Method for predicting hydrodynamic moment of underwater manipulator

Abstract

The application provides a method for predicting water dynamic moment of an underwater mechanical arm, which can be applied to the technical field of underwater robots. The method comprises the following steps: for the i-th section of the underwater mechanical arm, an initial water dynamic moment model of the i-th section is constructed based on flow parameters, mechanical arm parameters, dimensionless resistance coefficients and added mass coefficients, the underwater mechanical arm is evenly divided into N sections along the axial direction, i is an integer greater than or equal to 1 and less than or equal to N; the initial water dynamic moment model of the i-th section is converted into a linear model of the i-th section based on the correlation between the dimensionless stroke ratio and the dimensionless resistance coefficient and the added mass coefficient; the water dynamic coefficients in the linear model of each of the N sections are decoupled by using a least square algorithm, and a coefficient discrete sample set is obtained; the coefficient discrete sample set is spline fitted by using a least square algorithm based on a residual vector, and a target coefficient curve is obtained, so that the water dynamic moment of the underwater mechanical arm can be predicted by using the target coefficient curve.

Description

Technical Field

[0001] This invention relates to the field of underwater robots, and more specifically to a method for predicting the hydrodynamic torque of an underwater robotic arm. Background Technology

[0002] When performing heavy-duty operations, underwater robotic arms are subjected to significant unsteady hydrodynamic torques, the magnitude of which directly affects the joint drive torque requirements and motion control accuracy. The classic Morison equation decomposes hydrodynamics into two parts: viscous drag and added mass, which are described by drag coefficients and added mass coefficients.

[0003] In the classic Morison equation, the drag coefficient and the added mass coefficient are taken as constants, which makes it difficult to describe the transient changes in coefficients caused by the gradual evolution of the flow field structure (boundary layer establishment, near-wake vortex formation, vortex shedding, etc.) after startup. This results in a large prediction deviation of the hydrodynamic torque of the underwater robotic arm. Summary of the Invention

[0004] In view of the above problems, the present invention provides a method for predicting the hydrodynamic torque of an underwater robotic arm.

[0005] According to a first aspect of the present invention, a method for predicting the hydrodynamic torque of an underwater robotic arm is provided, comprising: for the i-th segment of the underwater robotic arm, constructing an initial hydrodynamic torque model for the i-th segment based on water flow parameters, robotic arm parameters, a dimensionless drag coefficient, and an added mass coefficient, wherein the underwater robotic arm is divided into N segments along the axial direction, and i is an integer greater than or equal to 1 and less than or equal to N; using the dimensionless stroke ratio as a state variable, and based on the correlation between the dimensionless stroke ratio and the dimensionless drag coefficient and the added mass coefficient, converting the initial hydrodynamic torque model of the i-th segment into a linear model of the i-th segment; and applying a least squares algorithm to each of the N segments' linear models. The hydrodynamic coefficients in the model are decoupled to obtain a discrete sample set of coefficients. The discrete sample set of coefficients includes sample points at multiple times in N segments, and each sample point includes multiple estimated values ​​of hydrodynamic coefficients. Based on the residual vector, spline fitting is performed on the discrete sample set of coefficients using the least squares algorithm to obtain the target coefficient curve. This target coefficient curve is then used to predict the hydrodynamic moment of the underwater manipulator. The residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment. The simulated hydrodynamic moment is obtained by performing fluid dynamics simulation on the underwater manipulator, while the model hydrodynamic moment is obtained by inputting multiple estimated values ​​of hydrodynamic coefficients into the initial hydrodynamic moment model.

[0006] A second aspect of the present invention provides a device for predicting the hydrodynamic torque of an underwater robotic arm, comprising: a construction module, configured to construct an initial hydrodynamic torque model for the i-th segment of the underwater robotic arm based on water flow parameters, robotic arm parameters, a dimensionless drag coefficient, and an added mass coefficient, wherein the underwater robotic arm is divided into N segments along the axial direction, and i is an integer greater than or equal to 1 and less than or equal to N; a conversion module, configured to convert the initial hydrodynamic torque model of the i-th segment into a linear model of the i-th segment based on the correlation between the dimensionless travel ratio and the dimensionless drag coefficient and the added mass coefficient, using the dimensionless travel ratio as a state variable; and a decoupling module, configured to decouple each of the N segments using a least squares algorithm. The hydrodynamic coefficients in the linear model are decoupled to obtain a discrete sample set of coefficients. The discrete sample set of coefficients includes sample points at multiple times in N segments, and each sample point includes multiple hydrodynamic coefficient estimates. The spline fitting module is used to perform spline fitting on the discrete sample set of coefficients based on the residual vector and using the least squares algorithm to obtain the target coefficient curve. This target coefficient curve is used to predict the hydrodynamic moment of the underwater manipulator. The residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment. The simulated hydrodynamic moment is obtained by performing hydrodynamic simulation on the underwater manipulator, and the model hydrodynamic moment is obtained by inputting multiple hydrodynamic coefficient estimates into the initial hydrodynamic moment model.

[0007] A third aspect of the present invention provides an electronic device comprising: one or more processors; and a memory for storing one or more computer programs, wherein the one or more processors execute the one or more computer programs to implement the steps of the method described above.

[0008] A fourth aspect of the present invention also provides a computer-readable storage medium having a computer program or instructions stored thereon, wherein the computer program or instructions, when executed by a processor, implement the steps of the above-described method.

[0009] A fifth aspect of the present invention also provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the above-described method.

[0010] According to an embodiment of the present invention, using the dimensionless travel ratio as the state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless drag coefficient and the added mass coefficient, the initial hydrodynamic moment model of the i-th segment is transformed into a linear model of the i-th segment. The hydrodynamic coefficients in each of the N segments' linear models are decoupled using a least squares algorithm, thereby transforming the identification of multiple hydrodynamic coefficients into a linear problem within the dimensionless travel ratio range. Based on the residual vector determined according to the difference between the simulated hydrodynamic moment and the model hydrodynamic moment, spline fitting is performed on the discrete sample set of coefficients using a least squares algorithm to obtain the target coefficient curve. This ensures that the identified target spline curve does not deviate from the actual simulated hydrodynamic moment curve, improving the fitting accuracy. The target coefficient curve can then be used for hydrodynamic moment prediction. Attached Figure Description

[0011] The above-described features, other objects, and advantages of the present invention will become clearer from the following description of embodiments of the invention with reference to the accompanying drawings, in which:

[0012] Figure 1 A flowchart illustrating a method for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown.

[0013] Figure 2 The graph shows the variation of the hydrodynamic torque of the underwater robotic arm over time under 12 different operating conditions.

[0014] Figure 3A A scatter plot showing the dimensionless drag coefficient as a function of the dimensionless stroke ratio according to an embodiment of the present invention is shown.

[0015] Figure 3B A scatter plot showing the variation of the additional mass coefficient with the dimensionless stroke ratio according to an embodiment of the present invention is shown.

[0016] Figure 4A A schematic diagram illustrating correct inflection point identification according to an embodiment of the present invention is shown.

[0017] Figure 4B A schematic diagram illustrating interference with inflection point identification according to an embodiment of the present invention is shown.

[0018] Figure 5A The diagram shows a comparison between the dimensionless drag coefficient samples obtained by least-squares decoupling according to an embodiment of the present invention and first-order / second-order splines.

[0019] Figure 5B The diagram shows a comparison between the additional quality coefficient samples obtained by least-squares decoupling according to an embodiment of the present invention and the first-order / second-order splines.

[0020] Figure 6The dimensionless drag coefficient curve and the added mass coefficient curve generated by the final control point according to an embodiment of the present invention are shown.

[0021] Figure 7 A structural block diagram of a device for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown.

[0022] Figure 8 A block diagram of an electronic device suitable for implementing a method for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown. Detailed Implementation

[0023] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings. However, it should be understood that these descriptions are exemplary only and are not intended to limit the scope of the invention. In the following detailed description, numerous specific details are set forth to provide a thorough understanding of the embodiments of the invention for ease of explanation. However, it will be apparent that one or more embodiments may be practiced without these specific details. Furthermore, descriptions of well-known structures and techniques are omitted in the following description to avoid unnecessarily obscuring the concept of the invention.

[0024] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the invention. The terms “comprising,” “including,” etc., as used herein indicate the presence of the stated features, steps, operations, and / or components, but do not exclude the presence or addition of one or more other features, steps, operations, or components.

[0025] All terms used herein (including technical and scientific terms) have the meanings commonly understood by those skilled in the art, unless otherwise defined. It should be noted that the terms used herein are to be interpreted in a manner consistent with the context of this specification, and not in an idealized or overly rigid way.

[0026] When using expressions such as "at least one of A, B and C", they should generally be interpreted in accordance with the meaning that is commonly understood by those skilled in the art (e.g., "a system having at least one of A, B and C" should include, but is not limited to, a system having A alone, a system having B alone, a system having C alone, a system having A and B, a system having A and C, a system having B and C, and / or a system having A, B and C, etc.).

[0027] Underwater robotic arms experience significant unsteady hydrodynamic torques during heavy-duty operations, the magnitude of which directly affects the joint drive torque requirements and motion control accuracy. The classic Morison equation decomposes hydrodynamic forces into viscous drag and added mass terms, described by drag and added mass coefficients. This equation is widely used in the design of cylindrical components in still water. However, in the actual rotational motion of underwater robotic arms, existing methods suffer from the following problems:

[0028] In the classic Morison equation, the drag coefficient and the added mass coefficient are taken as constants, which makes it difficult to describe the transient changes in coefficients caused by the gradual evolution of the flow field structure (boundary layer establishment, near-wake vortex formation, vortex shedding, etc.) after startup. This results in a large prediction deviation of the hydrodynamic torque of the underwater robotic arm.

[0029] While computational fluid dynamics can be used to solve the flow across the entire field online with high accuracy, the calculation time for a single operating condition can take several hours to several days, which is difficult to meet the requirements of real-time control.

[0030] The relevant unsteady coefficient identification methods usually adopt a single-point inversion method based on time or angular velocity, which is significantly affected by numerical noise and instantaneous signal fluctuations. It is difficult to simultaneously decouple viscous drag torque and additional mass torque, and the obtained coefficients lack transferability between different motion conditions.

[0031] For robotic arms with a large length-to-diameter ratio, the local Reynolds number and stroke vary significantly along different segments of the arm span. A single coefficient is insufficient to reflect the flow field state of each segment, and there is a lack of a unified systematic framework for processing multi-segment and multi-condition data.

[0032] Therefore, there is an urgent need for a decoupled identification method that can stably extract hydrodynamic coefficients that evolve with motion from limited fluid dynamics simulation data and has the ability to predict across working conditions.

[0033] In view of this, embodiments of the present invention provide a method for predicting the hydrodynamic torque of an underwater robotic arm, comprising: for the i-th segment of the underwater robotic arm, constructing an initial hydrodynamic torque model for the i-th segment based on water flow parameters, robotic arm parameters, dimensionless drag coefficient, and added mass coefficient, wherein the underwater robotic arm is divided into N segments along the axial direction, and i is an integer greater than or equal to 1 and less than or equal to N; using the dimensionless stroke ratio as a state variable, and based on the correlation between the dimensionless stroke ratio and the dimensionless drag coefficient and the added mass coefficient, converting the initial hydrodynamic torque model of the i-th segment into a linear model of the i-th segment; and applying a least squares algorithm to each of the N segments' linear models. The hydrodynamic coefficients in the model are decoupled to obtain a discrete sample set of coefficients. The discrete sample set of coefficients includes sample points at multiple times in N segments, and each sample point includes multiple estimated values ​​of hydrodynamic coefficients. Based on the residual vector, spline fitting is performed on the discrete sample set of coefficients using the least squares algorithm to obtain the target coefficient curve. This target coefficient curve is then used to predict the hydrodynamic moment of the underwater manipulator. The residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment. The simulated hydrodynamic moment is obtained by performing fluid dynamics simulation on the underwater manipulator, while the model hydrodynamic moment is obtained by inputting multiple estimated values ​​of hydrodynamic coefficients into the initial hydrodynamic moment model.

[0034] Figure 1A flowchart illustrating a method for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown.

[0035] like Figure 1 As shown, the method for predicting the hydrodynamic torque of the underwater robotic arm in this embodiment includes operations S110 to S140.

[0036] In operation S110, for the i-th segment of the underwater manipulator, an initial hydrodynamic moment model for the i-th segment is constructed based on the water flow parameters, manipulator parameters, dimensionless drag coefficient, and added mass coefficient. The underwater manipulator is divided into N segments along the axial direction, and i is an integer greater than or equal to 1 and less than or equal to N.

[0037] In operation S120, the dimensionless stroke ratio is used as the state variable. Based on the correlation between the dimensionless stroke ratio and the dimensionless resistance coefficient and the additional mass coefficient, the initial hydrodynamic moment model of the i-th segment is transformed into the linear model of the i-th segment.

[0038] In operation S130, the hydrodynamic coefficients in each of the N linear models are decoupled using the least squares algorithm to obtain a discrete sample set of coefficients. The discrete sample set of coefficients includes sample points at multiple times for each of the N segments, and the sample points include multiple hydrodynamic coefficient estimates.

[0039] In operation S140, based on the residual vector, spline fitting is performed on the discrete sample set of coefficients using the least squares algorithm to obtain the target coefficient curve, which is then used to predict the hydrodynamic moment of the underwater manipulator. The residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment. The simulated hydrodynamic moment is obtained by performing fluid dynamics simulation on the underwater manipulator, while the model hydrodynamic moment is obtained by inputting multiple hydrodynamic coefficient estimates into the initial hydrodynamic moment model.

[0040] According to embodiments of the present invention, water flow parameters are used to characterize the aquatic environment in which the underwater robotic arm operates. For example, water flow parameters may include fluid density, kinematic viscosity, boundary layer thickness, boundary layer transition location, water flow velocity, etc.

[0041] According to embodiments of the present invention, the robotic arm parameters are used to characterize the structural features of the underwater robotic arm. For example, the robotic arm parameters may include the geometry, dimensions, arm length, position of the rotation axis, number of joints, rotational speed, angular acceleration, moment of inertia, etc.

[0042] According to an embodiment of the present invention, the dimensionless drag coefficient can be a steady-state dissipative force related to the velocity, which plays a dominant role when the water flows around the component at a constant speed or when the component moves at a constant speed, and is used to describe the magnitude of the drag.

[0043] According to an embodiment of the present invention, the additional mass coefficient can be the inertial force related to acceleration, which plays a dominant role when the component accelerates, decelerates, swings, or vibrates, and is used to describe the additional inertial effect of the water body.

[0044] According to an embodiment of the present invention, the initial hydrodynamic torque model can be based on the resistance torque term of the i-th segment of the underwater robotic arm. With inertial torque term The sum is determined. The resistance torque term. It can be constructed based on water flow parameters, robotic arm parameters, and dimensionless drag coefficients. Inertial moment term. It can be built based on the robot arm parameters and the added mass coefficient.

[0045] A simplified model of a cylindrical underwater robotic arm is used as the research object. In the baseline example, the total length of the single arm is taken as L = 4 m, and the diameter is taken as d = 0.115 m; the fluid density is taken as ρ = 1025 kg / m³, and the kinematic viscosity is taken as ν = 1.0 × 10⁻⁶ m³. -6 m 2 / s. Subsequent single-arm spatial complex working case examples use the same geometric parameters, and the dual-arm model uses two identical single arms with the same geometric parameters. To provide segment-level identification capability for the hydrodynamic model, the robotic arm is divided into N segments along the axial direction, with each segment having a length ΔL as follows:

[0046] (1);

[0047] The representative position of the i-th segment is taken as the center of the segment, and its distance from the root axis of rotation is r. c,i for:

[0048] (2);

[0049] Based on the classic Morison equations, the hydrodynamic forces acting on it during motion can be decomposed into viscous drag and additional mass forces. The specific physical mechanisms of these two forces are as follows:

[0050] Resistance torque term As the underwater robotic arm moves relative to the water, fluid viscosity and separation create resistance, hindering its rotation. This resistance manifests as the sum of pressure drag and friction drag in the flow. In steady flow, a constant drag coefficient is commonly used to represent its intensity; however, in unsteady rotational situations, the drag coefficient changes with the development of the flow field.

[0051] Inertial torque term When an underwater robotic arm accelerates and rotates, it needs to push a certain volume of surrounding fluid to move along with it, which manifests as an inertial reaction torque imparted by the fluid to the robotic arm. Its magnitude is proportional to the angular acceleration, similar to the "additional inertia" of an object. In an ideal incompressible fluid, the additional mass coefficient of a slender cylinder can be taken as approximately 1 (i.e., the mass of displaced water affects inertia); however, in actual viscous flows, this coefficient will also change with the development of the flow field.

[0052] To obtain hydrodynamic coefficients that can change in real time with the development of the flow field, and ultimately decouple the two hydrodynamic coefficients mentioned above (such as the constant drag coefficient and the additional mass coefficient corresponding to the moment of inertia term), two additional assumptions are made based on the Morison equation. First, it is assumed that within the current Reynolds number range ( Under these conditions, the change in hydrodynamic coefficients is only related to the development of the flow field structure and is independent of the Reynolds number and angular acceleration. Assumptions: Second, it is assumed that each segment of the underwater robotic arm undergoes the same flow field development process, i.e., their hydrodynamic coefficient change curves are defined using the same Reynolds number definition method and the dimensionless parameter definition method that can represent the flow field development process.

[0053] The Reynolds number focuses on the flow separation, vortex development, and added mass effects around each segmented cross section. For the rotational motion of an underwater manipulator in water, from the perspective of local hydrodynamic mechanisms, the characteristic dimensions of these flow structures are mainly determined by the diameter of the cylindrical cross section. Therefore, the diameter d of the underwater manipulator is taken as the characteristic length dimension, and the tangential velocity U at the reference point is taken as the characteristic length dimension. C The Reynolds number is defined as a characteristic velocity measure. For typical motion conditions of underwater robotic arms, the Reynolds number is often around 10. 4 In the above turbulent regions, flow separation and vortex shedding are significant, therefore the drag coefficient cannot be simply obtained from the analytical results of laminar flow at low Reynolds number.

[0054] Figure 2 The graph shows the variation of the hydrodynamic torque of the underwater robotic arm over time under 12 different operating conditions.

[0055] like Figure 2 As shown, the operating parameters include angular acceleration and Reynolds number; different combinations of parameter values ​​result in different operating conditions. For example, angular acceleration = 0.05 rad / s². 2 Reynolds number = 2.86 × 10 4 This creates a working condition.

[0056] The 12 operating conditions are (0.05 rad / s) 2 2.86×10 4 (0.10 rad / s) 2 2.86×10 4 (0.15 rad / s) 2 2.86×104 (0.05 rad / s) 2 5.72×10 4 (0.10 rad / s) 2 , 5.72×10), (0.15rad / s 2 , 5.72×10), (0.05rad / s 2 8.58×10 4 (0.10 rad / s) 2 ,8.58×10), (0.15rad / s 2 , 8.58×10), (0.20rad / s 2 , 8.58×10), (0.25rad / s 2 ,8.58×10), (0.15rad / s 2 11.44×10 4 ). Figure 2 The horizontal axis represents time, and the vertical axis represents hydrodynamic torque, or simply torque. It includes curves showing the change of hydrodynamic torque over time for 12 different operating conditions.

[0057] According to an embodiment of the present invention, the dimensionless travel ratio is a dimensionless quantity describing the distance an object travels in a fluid relative to a characteristic dimension. In the problem of underwater robotic arm rotation, the arc length traversed by the i-th reference point within time t can be defined as the travel. The diameter d of the underwater robotic arm was made dimensionless to obtain the dimensionless stroke ratio. :

[0058] (3);

[0059] d is the diameter of the underwater robotic arm. Let be the angle of rotation of the underwater robotic arm relative to its initial position. Let be the distance from the i-th reference point to the base axis (i.e., the base axis is the rotation axis of the robotic arm). Also referred to as the distance between the i-th segment and the robotic arm's axis of rotation. For pure rotational motion, in the above formula... This represents the arc length by which the reference point moves along the circumference, therefore the dimensionless travel ratio. This reflects the distance the reference point moved relative to the boom diameter.

[0060] Select dimensionless stroke ratio As a variable describing the mechanical state, it originates from empirical and research observations: in many experimental and numerical studies, the torque coefficient curves measured under different motion conditions can often be expressed as a dimensionless stroke ratio. When the x-axis is used, it shows better consistency, but a weaker relationship with the actual time history. This finding makes the dimensionless travel ratio... It can serve as a universal state parameter, a dimensionless stroke ratio, and effectively summarize the coefficient curves measured under different motion conditions onto a single main curve. Selected as the primary state variable, assuming the i-th segment has a dimensionless drag coefficient. Additional quality coefficient Mainly depends on the dimensionless stroke ratio Evolution occurs regardless of the specific time scale. The physical explanation is that as an object travels a greater distance in a fluid, flow field separation, vortex development, and additional mass effects gradually occur; given the same dimensionless stroke ratio... At that time, the flow field state is often similar.

[0061] In summary, using dimensionless stroke ratio State parameters can unify hydrodynamic behavior under different rotational angular paths within a single framework. Coefficient curves obtained from one motion condition can potentially be reused under other conditions.

[0062] Based on the above theoretical model, basic assumptions, and dimensionless parameter settings, the final hydrodynamic coefficient model takes the form of two piecewise third-order spline curves based on control points, with the dimensionless travel ratio as the parameter. Using the development history of the flow field structure as the physical basis, the dynamic dimensionless drag coefficient is fully characterized as the independent variable. Additional quality coefficient parameters Curves. This spline representation under variables has two significant advantages. First, it can explicitly describe the different change mechanisms of coefficients in the initial and later stages, making it more suitable for unsteady transient problems. Second, it retains a sufficiently simple functional form, allowing each segment in complex motion scenarios to be represented only by its dimensionless travel ratio. By querying the same set of curves, hydrodynamic predictions can be completed, thus balancing physical expressiveness with ease of engineering use.

[0063] According to an embodiment of the present invention, based on the correlation between the dimensionless stroke ratio and the dimensionless drag coefficient and the added mass coefficient, the dimensionless drag coefficient and the added mass coefficient can be expressed as C. d ( C m ( ).

[0064] According to an embodiment of the present invention, the least squares algorithm can automatically decouple the drag torque term and the added mass term directly from the hydrodynamic torque data without relying on any prior empirical coefficients, thereby obtaining a purely data-driven discrete sample set of coefficients.

[0065] According to an embodiment of the present invention, the multiple hydrodynamic coefficient estimates may include dimensionless drag coefficient estimates and additional mass coefficient estimates.

[0066] According to an embodiment of the present invention, the simulated hydrodynamic torque is obtained by performing fluid dynamics simulation on the underwater manipulator. For example, a three-dimensional model of the underwater manipulator is selected, and a computational model (such as a turbulence model), mesh settings, and numerical simulation are performed to obtain the simulated hydrodynamic torque through fluid dynamics simulation.

[0067] The model hydrodynamic moment can be obtained by inputting multiple hydrodynamic coefficient estimates (such as dimensionless drag coefficient estimates and additional mass coefficient estimates) into the initial hydrodynamic moment model.

[0068] According to an embodiment of the present invention, spline fitting of the discrete sample set of coefficients using the least squares algorithm can be based on polynomial splines as fitting basis functions and the estimated hydrodynamic coefficients as optimization variables. Since the residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment, the identified target spline curve can fit the real hydrodynamic data.

[0069] According to an embodiment of the present invention, using the dimensionless travel ratio as the state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless drag coefficient and the added mass coefficient, the initial hydrodynamic moment model of the i-th segment is transformed into a linear model of the i-th segment. The hydrodynamic coefficients in each of the N segments' linear models are decoupled using a least squares algorithm, thereby transforming the identification of multiple hydrodynamic coefficients into a linear problem within the dimensionless travel ratio range. Based on the residual vector determined according to the difference between the simulated hydrodynamic moment and the model hydrodynamic moment, spline fitting is performed on the discrete sample set of coefficients using a least squares algorithm to obtain the target coefficient curve. This ensures that the identified target spline curve does not deviate from the actual simulated hydrodynamic moment curve, improving the fitting accuracy. The target coefficient curve can then be used for hydrodynamic moment prediction.

[0070] According to an embodiment of the present invention, the water flow parameters include at least one of the following: fluid density and projected area against the flow, and the robotic arm parameters include at least one of the following: the distance between the i-th segment and the robotic arm axis, instantaneous tangential velocity, angular acceleration and the equivalent moment of inertia of the water displaced by the i-th segment about the robotic arm axis.

[0071] The projected area of ​​the i-th segment facing the wind is A i for:

[0072] (4);

[0073] The drainage volume V of the i-th segment i for:

[0074] (5);

[0075] The distance between the i-th segment and the robotic arm's pivot is The instantaneous tangential velocity is U c,i Angular acceleration is The equivalent moment of inertia of the water displaced by the i-th segment about the axis of rotation of the robotic arm is: .

[0076] According to an embodiment of the present invention, an initial hydrodynamic torque model for the i-th segment is constructed based on water flow parameters, robotic arm parameters, dimensionless drag coefficient, and added mass coefficient. This includes: determining the initial drag torque term for the i-th segment based on fluid density, projected area facing the flow, dimensionless drag coefficient, instantaneous tangential velocity, and the distance between the i-th segment and the robotic arm's axis of rotation; determining the initial inertial torque term for the i-th segment based on the equivalent moment of inertia, added mass coefficient, and angular acceleration of the water displaced by the i-th segment about the robotic arm's axis of rotation; and constructing the initial hydrodynamic torque model for the i-th segment based on the initial drag torque term and the initial inertial torque term.

[0077] During the flow process, the flow field information derived from the fluid dynamics simulation is saved every 0.1 s, and piecewise integration is performed on each segment of the robotic arm to obtain the segment-level piecewise torque. Based on the time-varying total hydrodynamic torque and piecewise torque, and the establishment of the initial piecewise hydrodynamic torque model and state variable definitions, further decoupling and modeling of the hydrodynamic coefficient model curves are carried out. For the total hydrodynamic torque and piecewise torque, time axis alignment and necessary interpolation resampling are first performed, followed by integration with the torque obtained from the kinematic expression. Angular velocity, angular acceleration and They are then organized into a data table that can be used for inversion.

[0078] By combining piecewise geometry and the initial hydrodynamic model, the instantaneous torque of each segment can be calculated. Specifically, when the underwater manipulator rotates about its axis of rotation at an angular velocity w(t), the instantaneous tangential velocity at the reference point of the i-th segment is... For the case where a slender cylindrical arm rotates about one end of its axis of rotation, the initial resistance torque term of the i-th segment can be approximated. The square of the local relative velocity in the i-th segment Proportional, that is

[0079] (6);

[0080] Let be the dimensionless drag coefficient of the i-th segment, and t be the time variable.

[0081] The initial inertial torque term of the i-th segment reflects the fluid inertial effect propelled by accelerated motion. For rotational motion, the initial inertial torque term of the i-th segment... Usually related to angular acceleration Proportional:

[0082] (7);

[0083] in, Let be the equivalent moment of inertia of the water displaced by the i-th segment about the axis of rotation. is the additional mass coefficient for the i-th segment, used to correct the additional mass estimate under the assumption of ideal incompressible fluid.

[0084] Therefore, for the hydrodynamic torque of the i-th segment of the underwater robotic arm This can be expressed as the initial resistance torque term of the i-th segment. The initial moment of inertia term of the i-th segment The sum is as follows:

[0085] (8);

[0086] The total hydrodynamic torque of the underwater robotic arm is The sum of the hydrodynamic moments of each segment is as follows:

[0087] (9);

[0088] According to an embodiment of the present invention, using the dimensionless travel ratio as the state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless resistance coefficient and the added mass coefficient, the initial hydrodynamic torque model of the i-th segment is transformed into a linear model of the i-th segment. This includes: using the dimensionless travel ratio as the state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless resistance coefficient and the added mass coefficient, transforming the initial resistance torque term and the initial inertial torque term of the i-th segment into intermediate resistance torque term and intermediate inertial torque term of the i-th segment, respectively. The intermediate resistance torque term of the i-th segment is the product of the resistance motion kernel function and the dimensionless resistance coefficient, and the intermediate inertial torque term of the i-th segment is the product of the inertial force motion kernel function and the added mass coefficient; determining the intermediate hydrodynamic torque model of the i-th segment based on the intermediate resistance torque term and the intermediate inertial torque term of the i-th segment; and converting the intermediate hydrodynamic torque model of the i-th segment into matrix form to obtain the linear model of the i-th segment.

[0089] To facilitate the subsequent separation of the dimensionless drag coefficient and the added mass coefficient using the least squares method, the initial hydrodynamic moment model needs to be further rewritten into a linear form with respect to the coefficients to be identified.

[0090] The relationship between the dimensionless stroke ratio and the dimensionless drag coefficient and the added mass coefficient can be... , Based on formula (3), and taking the dimensionless stroke ratio as the state variable, the dimensionless drag coefficient can be expressed as: The additional quality coefficient can be expressed as .

[0091] Kernel function of resistance motion With dimensionless drag coefficient The product equals the intermediate resistance torque term of the i-th segment; inertial force motion kernel function. With additional quality coefficient The product of these terms equals the intermediate moment of inertia term of the i-th segment.

[0092] From formulas (8) to (9), it can be seen that, based on the intermediate resistance torque term and the intermediate inertia torque term of the i-th segment, the intermediate hydrodynamic torque model of the i-th segment is determined, that is, the hydrodynamic torque of the underwater manipulator in the i-th segment. It can also be expressed as follows:

[0093] (10);

[0094] in, , . It represents the normal velocity component of the reference point i relative to the surrounding fluid; for the single-arm planar rotation reference case, it can be degenerated into an expression equivalent to the tangential velocity.

[0095] If a narrow If the coefficients within the interval are approximated as constants, then for multiple time samples (t) falling within that interval... k A linear system can be established by converting the intermediate hydrodynamic moment model of the i-th segment into matrix form to obtain the linear model of the i-th segment, as follows:

[0096] (11);

[0097] Solving the linear model of the i-th segment using the least squares algorithm yields the discrete coefficient sample set within that bin. Along... By repeating the above process in different directions, a recognition interval covering the entire i-th segment can be gradually constructed. and Dotted clouds.

[0098] Although discrete sample sets of coefficients can reveal and Follow While the overall trend of point cloud changes can be observed, directly using point clouds for model calls can lead to problems such as discontinuities, local spikes, and noise differences between operating conditions. Therefore, it is necessary to further construct continuous, smooth, and physically constrained parametric curves. A piecewise spline method based on control points is used to... and Modeling is performed separately. Control points are located at... Several key control points are set on the axis, with the last control point x7 serving as a free control point. During the fitting process, it is automatically searched and determined and also serves as the starting point of the plateau segment. The coefficients of the region are taken as constant values ​​to reflect the physical fact that the coefficients tend to stabilize after the wake develops.

[0099] Regarding the constraints, the following physical requirements are imposed on the two curves: First, the initial state satisfies... , Secondly, in small Allowed within the range Build from scratch step by step The evolution should begin from near the potential flow value; thirdly, the curve should be as smooth as possible overall to avoid non-physical oscillations without data support; fourthly, the curve should remain constant after the plateau period to reduce high... The unstable effect of sparse samples on the distal shape. The spline interpolation form adopts the shape-preserving PCHIP (Piecewise Cubic Hermite Interpolating Polynomial) / piecewise cubic interpolation to balance smoothness and local monotonicity.

[0100] In optimizing the objective function design, a "two-layer consistency" approach is adopted. The first layer is... Spatial fitting coefficients on a discrete sample set, minimizing the spline curve and the global... , The first layer addresses the bias between samples; the second layer involves back-substituting the model's hydrodynamic torque in the time domain, requiring the total torque history reconstructed from the spline curves to be as consistent as possible with the simulated hydrodynamic torque. In summary, the optimization objective function can be written as:

[0101] (12);

[0102] This represents the fitting error of a discrete sample set. This indicates the back-substitution error of the model's hydrodynamic moment. This indicates the roughness penalty of the curve. and They are , The weights are determined by a grid search of these two types of weights. A balance can be struck between fitting accuracy and curve interpretability, ultimately solidifying a unified hydrodynamic moment model for widespread use.

[0103] According to an embodiment of the present invention, the above method further includes multi-dimensional sample quality control of the discrete sample set of coefficients, including: constructing the weight of each sample point based on the fitting residual of each sample point, wherein the weight represents the degree of influence of the sample point on the shape of the fitting curve; subjecting the value constraints of multiple hydrodynamic coefficient estimates in the sample points respectively; selecting the near-hinge segment from the N segments whose distance from the robotic arm pivot is less than a preset distance, and performing weight reduction or elimination processing on the sample points of the near-hinge segment.

[0104] All subsequent fitting, comparisons, and applications will be based on As the independent variable. Using "by" The actual calculation process using the "binning and intra-bin least squares inversion" method is as follows:

[0105] The time series of each working condition is arranged by Divide the area into several intervals (bins) from smallest to largest, with each interval covering a smaller area. .

[0106] Within each interval, assume that with Approximately constant, a pair of equivalent coefficients are obtained by fitting within this interval using linear least squares. , ).

[0107] The representative journey of this interval Using (such as the interval center or mean) as the independent variable, we obtain a discrete triplet sample { , , }

[0108] This method allows the inversion to find the optimal value within an interval scale, rather than calculating it step by step at every time step. This significantly suppresses the contamination of the coefficients by noise, while also preserving the main trend of coefficient evolution over time.

[0109] In the discrete sample set obtained by binning inversion, some points may deviate from the main trend due to weak local torque signals or numerical oscillations. If left untreated, this will directly skew the subsequent fitted curve.

[0110] Based on the fitting residuals of each sample point in the binning least squares stage, the larger the residual, the smaller the weight assigned in subsequent fitting, thereby weakening the pull of outliers on the curve shape.

[0111] Multiple hydrodynamic coefficients, including dimensionless drag coefficients and additional quality coefficient The estimation values ​​of multiple hydrodynamic coefficients in the sample points are subject to value constraints, including: and Apply hard lower bounds respectively At the same time, for Set a soft limit This is to prevent individual samples from generating unreasonable maxima due to weak signals, which could lead to numerical oscillations in subsequent spline fitting.

[0112] Because the segment near the hinge point has a smaller radius of motion when the robotic arm rotates around its axis, its local velocity, acceleration, and contribution to the total torque are significantly lower than those of the outer segment, and the vortex structure development is also relatively insufficient. Therefore, the inversion results of the segment near the hinge point are more susceptible to numerical noise. From the N segments, segments near the hinge point with a distance less than a preset distance from the robotic arm's axis are selected. The preset distance can be set according to actual conditions; for example, it can be 1 cm, 2 cm, etc., without limitation.

[0113] For the inner segments with smaller radii of motion, the local water velocity and acceleration are lower, resulting in smaller hydrodynamic moment amplitudes. During the inversion process, the effective signal is prone to approaching the numerical noise level, thus generating amplified anomalous coefficients. Sample points in these near-hinge segments are weighted or removed to prevent numerical artifacts from weak moment segments from entering subsequent fitting.

[0114] Figure 3A A scatter plot showing the dimensionless drag coefficient as a function of the dimensionless stroke ratio according to an embodiment of the present invention is shown.

[0115] Figure 3B A scatter plot showing the variation of the additional mass coefficient with the dimensionless stroke ratio according to an embodiment of the present invention is shown.

[0116] like Figure 3A and Figure 3B As shown, all working condition sample points are plotted as dimensionless drag coefficient C. d With the additional quality coefficient C m A scatter plot showing the dimensionless travel ratio S / d can be used to preliminarily determine the applicability of the single-variable representation of travel and the differences in sample quality (i.e., the dimensionless drag coefficient C in the plot). d With the additional quality coefficient C m This is obtained by decoupling the hydrodynamic coefficients in each of the N segments of the linear model using the least squares algorithm. As shown in the figure, with N=10, there are 10 segments of least squares samples. The segments near the hinge end (such as segment 1) have significantly higher sample dispersion due to their smaller radius of motion and weaker torque signal, with segment 1 showing the most significant deviation, and segments 2–4 also exhibiting some fluctuation. As the segment position moves outward, the local velocity and torque amplitudes increase, the sample points gradually converge, and a clearer common trend with S / d development is observed. These phenomena indicate that when performing C… d With C mWhen fitting curves, it is necessary to reduce the influence of low signal-to-noise ratio samples near the hinge end; after quality control, the sample points of the main force-bearing segments on the outer side can still show a relatively consistent evolution pattern on the same S / d axis, supporting the use of S / d as the dominant state variable for coefficient changes.

[0117] In terms of numerical trends, the dimensionless drag coefficient C d exist Initially, it rises rapidly, then gradually slows down after S / d enters a moderate range, and stabilizes at a level of approximately 0.5–0.6 in the larger S / d range. Additional mass coefficient C m In the initial stage, it is about 1. As S / d increases, it decreases and forms a trough. After that, it slowly recovers and stabilizes at about 1.5 in the larger S / d range.

[0118] According to an embodiment of the present invention, before obtaining the target coefficient curve by spline fitting of the discrete sample set of coefficients based on the residual vector using the least squares algorithm, the method further includes: obtaining the simulated hydrodynamic moment curve obtained by performing hydrodynamic simulation on the underwater manipulator, and smoothing the simulated hydrodynamic moment curve to obtain a smoothed simulated hydrodynamic moment curve; determining the positive peak inflection point and the negative peak inflection point of the smoothed simulated hydrodynamic moment curve based on the second derivative of the smoothed simulated hydrodynamic moment curve; and mapping the inflection point times corresponding to the positive peak inflection point and the negative peak inflection point to dimensionless lines respectively through kinematic relationships. The dimensionless travel ratio control points are obtained by using the travel ratio space; multiple preset control points are determined from the discrete sample set of coefficients; candidate positions of multiple platform control points are selected within the interval where the dimensionless travel ratio is greater than the multiple preset control points and the dimensionless travel ratio control points; the legality of the candidate positions is verified by hydrodynamic physical constraints, and iterative search of hyperparameters by spline fitting is performed to determine the platform control points; sample points with dimensionless travel ratios greater than the platform control points are determined as cutoff points, and the hydrodynamic coefficients corresponding to the cutoff points are taken as the hydrodynamic coefficients corresponding to the platform control points.

[0119] First, the simulated hydrodynamic moment curve is smoothed to obtain a physically smooth and reasonably shaped simulated hydrodynamic moment smooth curve in space.

[0120] Then, a consistency constraint on the total torque time history is introduced for joint correction, so that the spline number curve not only fits the scatter trend, but also can stably reconstruct the total torque response in the time domain.

[0121] The ordinate value of the control point of the fitted simulation hydrodynamic moment smooth curve ( or Before that, first adjust the horizontal axis of the simulated hydrodynamic torque smoothing curve ( Preprocessing is performed. The horizontal coordinate control points of the spline curve ( The location of the control points determines the distribution of degrees of freedom in different intervals of the model. If the location of the control points is treated as a free variable, the fitting process is easily influenced by sample discrepancies and noise, resulting in unnecessary local oscillations. Conversely, fixing key nodes at locations with significant physical transitions and consistent performance across operating conditions can significantly improve the stability and transferability of the fitting. Therefore, before spline fitting, two representative curvature change points are automatically extracted from the total moment curve and placed in... The clustering location in the space is used as a key reference x-coordinate to determine (or initialize) spline nodes.

[0122] To address the typical pattern of a monotonically increasing total torque that initially "steeps steeply and then gradually slows down," two inflection points are defined as: the simulated hydrodynamic torque smoothing curve. The second derivative Above, in chronological order, there is a positive wide peak (beginning to steepen) and a negative wide peak (beginning to soften).

[0123] For example, the extreme value of the second derivative is the peak of the second derivative. A candidate positive peak inflection point can be a local maximum of the curve, i.e., the highest point where the curve bulges upwards. A candidate negative peak inflection point can be a local minimum of the curve, i.e., the lowest point where the curve dips downwards.

[0124] By analyzing the inflection points corresponding to the positive and negative peaks, the instantaneous angular displacement can be obtained through kinematic relationships. Based on formula (3), the coordinates are mapped to the dimensionless travel ratio space to obtain the dimensionless travel ratio control points.

[0125] Multiple preset control points are determined from the discrete sample set of coefficients.

[0126] For example, to avoid the attraction of deceleration or later oscillations to inflection point location, inflection point search is performed by default only within the first 75% of the acceleration phase. The main inflection point result is primarily extracted from the total hydrodynamic torque curve output by fluid dynamics simulation; piecewise torques are only used for comparison and verification and for segmentation reference.

[0127] Fluid simulations were performed on the underwater robotic arm under various working conditions, and the hydrodynamic coefficients were decoupled and identified based on the simulated hydrodynamic torque curves under various working conditions, resulting in a discrete sample set of coefficients under various working conditions.

[0128] Statistical analysis of effective cases using discrete sample sets of coefficients under various operating conditions yielded two inflection points. The space exhibits a strong tendency to cluster:

[0129] (13);

[0130] Furthermore, the dispersion is relatively controllable (the standard deviation is on the order of 0.1~0.2). The statistical means of P1 and P2 are used as the key reference x-axis of the spline curve (two preset control points, such as...). , These correspond to the phase boundaries of "initial transition (steepening)" and "gradual transition (gradualizing)". The remaining preset control points are configured according to the sample distribution density and the requirements of the platform segment.

[0131] In terms of implementation, The positions of the first six preset control points can be:

[0132] (14);

[0133] Within the range where the dimensionless stroke ratio is greater than multiple preset control points (e.g., greater than 4.650) and the dimensionless stroke ratio control point, the stroke is large and the hydrodynamics tends to be saturated. Within this large range, multiple candidate positions of platform control points are selected in batches.

[0134] The legality of candidate locations is verified by hydrodynamic physical constraints, and the platform control points are determined by iterative search using spline fitting hyperparameter grids.

[0135] Hydrodynamic physical constraints can be hydrodynamic coefficients ( or The points must not exhibit abrupt changes, reverse abrupt changes, or exceed theoretical extremes; their concavity / convexity must conform to the torque curve law. Candidate points that violate hydrodynamic physical constraints are eliminated, retaining only physically reasonable locations. A mesh traversal is performed on the valid candidate point combinations and spline hyperparameters, and the fitting error is iteratively compared to select a set of optimal coordinates as the platform control points.

[0136] For example, if we introduce platform control point x7 as the end of the development segment / start of the platform segment, and its position is used as a variable to be optimized, and the interval of the dimensionless travel ratio control point is set to [0, 8], then a range constraint can be applied to platform control point x7. Hydrodynamic physical constraints can be defined by fixing the boundary conditions of the platform control point's ordinate as follows: .

[0137] The hydrodynamic coefficients corresponding to multiple preset control points, dimensionless stroke ratio control points, and platform control points are extracted from the discrete coefficient samples, i.e., control point pairing data. , )or( , The hydrodynamic coefficients at each location within the interval between two adjacent control points are calculated using piecewise cubic interpolation.

[0138] Sample points with a dimensionless travel ratio greater than the platform control point are identified as cutoff points. Saturation truncation means that interpolation calculations are no longer performed, and the cutoff point is directly considered. The value is forcibly assigned to the hydrodynamic coefficient corresponding to the platform control point.

[0139] To reduce overshoot and maintain the interpretability of the curve shape, a shape-preserving piecewise cubic Hermitian interpolation polynomial was used for interpolation between control points. Simultaneously, to explicitly express the plateauing characteristics of the medium-to-long stroke intervals, the independent variables were truncated.

[0140] (15);

[0141] s represents and the cutoff point corresponding As the actual evaluation value. When The time coefficient remains constant (platform extrapolation), thereby avoiding non-physical divergence caused by control point extrapolation.

[0142] According to an embodiment of the present invention, determining the positive and negative peak inflection points of the simulated hydrodynamic moment smoothing curve based on the second derivative of the simulated hydrodynamic moment smoothing curve includes: calculating the second derivative of the simulated hydrodynamic moment smoothing curve in chronological order, and determining at least one candidate positive peak inflection point and at least one candidate negative peak inflection point from the simulated hydrodynamic moment smoothing curve based on the extreme values ​​of the second derivatives; determining the positive peak confidence level of each of the at least one candidate positive peak inflection point based on the duration length of each of the at least one candidate positive peak inflection point and the second derivative value of the simulated hydrodynamic moment value, and taking the candidate positive peak inflection point with a positive peak confidence level greater than a first preset confidence level as a positive peak inflection point; determining the negative peak confidence level of each of the at least one candidate negative peak inflection point based on the duration length of each of the at least one candidate negative peak inflection point and the second derivative value of the simulated hydrodynamic moment value, and taking the candidate negative peak inflection point with a negative peak confidence level greater than a second preset confidence level as a negative peak inflection point.

[0143] The extreme values ​​of the second derivative are the peaks of the second derivative. A candidate positive peak inflection point can be a local maximum of the curve, i.e., the highest point where the curve bulges upwards. A candidate negative peak inflection point can be a local minimum of the curve, i.e., the lowest point where the curve dips downwards.

[0144] Because the second derivative of the simulated hydrodynamic moment smoothing curve exhibits multiple local extrema (such as a sawtooth-like curve), these local extrema are not true inflection points (i.e., spike noise) over a certain time period. Therefore, identifying at least one candidate positive peak inflection point and at least one candidate negative peak inflection point that satisfy a time length condition from the simulated hydrodynamic moment smoothing curve can filter out spike noise. The time length condition can be set to be equal to a preset time length (i.e., taking a broad peak), and the preset time length can be determined based on actual conditions (e.g., the time length between a local maximum extremum and a local minimum extremum).

[0145] First, fit a smooth spline at the original non-uniform time points. Then evaluate on a uniform time grid; subsequently in Peak retrieval is performed, and spike noise is filtered out based on the peak height and peak width, ultimately resulting in...

[0146] The confidence level of the positive peak = the second derivative of the simulated hydrodynamic moment value × the duration of the candidate positive peak inflection point (16);

[0147] The first preset confidence level can be determined based on the maximum positive peak confidence level among the at least one candidate positive peak inflection point. For example, the maximum positive peak confidence level among the at least one candidate positive peak inflection point can be 1, and the first preset confidence level can be 0.999. It should be noted that the positive peak inflection point is the inflection point with the highest positive peak confidence level among the at least one candidate positive peak inflection point.

[0148] Negative peak confidence = second derivative of simulated hydrodynamic moment value × duration of candidate negative peak inflection point (17);

[0149] The second preset confidence level can be determined based on the maximum value among the positive peak confidence levels of at least one candidate negative peak inflection point. It should be noted that the negative peak inflection point is the inflection point with the highest negative peak confidence level among at least one candidate negative peak inflection point.

[0150] The most reliable positive and negative peaks are selected as the two inflection points. The criterion emphasizes that the curvature change needs to persist over a certain time scale, which can suppress false detections caused by local high-frequency fluctuations and avoid the tendency to select accidental spikes based solely on peak height under "broad peak / broad plateau" transition.

[0151] Figure 4A A schematic diagram illustrating correct inflection point identification according to an embodiment of the present invention is shown.

[0152] like Figure 4A As shown, with an angular velocity of 0.25 and an angular acceleration of 0.10, the total hydrodynamic moment curve shows a clear transition and a significant second-order broad peak. The two inflection points (inflection point 1 and inflection point 2) are stably located and consistent with the physical stage boundary.

[0153] Figure 4B A schematic diagram illustrating interference with inflection point identification according to an embodiment of the present invention is shown.

[0154] like Figure 4B As shown, with an angular velocity of 0.375 and an angular acceleration of 0.05, the transition of the total hydrodynamic moment curve is characterized by a "broad peak" (duration) and is accompanied by later fluctuations. If only the peak height is relied upon, it is easily interfered with by sharp noise. However, the credibility score can more robustly lock the "continuous curvature change range", which reflects the necessity of the width factor in complex morphology.

[0155] According to an embodiment of the present invention, the inflection point times corresponding to the positive peak inflection point and the negative peak inflection point are mapped to the dimensionless travel ratio space through the kinematic relationship of the underwater manipulator to obtain the dimensionless travel ratio control point. This includes: obtaining the instantaneous angular displacement of the positive peak inflection point and the negative peak inflection point according to the inflection point times corresponding to the positive peak inflection point and the negative peak inflection point through the kinematic relationship of the underwater manipulator; mapping the instantaneous angular displacement of the positive peak inflection point and the negative peak inflection point to the dimensionless travel space to obtain the dimensionless travel of the positive peak and the dimensionless travel of the negative peak; and determining the dimensionless travel ratio control point based on the dimensionless travel of the positive peak and the dimensionless travel of the negative peak.

[0156] By taking the inflection point times corresponding to the positive and negative peaks, the instantaneous angular displacements of the positive and negative peaks are obtained through kinematic relationships. Based on formula (3), the instantaneous angular displacements of the positive and negative peaks are mapped to the dimensionless travel ratio space to obtain the dimensionless travel of the positive peak and the dimensionless travel of the negative peak.

[0157] When there are multiple dimensionless travel distances for both positive and negative peaks, the average value within each of the multiple positive peaks is used to determine a dimensionless travel ratio control point; the average value within each of the multiple negative peaks is used to determine a second dimensionless travel ratio control point.

[0158] According to an embodiment of the present invention, the residual vector includes a first residual vector and a second residual vector; based on the residual vector, spline fitting is performed on the discrete sample set of coefficients using the least squares algorithm to obtain the target coefficient curve, including: based on the first residual vector, performing first-level sample domain fitting on the discrete sample set of coefficients using the least squares algorithm to obtain an intermediate coefficient curve, wherein the first residual vector is composed of at least one of the following: fitting residuals obtained from linear model decoupling, smoothness regularization used to ensure curve smoothness, and physical soft penalty used to constrain the reasonable shape of the coefficients; based on the second residual vector, performing second-level torque domain fitting on the intermediate coefficient curve using the least squares algorithm to obtain the target coefficient curve, wherein the second residual vector is constructed based on the first residual vector and the torque residual vector, and the torque residual vector is determined based on the difference between the simulated hydrodynamic torque and the model hydrodynamic torque.

[0159] After the abscissa of the spline curve corresponding to the discrete sample set of coefficients is initialized, the ordinate identification and fitting of the two-level spline curve (Level-1 sample domain, Level-2 torque domain) can be performed.

[0160] The goal of the Level-1 sample domain is the discrete sample set of coefficients obtained from least squares decomposition. The principal curve shape is extracted. To suppress the destructive effect of poor-quality samples on the fit, the fitting residuals rel_rmse recorded by the sample points during the least squares decomposition (hydrodynamic coefficient decoupling) stage are used. i Construct weight wi :

[0161] (18);

[0162] in This is a very small numerical stability constant, used to avoid weight divergence when residuals approach zero and to limit the over-dominance of individual low-residual samples on the fitting results. Based on this, for larger... An attenuation coefficient is introduced into the interval, for example, the weight is halved when s>8, in order to reduce the pull of sparse far endpoints on the curve shape.

[0163] Interpolating point by point only in the discrete sample set of coefficients often solidifies the inversion noise into curve oscillations, thereby reducing the stability of torque reconstruction.

[0164] To extract transferable dominant laws, a two-level fitting strategy of Level-1 (sample domain) and Level-2 (torque domain) is adopted. In implementation, the Level-1 (sample domain) fitting optimization uses the least_squares function, and the TRF (trust-region reflective) algorithm is used to solve the nonlinear least squares problem with boundary constraints.

[0165] The fitting residuals obtained from decoupling the linear model include the fitting residuals of the dimensionless drag coefficient. Fitting residuals with and additional quality coefficients .

[0166] (19);

[0167] Let be the sample weights corresponding to the dimensionless drag coefficient and the added mass coefficient of the i-th sample point, respectively. These are the relative weights of the dimensionless drag coefficient and the additional mass coefficient, respectively (the default value for Level-1 is 1). This is the dimensionless drag coefficient after spline interpolation. The coefficient is the dimensionless resistance coefficient estimate for sample points in the discrete sample set. This is the additional quality coefficient after spline interpolation. This represents the estimated additional quality coefficients for sample points in the discrete sample set.

[0168] Smoothness regularization applies a second-order difference penalty to the ordinate of the control points to suppress unnecessary oscillations between control points. The smoothness penalty residual includes the smoothness penalty residual of the j-th control point on the dimensionless drag coefficient curve. The penalized residual at the j-th control point of the additional quality coefficient curve .

[0169] (20);

[0170] The ordinate of the j-th control point on the dimensionless drag coefficient curve (corresponding to the dimensionless drag coefficient value). The ordinate of the j-th control point of the additional quality coefficient curve (corresponding to the additional quality coefficient value).

[0171] The ordinates are the ordinates of three consecutive adjacent control points. It is a second-order difference operator. . It is the smoothing regularization weight coefficient, used to control the smoothness of the curve.

[0172] Physical soft punishment includes Apply a hard lower bound to An upper-bound soft penalty is introduced at the control points to suppress excessively large non-physical peaks caused by local noise or ill-conditioned decomposition. Upper-bound soft-penalty residuals. as follows:

[0173] (twenty one);

[0174] For the j-th Function values ​​of control points This is a soft upper bound threshold. You can choose 3.0. Let be the weighting coefficient of this penalty term in the total residual sum of squares, which can be taken as 5.0, indicating that when Control points exceeded When the limit is exceeded, the square of the excess portion will be included in the optimization objective with a weight of 5.0.

[0175] The above is the first residual vector. It consists of the fitting residuals obtained from the decoupling of the linear model, the smoothness regularization, and the physical soft penalty.

[0176] Level-1 (sample domain) fitting It can fit the overall trend of the sample point cloud while avoiding non-physical oscillations caused by local discrete points, thus providing a stable initial value for the subsequent introduction of torque constraints.

[0177] Only Spatial fitting of scatter points may cause the curve to be over-constrained by numerical algorithms, deviating from the true continuous torque curve. Level-2 (torque domain) fitting introduces the total torque time history residual on the basis of the Level-1 (sample domain) fitting solution and performs joint least squares optimization.

[0178] The second residual vector of the Level-2 (moment domain) fitting for:

[0179] (twenty two);

[0180] Let be the first residual vector. Controlling the weight of time-domain torque consistency in joint fitting The model hydrodynamic moment is obtained by inputting the estimated dimensions of the drag coefficient and the estimated values ​​of the added mass coefficient into the initial hydrodynamic moment model. To simulate hydrodynamic torque.

[0181] Level-1 ensures that the overall shape of the curve is consistent with the sample cloud, while Level-2 ensures that the model can stably reconstruct the total torque response in the time domain. The two work together to avoid the problem of "oversmoothing or local deviation from the actual torque curve" caused by fitting only in the sample domain.

[0182] According to an embodiment of the present invention, the multiple hydrodynamic coefficient estimates include dimensionless drag coefficient estimates and added mass coefficient estimates, and the target coefficient curves include dimensionless drag coefficient curves and added mass coefficient curves; the above method further includes: predicting the hydrodynamic torque of the underwater manipulator using the dimensionless drag coefficient curves and added mass coefficient curves based on the kinematic relationship of the underwater manipulator.

[0183] The identified dimensionless drag coefficient curve and additional mass coefficient curve can be combined with the kinematics of the robotic arm, under given working parameters (such as...). In the case of w(t), the torque of each segment and the total hydrodynamic torque of the joint can be calculated in real time without any flow field solution.

[0184] Figure 5A The diagram shows a comparison between the dimensionless drag coefficient samples obtained by least-squares decoupling according to an embodiment of the present invention and first-order / second-order splines.

[0185] Figure 5B The diagram shows a comparison between the additional quality coefficient samples obtained by least-squares decoupling according to an embodiment of the present invention and the first-order / second-order splines.

[0186] like Figure 5A and 5B In the figure, the dimensionless drag coefficient C of the 10 segments is shown. d With the additional quality coefficient C m The hydrodynamic coefficients in each of the 10 linear models were decoupled using the least squares algorithm (and...). Figure 3A and Figure 3B same).

[0187] A first-order spline can refer to the dimensionless drag coefficient C.d The intermediate coefficient curve or the additional quality coefficient C m The intermediate coefficient curve is obtained by fitting the first-level sample domain of the discrete sample set of coefficients using the least squares algorithm based on the first residual vector.

[0188] Second-order splines can refer to the dimensionless drag coefficient C. d Target coefficient curve or additional quality coefficient C m The target coefficient curve is obtained by fitting the intermediate coefficient curve in the second torque domain using the least squares algorithm based on the second residual vector.

[0189] Figure 6 The dimensionless drag coefficient curve and the added mass coefficient curve generated by the final control point according to an embodiment of the present invention are shown.

[0190] In joint fitting, and The trade-off between the scatter sample values ​​of the directly controlled torque and the total hydrodynamic torque value in the numerical simulation is addressed. For this set of hyperparameters, a grid search is employed: and Level-1 and Level-2 fitting were performed group by group, and the relative RMSE (Root Mean Square Error) of the total torque was calculated for the acceleration phase of all operating conditions. Simultaneously, a roughness index was constructed using the second-order difference of the control points (the smaller the index, the smoother the curve). Finally, a recommended model was selected from the combination with errors close to the optimal (within 1 percentage point of the optimal mean error) and the smallest roughness, for unified use in subsequent evaluation and plotting scripts. Among the 12 operating conditions used, the optimal combination was... , Under this optimal combination, the control point parameters of the hydrodynamic moment model finally identified are shown in Table 1, and the dimensionless drag coefficient curve and the added mass coefficient curve are shown in Table 1. Figure 6 As shown.

[0191] Table 1

[0192]

[0193] Figure 7 A structural block diagram of a device for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown.

[0194] like Figure 7 As shown, the underwater robotic arm hydrodynamic torque prediction device 700 of this embodiment includes a construction module 710, a conversion module 720, a decoupling module 730, and a spline fitting module 740.

[0195] The construction module 710 is used to construct an initial hydrodynamic moment model for the i-th segment of the underwater manipulator based on water flow parameters, manipulator parameters, dimensionless drag coefficient, and added mass coefficient. The underwater manipulator is divided into N segments along its axial direction, and i is an integer greater than or equal to 1 and less than or equal to N. In one embodiment, the construction module 710 can be used to perform the operation S110 described above, which will not be repeated here.

[0196] The conversion module 720 is used to convert the initial hydrodynamic moment model of the i-th segment into a linear model of the i-th segment, using the dimensionless stroke ratio as the state variable and based on the correlation between the dimensionless stroke ratio and the dimensionless drag coefficient and the added mass coefficient. In one embodiment, the conversion module 720 can be used to perform the operation S120 described above, which will not be repeated here.

[0197] The decoupling module 730 is used to decouple the hydrodynamic coefficients in each of the N segments of the linear model using a least squares algorithm, thereby obtaining a discrete sample set of coefficients. This discrete sample set includes sample points at multiple time points for each of the N segments, and each sample point includes multiple estimated hydrodynamic coefficient values. In one embodiment, the decoupling module 730 can be used to perform the operation S130 described above, which will not be repeated here.

[0198] The spline fitting module 740 is used to perform spline fitting on the discrete sample set of coefficients based on the residual vector using the least squares algorithm to obtain the target coefficient curve. This target coefficient curve is then used to predict the hydrodynamic moment of the underwater robotic arm. The residual vector is determined based on the difference between the simulated hydrodynamic moment and the model hydrodynamic moment. The simulated hydrodynamic moment is obtained through hydrodynamic simulation of the underwater robotic arm, while the model hydrodynamic moment is obtained by inputting multiple estimated hydrodynamic coefficient values ​​into an initial hydrodynamic moment model. In one embodiment, the spline fitting module 740 can be used to perform the operation S140 described above, which will not be repeated here.

[0199] According to embodiments of the present invention, any plurality of modules among the construction module 710, conversion module 720, decoupling module 730, and spline fitting module 740 may be combined into one module, or any one of these modules may be split into multiple modules. Alternatively, at least a portion of the functionality of one or more of these modules may be combined with at least a portion of the functionality of other modules and implemented in one module. According to embodiments of the present invention, at least one of the construction module 710, conversion module 720, decoupling module 730, and spline fitting module 740 may be at least partially implemented as hardware circuitry, such as a field-programmable gate array (FPGA), a programmable logic array (PLA), a system-on-a-chip, a system-on-a-substrate, a system-on-package, an application-specific integrated circuit (ASIC), or any other reasonable means of integrating or packaging circuitry, or implemented in software, hardware, or firmware, or in any suitable combination of any of these three implementation methods. Alternatively, at least one of the building module 710, the transformation module 720, the decoupling module 730, and the spline fitting module 740 may be implemented at least partially as a computer program module, which can perform corresponding functions when the computer program module is run.

[0200] Figure 8 A block diagram of an electronic device suitable for implementing a method for predicting the hydrodynamic torque of an underwater robotic arm according to an embodiment of the present invention is shown.

[0201] like Figure 8 As shown, an electronic device 800 according to an embodiment of the present invention includes a processor 801, which can perform various appropriate actions and processes according to a program stored in a read-only memory (ROM) 802 or a program loaded from a storage portion 808 into a random access memory (RAM) 803. The processor 801 may include, for example, a general-purpose microprocessor (e.g., a CPU), an instruction set processor and / or an associated chipset and / or a special-purpose microprocessor (e.g., an application-specific integrated circuit (ASIC)), etc. The processor 801 may also include onboard memory for caching purposes. The processor 801 may include a single processing unit or multiple processing units for performing different actions of the method flow according to an embodiment of the present invention.

[0202] RAM 803 stores various programs and data required for the operation of electronic device 800. Processor 801, ROM 802, and RAM 803 are interconnected via bus 804. Processor 801 executes various operations of the method flow according to embodiments of the present invention by executing programs in ROM 802 and / or RAM 803. It should be noted that the programs may also be stored in one or more memories other than ROM 802 and RAM 803. Processor 801 may also execute various operations of the method flow according to embodiments of the present invention by executing programs stored in said one or more memories.

[0203] According to an embodiment of the present invention, the electronic device 800 may further include an input / output (I / O) interface 805, which is also connected to a bus 804. The electronic device 800 may also include one or more of the following components connected to the input / output (I / O) interface 805: an input section 806 including a keyboard, mouse, etc.; an output section 807 including a cathode ray tube (CRT), liquid crystal display (LCD), etc., and a speaker, etc.; a storage section 808 including a hard disk, etc.; and a communication section 809 including a network interface card such as a LAN card, modem, etc. The communication section 809 performs communication processing via a network such as the Internet. A drive 810 is also connected to the input / output (I / O) interface 805 as needed. A removable medium 811, such as a disk, optical disk, magneto-optical disk, semiconductor memory, etc., is installed on the drive 810 as needed so that computer programs read from it can be installed into the storage section 808 as needed.

[0204] The present invention also provides a computer-readable storage medium, which may be included in the device / apparatus / system described in the above embodiments; or it may exist independently and not assembled into the device / apparatus / system. The computer-readable storage medium carries one or more programs, which, when executed, implement the method according to the embodiments of the present invention.

[0205] According to embodiments of the present invention, a computer-readable storage medium may be a non-volatile computer-readable storage medium, such as including, but not limited to: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. In the present invention, a computer-readable storage medium may be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, apparatus, or device. For example, according to embodiments of the present invention, a computer-readable storage medium may include ROM 802 and / or RAM 803 and / or one or more memories other than ROM 802 and RAM 803 described above.

[0206] Embodiments of the present invention also include a computer program product comprising a computer program containing program code for performing the methods shown in the flowchart. When the computer program product is run on a computer system, the program code is used to enable the computer system to implement the method for predicting the hydrodynamic torque of an underwater robotic arm provided in the embodiments of the present invention.

[0207] When the computer program is executed by the processor 801, it performs the functions defined in the system / apparatus of this invention. According to embodiments of the invention, the systems, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0208] In one embodiment, the computer program may rely on a tangible storage medium such as an optical storage device or a magnetic storage device. In another embodiment, the computer program may also be transmitted and distributed in the form of signals over a network medium, and may be downloaded and installed via the communication section 809, and / or installed from a removable medium 811. The program code contained in the computer program can be transmitted using any suitable network medium, including but not limited to: wireless, wired, etc., or any suitable combination thereof.

[0209] In such an embodiment, the computer program can be downloaded and installed from a network via communication section 809, and / or installed from removable medium 811. When the computer program is executed by processor 801, it performs the functions defined in the system of this embodiment of the invention. According to embodiments of the invention, the systems, devices, apparatuses, modules, units, etc., described above can be implemented by computer program modules.

[0210] According to embodiments of the present invention, program code for executing the computer programs provided in the embodiments of the present invention can be written in any combination of one or more programming languages. Specifically, these computational programs can be implemented using high-level procedural and / or object-oriented programming languages, and / or assembly / machine languages. Programming languages ​​include, but are not limited to, languages ​​such as Java, C++, Python, "C", or similar programming languages. The program code can be executed entirely on the user's computing device, partially on the user's device, partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0211] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of code containing one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions indicated in the blocks may occur in a different order than those indicated in the drawings. For example, two consecutively indicated blocks may actually be executed substantially in parallel, and they may sometimes be executed in reverse order, depending on the functions involved. It should also be noted that each block in a block diagram or flowchart, and combinations of blocks in a block diagram or flowchart, may be implemented using a dedicated hardware-based system that performs the specified function or operation, or using a combination of dedicated hardware and computer instructions.

[0212] Those skilled in the art will understand that the features described in the various embodiments of the present invention can be combined and / or combined in various ways, even if such combinations or combinations are not explicitly described in the present invention. In particular, the features described in the various embodiments of the present invention can be combined and / or combined in various ways without departing from the spirit and teachings of the present invention. All such combinations and / or combinations fall within the scope of the present invention.

[0213] The embodiments of the present invention have been described above. However, these embodiments are merely illustrative and not intended to limit the scope of the invention. Although various embodiments have been described above, this does not mean that the measures in the various embodiments cannot be used advantageously in combination. Various substitutions and modifications can be made by those skilled in the art without departing from the scope of the invention, and all such substitutions and modifications should fall within the scope of the invention.

Claims

1. A method for predicting the hydrodynamic torque of an underwater robotic arm, characterized in that, The method includes: For the i-th segment of the underwater manipulator, an initial hydrodynamic moment model for the i-th segment is constructed based on the water flow parameters, manipulator parameters, dimensionless drag coefficient, and added mass coefficient. The underwater manipulator is divided into N segments along the axial direction, where i is an integer greater than or equal to 1 and less than or equal to N. The water flow parameters are used to characterize the water environment in which the underwater manipulator is located, the manipulator parameters are used to characterize the structural features of the underwater manipulator, the dimensionless drag coefficient is used to describe the magnitude of the drag, and the added mass coefficient is used to describe the added inertial effect of the water body. Using the dimensionless stroke ratio as the state variable, and based on the correlation between the dimensionless stroke ratio and the dimensionless drag coefficient and the additional mass coefficient, the initial hydrodynamic moment model of the i-th segment is transformed into a linear model of the i-th segment. Here, the dimensionless stroke ratio is a dimensionless quantity that describes the distance an object travels in a fluid relative to its characteristic size. The hydrodynamic coefficients in each of the N linear models are decoupled using the least squares algorithm to obtain a discrete sample set of coefficients. The discrete sample set of coefficients includes sample points at multiple times for each of the N segments, and the sample points include multiple hydrodynamic coefficient estimates. Based on the residual vector, the least squares algorithm is used to perform spline fitting on the discrete sample set of coefficients to obtain the target coefficient curve, so as to use the target coefficient curve to predict the hydrodynamic torque of the underwater manipulator. The residual vector includes a first residual vector and a second residual vector. The first residual vector is composed of the fitting residual obtained by decoupling the linear model, smoothness regularization, and physical soft penalty. The second residual vector includes a torque residual vector determined according to the difference between the simulated hydrodynamic torque and the model hydrodynamic torque. The simulated hydrodynamic torque is obtained by performing hydrodynamic simulation on the underwater manipulator. The model hydrodynamic torque is obtained by inputting the estimated values ​​of the multiple hydrodynamic coefficients into the initial hydrodynamic torque model.

2. The method according to claim 1, characterized in that, The water flow parameters include at least one of the following: fluid density and projected area facing the flow. The robotic arm parameters include at least one of the following: the distance between the i-th segment and the robotic arm axis, instantaneous tangential velocity, angular acceleration, and the equivalent moment of inertia of the water displaced by the i-th segment about the robotic arm axis.

3. The method according to claim 2, characterized in that, The initial hydrodynamic moment model for the i-th segment is constructed based on water flow parameters, robotic arm parameters, dimensionless drag coefficient, and added mass coefficient, including: The initial resistance torque term of the i-th segment is determined based on the fluid density, the projected area facing the flow, the dimensionless drag coefficient, the instantaneous tangential velocity, and the distance between the i-th segment and the robotic arm's axis of rotation. Based on the equivalent moment of inertia of the water displaced by the i-th segment about the mechanical arm's axis of rotation, the additional mass coefficient, and the angular acceleration, the initial inertial torque term of the i-th segment is determined. Based on the initial resistance torque term and the initial inertial torque term of the i-th segment, the initial hydrodynamic torque model of the i-th segment is constructed.

4. The method according to claim 3, characterized in that, The method of using the dimensionless travel ratio as a state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless drag coefficient and the additional mass coefficient, transforming the initial hydrodynamic moment model of the i-th segment into a linear model of the i-th segment, includes: Using the dimensionless travel ratio as a state variable, and based on the correlation between the dimensionless travel ratio and the dimensionless resistance coefficient and the additional mass coefficient, the initial resistance torque term and the initial inertial torque term of the i-th segment are respectively converted into the intermediate resistance torque term and the intermediate inertial torque term of the i-th segment. The intermediate resistance torque term of the i-th segment is the product of the resistance motion kernel function and the dimensionless resistance coefficient, and the intermediate inertial torque term of the i-th segment is the product of the inertial force motion kernel function and the additional mass coefficient. Based on the intermediate resistance torque term and the intermediate inertia torque term of the i-th segment, the intermediate hydrodynamic torque model of the i-th segment is determined. The intermediate hydrodynamic moment model of the i-th segment is converted into matrix form to obtain the linear model of the i-th segment.

5. The method according to claim 1, characterized in that, Before obtaining the target coefficient curve by performing spline fitting on the discrete sample set of coefficients based on the residual vector and using the least squares algorithm, the method further includes: Obtain the simulated hydrodynamic torque curve obtained from the fluid dynamics simulation of the underwater robotic arm, and smooth the simulated hydrodynamic torque curve to obtain the smoothed simulated hydrodynamic torque curve; Based on the second derivative of the simulated hydrodynamic moment smoothing curve, determine the positive and negative peak inflection points of the simulated hydrodynamic moment smoothing curve. The inflection points corresponding to the positive peak inflection point and the negative peak inflection point are mapped to the dimensionless travel ratio space through kinematic relationships to obtain the dimensionless travel ratio control point. Multiple preset control points are determined from the discrete sample set of coefficients; Within the range where the dimensionless travel ratio is greater than the multiple preset control points and the dimensionless travel ratio control point, candidate positions of multiple platform control points are selected; the legality of the candidate positions is verified by hydrodynamic physical constraints, and the platform control points are determined by iterative search of hyperparameter mesh with spline fitting. Sample points with a dimensionless stroke ratio greater than that of the platform control point are identified as cut-off points, and the hydrodynamic coefficient corresponding to the cut-off point is set to the hydrodynamic coefficient corresponding to the platform control point.

6. The method according to claim 5, characterized in that, The step of determining the positive and negative peak inflection points of the simulated hydrodynamic moment smoothing curve based on the second derivative of the simulated hydrodynamic moment smoothing curve includes: Calculate the second derivative of the simulated hydrodynamic moment smoothing curve in chronological order, and determine at least one candidate positive peak inflection point and at least one candidate negative peak inflection point from the simulated hydrodynamic moment smoothing curve based on the extreme values ​​of the second derivative, satisfying the time length condition. Based on the duration of each candidate positive peak inflection point and the second derivative of the simulated hydrodynamic torque value, the positive peak confidence of each candidate positive peak inflection point is determined, and the candidate positive peak inflection point whose positive peak confidence is greater than the first preset confidence is taken as the positive peak inflection point. Based on the duration of each candidate negative peak inflection point and the second derivative of the simulated hydrodynamic torque value, the negative peak confidence level of each candidate negative peak inflection point is determined, and the candidate negative peak inflection point whose negative peak confidence level is greater than a second preset confidence level is taken as the negative peak inflection point.

7. The method according to claim 5, characterized in that, The step of mapping the inflection point times corresponding to the positive peak inflection point and the negative peak inflection point to the dimensionless stroke ratio space through the kinematic relationship of the underwater robotic arm to obtain the dimensionless stroke ratio control point includes: Based on the inflection point times corresponding to the positive peak inflection point and the negative peak inflection point, the instantaneous angular displacements of the positive peak inflection point and the negative peak inflection point are obtained through the kinematic relationship of the underwater robotic arm. The instantaneous angular displacements of the positive peak inflection point and the negative peak inflection point are respectively mapped to the dimensionless travel space to obtain the dimensionless travel of the positive peak and the dimensionless travel of the negative peak. The dimensionless travel distance of the positive peak and the dimensionless travel distance of the negative peak are used to determine the dimensionless travel distance ratio control point.

8. The method according to any one of claims 1 to 7, characterized in that, The step of performing spline fitting on the discrete sample set of coefficients based on the residual vector using the least squares algorithm to obtain the target coefficient curve includes: Based on the first residual vector, the least squares algorithm is used to perform a first-level sample domain fitting on the discrete sample set of coefficients to obtain an intermediate coefficient curve. The first residual vector consists of the fitting residual, the smoothness regularization used to ensure the smoothness of the curve, and the physical soft penalty used to constrain the reasonable shape of the coefficients. Based on the second residual vector, the intermediate coefficient curve is fitted to the second-level torque domain using the least squares algorithm to obtain the target coefficient curve. The second residual vector is constructed based on the first residual vector and the torque residual vector.

9. The method according to claim 8, characterized in that, The method further includes multi-dimensional sample quality control of the discrete sample set of coefficients, including: The weights of each sample point are constructed based on the fitting residuals of each sample point, wherein the weights represent the degree of influence of the sample point on the shape of the fitting curve. Each of the estimated hydrodynamic coefficients in the sample points is subject to value constraints. From the N segments, select the near-hinge segments whose distance from the robotic arm pivot is less than a preset distance, and then reduce the weight or remove the sample points of the near-hinge segments.

10. The method according to claim 1, characterized in that, The multiple hydrodynamic coefficient estimates include dimensionless drag coefficient estimates and added mass coefficient estimates, and the target coefficient curves include dimensionless drag coefficient curves and added mass coefficient curves; the method further includes: Based on the kinematic relationship of the underwater manipulator, the hydrodynamic torque of the underwater manipulator is predicted using the dimensionless drag coefficient curve and the additional mass coefficient curve.

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