layer FLC adaptively maps actuator commands based on fuzzy rules, compensating for nonlinear fluid coupling and unmodeled environmental disturbances in real time; and it constructs a high-fidelity hydrodynamic environment using a dynamics simulator, significantly reducing parametric simulation distortion. Comparison between U-shaped and figure-eight paths verifies that this method possesses higher control accuracy, stronger anti-interference robustness, and more agile dynamic response capabilities in complex trajectory tracking.

[0080] The following further describes specific embodiments of the present invention, algorithm examples, and experimental verification.

[0081] A path tracking method for a turtle-inspired amphibious robot based on model predictive control (MPC) mainly includes the following aspects: establishing a multi-rigid-body dynamic model of the turtle-inspired amphibious robot and deriving a simplified dynamic model with hydrodynamic coefficients; building a mesh computing model of the robot in STAR-CCM+ and conducting steady-state towing tests, rotating arm tests, and analysis of unsteady planar motion mechanisms (PMMs); ​​importing the robot's CAD model into the dynamics simulator Webots and building a simulated pool environment; designing a fuzzy rule table; establishing a model predictive control-fuzzy logic cascaded controller; and conducting U-shaped path tracking control and figure-eight-shaped path tracking control simulations in the dynamics simulator. This invention proposes a two-layer control strategy for the turtle-inspired amphibious robot path tracking method, combining model predictive control and fuzzy logic control. The upper-layer MPC optimizes reference forces and torques through rolling time-domain optimization, while the lower-layer FLC adaptively maps these reference values ​​to actuator commands based on fuzzy rules, effectively addressing hydrodynamic nonlinear coupling, coefficient uncertainty, and unmodeled environmental disturbances.

[0082] This invention provides a path tracking method for a turtle-like amphibious robot based on model predictive control, comprising the following steps:

[0083] S1. Establish a turtle-inspired coordinate system for the amphibious robot, and define its spatial configuration relative to the global coordinate system. Based on established ship dynamics methods, the dynamic behavior of the system in the robot's fixed coordinate system can be represented as follows:

[0084]

[0085] in, and These correspond to the rigid body inertia matrix and the Coriolis-centripetal force matrix, respectively. and This is used to describe the additional mass effect in fluid mechanics. and These represent viscous damping force and hydrostatic restoring force, respectively. This represents the generalized control input vector.

[0086] Considering the robot's dual symmetry about the longitudinal-lateral (xoy) and planar longitudinal-vertical (xoz) axes, the coupled motions between longitudinal, lateral, and yaw motions are deduced to be negligible. Based on biological observations of tortoise locomotion patterns and engineering simplification goals, it is determined that only longitudinal and yaw motions need active control during planar motion. Following this biomechanical principle and incorporating the latest research findings, a simplified dynamic model for horizontal navigation is expressed as follows:

[0087]

[0088] in, , and The added quality effect was quantified, while , and This corresponds to the linear damping coefficient. The quadratic damping term... , and Capture nonlinear fluid dynamic drag. Control input quantity. and Control propulsion and yaw moment separately (reference) Figure 5 ). This indicates the total mass of the robot. Let be the moment of inertia of rotation along the vertical axis.

[0089] S2. The established dynamic equations are obtained through the matrix. and This invention integrates key fluid-structure interaction components, encompassing hydrodynamic inertia and viscous dissipation characteristics. To construct a high-precision parametric model, computational fluid dynamics (CFD) simulations were systematically used to identify these undetermined hydrodynamic parameters. This invention focuses on quantifying three main hydrodynamic dependencies: 1) velocity-dependent damping factors, 2) rotational drag, and 3) inertial effects caused by acceleration. The parameter identification process combines two complementary computational frameworks: steady-state hydrodynamic simulations including towing and rotating arm tests, and unsteady-state hydrodynamic simulations using the standardized PMM testing protocol. This invention employs STAR-CCM+ software for CFD simulations.

[0090] S3. Sea turtles achieve efficient movement in unstable media through flexible, three-degree-of-freedom flippers, while maintaining stability through a low center of gravity and intermittent ground contact. These biological principles have inspired the mechanical design of biomimetic robotic systems. This experimental platform employs a biomimetic propulsion system, has a compact overall size, with a total length of only 430 mm, and achieves a net weight of 10.48 kg through optimized structural design.

[0091] The hardware architecture comprises two core subsystems: a central carapace structure and three three-degree-of-freedom bionic fin-like limb mechanisms. The central carapace is a waterproof hull constructed with a carbon fiber support frame, housing all electronic components. The main control system is powered by a Raspberry Pi 4B single-board computer with 4 GB of memory, enabling wireless communication only when floating on water or moving on land; underwater propulsion is achieved through pre-programmed commands.

[0092] A nine-axis inertial measurement unit (IMU) MPU-9250 was used to monitor triaxial acceleration and angular velocity (X, Y, Z directions) at a sampling frequency of 100 Hz, enabling quantitative analysis of disturbance dynamics during motion. The drive system is equipped with twelve custom-designed GXYipin waterproof servo motors, each with a maximum output torque of 130 kg·cm. This wireless electronic architecture supports autonomous movement and real-time data transmission.

[0093] Based on the above robot design, the simplified model is imported into the dynamics simulator Webots.

[0094] S4. When performing path tracking tasks, the robot obtains the desired position using the line-of-sight (LOS) method. X d , Y d and expected heading angle For executor instructions (see reference) Figure 5 This invention proposes a hierarchical model predictive control-fuzzy logic controller (MPC-FLC) architecture. In this architecture, the MPC acts as the upper-level controller, responsible for generating reference forces and torques, while the lower-level fuzzy logic controller maps these reference quantities to instructions for specific actuators. This two-layer framework ensures optimal path tracking through predictive optimization while addressing dynamic uncertainties through adaptive signal allocation.

[0095] Model predictive control provides a systematic framework for regulating nonlinear multivariable systems through finite-time optimization methods. At each discrete time point... The algorithm solves a finite-time optimal control problem, in The optimal control sequence is calculated within one prediction step. The control actions implemented. It is the first element of the sequence, embodying the signature moving horizon strategy in predictive control.

[0096] To overcome key technical challenges in the motion control of biomimetic turtle robots, this invention proposes a dual-module control architecture based on fuzzy logic control. The system complexity stems primarily from three core issues: 1) the inherent nonlinear hydrodynamic characteristics of the mechanical structure; 2) the uncertainty of hydrodynamic coefficients; and 3) the unpredictability of marine environmental disturbances. Notably, the strongly nonlinear relationship between the three-dimensional fin motion and dynamic response has not yet been modeled with sufficient precision. This hierarchical control framework aims to achieve path tracking. The upper-level model predictive controller decouples the output into two independent channels: a thrust channel for speed regulation by adjusting the oscillation frequency of the fore and aft fins, and a torque channel for directional adjustment through the amplitude difference between the fore and aft fins in conjunction with the tail fin rudder angle. At the lower-level control layer, a fuzzy logic compensation mechanism is employed to address the uncertainties in system modeling.

[0097] S5. A U-shaped path consisting of two straight lines and a semicircular arc is suitable for evaluating a robot's straight-line tracking and turning capabilities. This invention designs a U-shaped trajectory, comprising a 5-meter-long straight line and a semicircular arc with a radius of 2.5 meters. For this trajectory, three controllers—a PID controller, a sliding mode controller, and the proposed MPC-FLC strategy—were compared and evaluated. To examine the robustness of the controllers, random hydrodynamic disturbances were applied to the robot system, and their modeling expression is as follows:

[0098]

[0099] in This represents a normally distributed noise signal with a mean of 0 and a variance of 1.

[0100] The figure-eight path, characterized by bidirectional steering and continuous curvature variation, serves as a rigorous benchmark for evaluating a controller's lateral error suppression capability and transient stability. The figure-eight path is mathematically defined in parametric equation form as follows:

[0101]

[0102] in Indicates the rotation angle.

[0103] A path tracking method for a turtle-inspired amphibious robot based on model predictive control mainly includes the following steps:

[0104] S1, The robot's spatial pose coordinates are defined as follows: , Represents three-axis coordinates. Indicates the corresponding attitude angle, in body coordinate system OB-XBYBZB The motion speed of the robot is defined as , Indicates the velocity along three axes. This represents the corresponding angular velocity. Specifically, Figure 1 The coordinate system configuration of the amphibious robot system in an embodiment of the present invention is shown, and the local body attachment reference frame and the global inertial coordinate system are described in detail. OE-XEYEZE Based on legged robots, autonomous underwater vehicles (AUVs), and biomimetic underwater robots (BURs), the full-body dynamic model of the turtle-inspired amphibious robot is as follows: Figure 2 As shown. Figure 2 The dynamic model and joint configuration of the amphibious robot system in an embodiment of the present invention are shown, wherein, Indicates the center of the robot body. This indicates the robot's center of gravity. f 0s 、f 1s 、f 2s 、f 3s These are the contact force vectors of the four flippers. P 0 、P 1 、P 2 、P 3 These are the position vectors of the four fin tips. O-XsYsZs It is a global inertial coordinate system, where g represents gravity.

[0105] S2. To quantify the axial motion resistance characteristics, hydrodynamic experiments were conducted by simulating dragging conditions along the main translational axes. In these numerical studies, the biomimetic robot platform remained stationary within a controlled fluid domain, while the ambient flow velocity gradually increased from a static state to 1 m / s. The computational domain size was optimized according to the standardization specifications of the International Commission on Experimental Marine Hydrodynamics (ITTC) to minimize wall interference effects and ensure solution accuracy.

[0106] See Figure 3 , Figure 3 The following diagram illustrates the fluid dynamics characteristics of the biomimetic turtle robot in this embodiment of the invention (based on CFD simulation): (a) Velocity distribution in the oscillating direction at an inflow velocity of 0.6 m / s. (b) Velocity distribution around the biomimetic turtle at a computational domain radius of 3.3 m at an inflow velocity of 0.5 m / s. (c) Velocity distribution in the flow field during pure pitching motion at 1 Hz. (d) Relationship between pitching drag and flow velocity. (e) Relationship between yaw rate and yaw moment. (f) Time-dependent drag and Fourier fitting results during pure pitching motion at 1 Hz.

[0107] Specifically, Figure 3(a) illustrates the vortex structure and velocity profile formed under lateral flow conditions of 0.6 m / s. By systematically scanning different incoming flow velocities, longitudinal drag was recorded and correlated with the corresponding flow velocity magnitudes, such as... Figure 3 As shown in (d). Subsequently, based on the force-velocity characteristic curve, multinomial regression analysis was used to extract coefficients, thereby determining the velocity-dependent hydrodynamic parameters.

[0108] To determine the hydrodynamic coefficients dependent on the yaw angle This invention simulates a rotating arm test by changing the computational domain radius R during the rotational motion of a mechanical turtle. While maintaining a constant inflow velocity of 0.5 m / s, the computational domain radius was gradually increased from 2.2 m to 4.4 m in increments of 0.55 m. Figure 3 (b) shows the uniform flow field distribution of the mechanical turtle at a typical radius of 3.3 meters. Figure 3 As shown in (e), this invention quantifies the influence of rotational parameters on the flow field by least-squares fitting of torque and yaw rate r.

[0109] To simulate unsteady motion in PMM testing, this invention employs a superimposed mesh technique, achieving efficient flow field analysis under complex geometries through overlapping mesh regions. By constructing a multi-region computational domain including inner and outer areas, data transmission between meshes is effectively facilitated. Given the inherent transient characteristics of PMM simulation, this invention utilizes a time-dependent solver. Figure 3 (c) shows the transient velocity field at a frequency f = 1Hz during pure swell motion, while Figure 3 (f) shows the drag variation over time at the same frequency. By performing Fourier analysis on the time-varying drag data, this invention obtains the hydrodynamic coefficients related to acceleration.

[0110] S3. Build a water tank environment in the dynamics simulator, see [reference needed]. Figure 4 This illustrates a turtle-inspired robot in the Webots dynamics simulator of an embodiment of the present invention.

[0111] S4. A hierarchical Model Predictive Control-Fuzzy Logic Controller (MPC-FLC) architecture was established. In this architecture, the MPC acts as the upper-level controller, responsible for generating reference forces and torques, while the lower-level fuzzy logic controller maps these reference quantities to instructions for specific actuators. This two-layer framework ensures optimal path tracking through predictive optimization while addressing dynamic uncertainties through adaptive signal allocation. For details, please refer to the turtle-inspired robot path tracking control block diagram. Figure 5 The asymmetric surface of the fuzzy rule graph highlights the system's ability to handle nonlinear and transient states, thus balancing response and stability under different operating conditions. For details, see [link to relevant documentation]. Figure 6 The input and output rules of the fuzzy inference system in an embodiment of the present invention are shown. (a) Input and With output The relationship between them. (b) Input and With output The relationship between them.

[0112] S5. Starting from a stationary state, the proposed controller enables the turtle-like robot to smoothly converge to the target trajectory without overshoot. Subsequently, the robot travels along the reference path, with minimal deviation between the desired and actual trajectories. (See also...) Figure 7 This paper illustrates the performance evaluation of the path tracking controller in an embodiment of the present invention, including (a) a visualization comparing the U-shaped path with the actual motion; and (b) the temporal evolution of the lateral path tracking deviation. The lateral error, defined quantitatively as the shortest distance from the centroid to the reference path, is... Figure 7 (b) illustrates the timing variations of the lateral error for the three controllers. The SMC controller completed the path in 317.4 seconds with an average lateral error of 0.112 meters; the PID controller took 244.0 seconds with an average error of 0.097 meters. In comparison, this control strategy performs better, completing the task in only 145.4 seconds, significantly reducing the lateral error to 0.0295 meters, and demonstrating superior tracking accuracy and efficiency. Specifically, the signal output of the MPC-FLC control strategy in the U-shaped trajectory tracking of this embodiment of the invention is described in [reference needed]. Figure 8 The paper further demonstrates the smooth and stable control signal generated by this strategy, verifying its operational reliability.

[0113] Figure 9 The performance evaluation of the path tracking controller in this embodiment of the invention is shown: (a) Visual comparison of the figure-eight path with the actual motion. (b) Temporal evolution of the lateral path tracking deviation. Figure 9 (a) demonstrates the path-following performance of the turtle-inspired robot under three control strategies, all of which implement basic path-following functionality. Figure 9 (b) quantifies the trend of lateral error of each controller over time: the SMC controller completed the trajectory in 663.3 seconds with an average lateral error of 0.122 meters; the PID controller reduced the time to 518.1 seconds while maintaining an average error of 0.101 meters. Notably, the MPC-FLC strategy proposed in this invention performs better, completing the task in only 429.1 seconds with a significantly reduced average error of 0.0393 meters, demonstrating superior tracking accuracy and operating efficiency.

[0114] The signal output of the MPC-FLC control strategy in figure-eight trajectory tracking in this embodiment of the invention is described in the following reference. Figure 10The document further demonstrates the controller's drive signals, which achieve stable motion control by adjusting the propulsion speed and steering angle in real time.

[0115] In summary, this invention proposes a path tracking method for a turtle-inspired amphibious robot based on model predictive control. A high-fidelity hydrodynamic model is established through CFD-driven parameterization, and a hierarchical control framework integrating MPC and FLC is proposed. This turtle-inspired amphibious robot path tracking method combines the high-fidelity hydrodynamic model obtained through CFD numerical calculations with the stability of MPC-FLC cascade control, effectively solving problems such as nonlinear hydrodynamic interactions, coefficient uncertainties, and unmodeled environmental disturbances. Specifically, the above technical solution has the following advantages:

[0116] 1. The turtle-inspired amphibious robot platform is reliable. As a biomimetic prototype, the turtle's propulsion and steering can be decoupled during movement, providing a stable foundation for the robot's path tracking algorithm.

[0117] 2. The simplified dynamic model based on the multi-rigid-body dynamics model parameterizes the influence of the water environment on the robot into hydrodynamic coefficients, and simplifies the control dimension through mechanical structure analysis and biomimetic prototype analysis;

[0118] 3. By integrating steady-state towing tests, rotating arm simulations, and PMM analysis, CFD simulations were performed using STAR-CCM+ software to obtain the corresponding hydrodynamic coefficients, and a high-fidelity hydrodynamic model was established.

[0119] 4. A hydrodynamic environment model suitable for turtle-like robots was established in the dynamics simulator Webots, which reduces the model distortion problem in the ordinary parametric simulation process and can more systematically evaluate the performance of the constructed path tracking framework.

[0120] 5. The model predictive control framework adopted in this invention explicitly integrates kinematic constraints through back-look optimization, which is more effective than traditional PID control strategies and sliding mode controllers in dealing with problems such as parameter uncertainty, lack of operator experience, and control output oscillation.

[0121] 6. This invention proposes a two-layer control strategy combining model predictive control and fuzzy logic control. The upper-layer MPC optimizes the reference force and torque using a rolling time-domain optimization method, while the lower-layer FLC adaptively maps the aforementioned reference signals into actuator commands based on fuzzy rules, effectively addressing nonlinear hydrodynamic interactions, parameter uncertainties, and unmodeled environmental disturbances;

[0122] 7. The proposed control framework has been compared with U-shaped path tracking simulation and figure-eight path tracking simulation, achieving higher control accuracy and more agile robot movement.

[0123] This invention also provides a storage medium for storing a computer program, which, when executed, performs at least the methods described above.

[0124] This invention also provides a control device, including a processor and a storage medium for storing a computer program; wherein the processor executes the computer program by performing at least the method described above.

[0125] This invention also provides a processor that executes a computer program, at least performing the methods described above.

[0126] The storage medium can be implemented by any type of non-volatile storage device, or a combination thereof. The non-volatile memory can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), magnetic random access memory (FRAM), flash memory, magnetic surface memory, optical disc or CD-ROM; magnetic surface memory can be disk storage or magnetic tape storage. The storage media described in the embodiments of this invention are intended to include, but are not limited to, these and any other suitable types of memory.

[0127] In the several embodiments provided by this invention, it should be understood that the disclosed systems and methods can be implemented in other ways. The device embodiments described above are merely illustrative. For example, the division of units is only a logical functional division, and in actual implementation, there may be other division methods, such as: multiple units or components can be combined, or integrated into another system, or some features can be ignored or not executed. In addition, the coupling or direct coupling or communication connection between the various components shown or discussed can be through some interfaces, and the indirect coupling or communication connection between devices or units can be electrical, mechanical, or other forms.

[0128] The units described above as separate components may or may not be physically separate. The components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the units may be selected to achieve the purpose of this embodiment according to actual needs.

[0129] In addition, in the various embodiments of the present invention, each functional unit can be integrated into one processing unit, or each unit can be a separate unit, or two or more units can be integrated into one unit; the integrated unit can be implemented in hardware or in the form of hardware plus software functional units.

[0130] Those skilled in the art will understand that all or part of the steps of the above method embodiments can be implemented by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps of the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0131] Alternatively, if the integrated units of this invention are implemented as software functional modules and sold or used as independent products, they can also be stored in a computer-readable storage medium. Based on this understanding, the technical solutions of the embodiments of this invention, or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the methods described in the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, ROM, RAM, magnetic disks, or optical disks.

[0132] The methods disclosed in the several method embodiments provided by this invention can be arbitrarily combined without conflict to obtain new method embodiments.

[0133] The features disclosed in the several product embodiments provided by this invention can be arbitrarily combined without conflict to obtain new product embodiments.

[0134] The features disclosed in the several method or device embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method or device embodiments.

[0135] The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various equivalent substitutions or obvious modifications can be made without departing from the concept of the present invention, and all such modifications, achieving the same performance or application, should be considered within the scope of protection of the present invention.

Claims

1. A path tracking method for a turtle-inspired amphibious robot based on model predictive control, characterized in that, Includes the following steps: S1. Establish a multi-rigid-body dynamics model of the turtle-like amphibious robot, derive simplified dynamic equations including hydrodynamic coefficients, and simplify the control dimension based on biological motion characteristics. S2. Through computational fluid dynamics simulation, combined with steady-state towing test, rotating arm test and analysis of unsteady planar motion mechanism, the hydrodynamic parameters are quantified. The quantification of the hydrodynamic parameters specifically includes: identifying velocity-related damping factors through towing tests, correlating resistance data at different flow velocities, and fitting the relationship between flow velocity and resistance data; identifying rotational motion resistance through rotating arm tests and fitting the relationship between yaw moment and angular velocity; identifying inertial effects through analysis of unsteady planar motion mechanisms and extracting acceleration-related hydrodynamic coefficients; and using superimposed mesh technology to analyze the transient flow field and realizing inter-mesh data transmission through multi-domain computation. S3. Construct robot models and underwater environments in a dynamics simulator; S4. Design Model Predictive Control - Fuzzy Logic Cascade Controller: The upper-level model predictive controller generates reference forces and torques through rolling time-domain optimization; The lower-level fuzzy logic controller adaptively maps reference forces and torques into actuator commands based on fuzzy rules; The input generation of the model predictive controller in step S4 specifically includes: obtaining the desired position and heading angle; decoupling the path tracking problem into a thrust channel and a torque channel: the thrust channel controls the longitudinal velocity, and the torque channel realizes the direction adjustment; the design of the fuzzy logic controller specifically includes: constructing a fuzzy rule table, defining the mapping relationship between reference force and torque errors and actuator commands; handling nonlinear and transient conditions, balancing dynamic response and stability; the cooperation mechanism of the model predictive control-fuzzy logic cascade controller specifically includes: the upper-level model predictive controller explicitly integrates kinematic constraints, optimizes reference force and torque to cope with parameter uncertainties; the lower-level fuzzy logic controller compensates for unmodeled hydrodynamic coupling and environmental disturbances through adaptive mapping; S5. Conduct path tracking control verification in a simulation environment to evaluate the robustness of the controller and the tracking accuracy.

2. The path tracking method as described in claim 1, characterized in that, The simplified establishment of the multi-rigid-body dynamics model in step S1 specifically includes: By utilizing the robot's dual symmetry about the longitudinal-lateral plane and the longitudinal-vertical plane, the coupling effect of longitudinal, lateral and yaw motions is ignored; Based on biomimetic motion characteristics, the active control dimension is limited to longitudinal propulsion and yaw moment; The additional mass effect, linear damping term, and nonlinear fluid resistance term are integrated into the dynamic equation.

3. The path tracking method as described in claim 1, characterized in that, The hydrodynamic parameter quantification in step S2 specifically includes: fitting the relationship between flow velocity and resistance data by fitting polynomial coefficients; and extracting acceleration-related hydrodynamic coefficients using Fourier analysis.

4. The path tracking method as described in claim 1, characterized in that, Step S3 specifically includes: The robot hardware architecture is designed using a biomimetic propulsion system, which includes a central waterproof cabin and multiple three-degree-of-freedom fin-like limbs. Importing a simplified robot model into the dynamics simulator and constructing an underwater environment reduces distortion in parametric simulations.

5. The path tracking method as described in claim 1, characterized in that, The specific steps for generating the input to the model prediction controller in step S4 include: The desired position and heading angle are obtained using the line-of-sight method; The thrust channel controls the longitudinal velocity by adjusting the oscillation frequency of the flippers; The torque channel achieves directional adjustment by coordinating the amplitude difference of the fins and the rudder angle of the tail fin.

6. The path tracking method as described in claim 1 or 5, characterized in that, The optimization process of the model prediction controller in step S4 specifically includes: Solve the finite-time optimal control problem at discrete time points and generate the optimal control sequence within the prediction step size; A moving view strategy is adopted, and the first element of the sequence is used as the current control action output.

7. The path tracking method as described in claim 1, characterized in that, The design of the fuzzy logic controller in step S4 specifically includes: By using asymmetric response surfaces to handle nonlinear and transient conditions, dynamic response and stability are balanced.

8. The path tracking method as described in claim 1, characterized in that, The path tracking control verification in step S5 specifically includes: Design a U-shaped path containing straight segments and arcs, and evaluate straight-line tracking and steering capabilities; Design a figure-eight path with bidirectional curvature variation to evaluate lateral error suppression and transient stability; A random hydrodynamic disturbance model was applied to test the controller's robustness against disturbances.

9. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by the processor, it implements the model predictive control-based path tracking method for a turtle-like amphibious robot as described in any one of claims 1 to 8.

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