Sensor chain body dynamic space positioning method and system of ship drag chain

Abstract

The invention relates to the technical field of ship underwater positioning, in particular to a dynamic space positioning method and system for a sensor chain body of a ship drag chain, and the method comprises the steps: dispersing the drag chain into a plurality of hinged rigid sections in a two-dimensional inertial reference system, connecting the sensor chain body to the tail end, and taking the motion of a ship as a traction point boundary; according to the given traction speed, the initial pitch angle, depth and tension distribution are obtained through static balance; calculating local speeds in the sections according to translation and rotation states of upstream nodes of all the sections, obtaining distributed hydrodynamic loads in combination with fluid parameters, establishing moment balance equations at all the upstream nodes, and forming a second-order ordinary differential equation set with a pitch angle as an unknown quantity; and numerical solution is carried out under a unified time step length, and the mass center positions of each node and a sensor chain body are obtained according to geometric recursion, so that dynamic positioning in an inertial reference system is realized. According to the invention, the precision and reliability of underwater target motion analysis and target tracking can be improved.

Description

Technical Field

[0001] This invention relates to the field of underwater positioning technology for ships, and in particular to a method and system for dynamic spatial positioning of the sensor chain of a ship's towing chain. Background Technology

[0002] In existing technologies, shipborne towed chains are based on the dynamic profile scanning principle of a ship towing a multi-node sensor chain, achieving continuous upper-level ocean observation through the coupling of the tow cable's pitch angle and speed. Specifically, the shipborne towed chain is released by a shipborne winch, and a sensor array chain is mounted on the tow cable. The chain can carry temperature, salinity, and depth (CTD) sensors, ecological sensors (chlorophyll, CDOM, turbidity, etc.); the chain moves within a specific water layer; and the multiple sensor observation nodes spaced apart on the chain can measure hydrodynamic data of their location in real time based on the onboard sensors.

[0003] To accurately estimate the target's position and motion state from the orientation information in hydrodynamic data, it is necessary to know the actual positions of each sensor unit in inertial space beforehand. However, the tow chain is subjected to gravity, buoyancy, and hydrodynamic forces that vary with ship speed during navigation, causing its attitude and spatial shape to change continuously over time. Especially under maneuvering conditions such as ship acceleration, deceleration, and turning, the spatial position of the sensor chain is not a simple geometric relationship with the ship's trajectory. In existing technologies, the chain shape is often estimated by simplifying it to a static catenary, empirical curves, or by interpolating with a small number of depth / attitude sensors. These methods are mostly based on steady-state or weak dynamic assumptions and are prone to significant deviations under strong maneuvers, large speed changes, or significant hydrodynamic influences. Summary of the Invention

[0004] In view of the above technical problems, the present invention provides a dynamic spatial positioning method and system for the sensor chain of a ship towing chain. Under the condition of known ship motion, by establishing the dynamic relationship between the towing chain and the sensor chain, the spatial position of the sensor chain in the inertial reference frame as a function of time can be obtained with high accuracy and continuously, thereby improving the accuracy and reliability of underwater target motion analysis and target tracking.

[0005] Other features and advantages of this disclosure will become apparent from the following detailed description, or may be learned in part from practice of this disclosure.

[0006] According to one aspect of the present invention, a method for dynamic spatial positioning of sensor chain members of a ship towing chain is proposed, the method comprising: A model of the towing chain and sensor chain is established in a two-dimensional inertial reference frame. The towing chain is discretized along its length into several rigid towing cable segments that are hinged to each other. The parameters of each towing cable segment include the segment length, mass, and equivalent diameter. The upstream end of the first towing cable segment is defined as the traction point, and the adjacent connection points between each towing cable segment are defined as nodes. The sensor chain is hinged to the downstream end of the downstream towing cable segment and records the position, velocity, and acceleration of the ship at each moment as the motion input of the traction point. At a given traction speed, static balance calculations are performed on the towing chain and the sensor chain to obtain the initial pitch angle, initial depth, and tension distribution, which are then used as the initial conditions for the dynamic solution. For any of the aforementioned tow cable segments, the local velocities of each position point within the segment are obtained based on the translational and rotational states of its upstream node, and the local velocities relative to the axis of the tow cable segment are decomposed into normal velocity components and tangential velocity components. Based on the normal velocity component, the tangential velocity component, and fluid density, equivalent diameter, and resistance parameters, the distributed hydrodynamic loads acting on each of the tow cable segments and the sensor chain are determined. Taking the upstream nodes of each tow cable segment and sensor chain as reference points, a moment balance equation including the distributed hydrodynamic load and downstream reaction force is established. The moment balance equation is written as a set of second-order ordinary differential equations with the pitch angle of each segment as unknown. The second-order ordinary differential equations are solved synchronously with a uniform time step to obtain the pitch angle and angular velocity of each tow cable segment and the sensor chain as a function of time. Based on the pitch angle and length of each tow cable segment, the position and velocity of each node are calculated using a geometric recursive method, and the position of the sensor chain relative to the two-dimensional inertial reference frame is output from the centroid coordinates of the sensor chain.

[0007] Furthermore, the towing chain is discretized along its length into two rigid towing cable segments, namely the first towing cable segment and the second towing cable segment. The upstream end of the first towing cable segment is the traction point A connected to the ship, and the connection node between the first towing cable segment and the second towing cable segment is node C. The downstream end of the second towing cable segment is hinged to the upstream end node E of the sensor chain. Torque balance equations are established at the traction point A, node C, and node E, respectively, and the three torque balance equations are solved simultaneously.

[0008] Furthermore, the calculation of the local velocity and the distributed hydrodynamic load satisfies the following rules: For any infinitesimal element within any of the aforementioned cable segments, whose distance from its upstream node is an arc length of r, its local velocity is the sum of the translational velocity and angular velocity of the upstream node and the vector product of the position vector pointing from the upstream node to the infinitesimal element. The distributed hydrodynamic loads include normal towing and tangential towing, wherein: The normal drag acts along the normal direction, and its element strength is calculated from the fluid density, normal drag coefficient, equivalent diameter of the drag cable segment, element length, and the product of the normal velocity component and the velocity magnitude. The tangential drag acts along the axial direction of the tow cable segment, and its micro-element strength is calculated by the fluid density, tangential drag coefficient, equivalent force-bearing area of ​​the micro-element, and half of the product of the tangential velocity component and the velocity magnitude. The directions of both the normal drag and the tangential drag are opposite to the corresponding local velocity directions.

[0009] Furthermore, the process of establishing the torque balance equation also includes: Taking the upstream node of each tow cable segment as the moment center, the torque integral is performed on the normal towing of the tow cable segment along the length direction to obtain the hydrodynamic torque of the tow cable segment about the upstream node, and the tangential towing does not generate a torque about the upstream node. The reaction force and reaction torque transmitted from the downstream adjacent tow cable segment and the sensor chain to the upstream node of the tow cable segment through the connecting node are included in the external torque of the tow cable segment. The integral of the torque caused by the normal drag along the length of the tow cable segment is numerically approximated using the Gauss-Legendre five-point quadrature method to obtain the total external torque of each tow cable segment and the sensor chain about its upstream node.

[0010] Furthermore, in establishing the torque balance equation, the method further includes: for each tow cable segment and the sensor chain, taking its upstream node as a reference point, establishing the angular momentum balance relationship according to the angular momentum theorem, that is: The external torque of the tow cable segment or the sensor chain is expressed as the sum of the inertial torque term about its center of mass and the torque term caused by the translational acceleration of the center of mass relative to the upstream node; By setting the external torque to the derivative of the angular momentum with respect to time, a second-order ordinary differential equation is obtained with the pitch angle of the tow cable segment or the sensor chain as the unknown quantity.

[0011] Furthermore, the static equilibrium calculation includes: In the vertical direction, the sum of the vertical components of the gravity, buoyancy, and traction tension of the drag chain and the sensor chain is made zero to determine the initial pitch angle; In the horizontal direction, the total drag force of all the tow cable segments and the sensor chain is balanced with the horizontal component of the traction tension to determine the initial depth and the tension distribution along the tow chain; The obtained initial pitch angle, initial depth, and tension distribution serve as the initial conditions for the second-order ordinary differential equation system.

[0012] Furthermore, the boundary conditions of the second-order ordinary differential equation system are set as follows: The displacement, velocity, and acceleration of the traction point of the towing chain at any given moment are the same as those of the ship. The downstream end of the sensor chain is a free end, and its reaction force and reaction torque are taken as zero. And, in solving the aforementioned system of second-order ordinary differential equations, the following are included: The second-order ordinary differential equations are numerically integrated at a uniform time step, and the pitch angle and angular velocity of all tow cable segments and sensor chains are updated simultaneously at each time step using the fourth-order Runge-Kutta method.

[0013] Furthermore, the method also includes: The centroid position of the output sensor chain and the azimuth measurement obtained by the passive detector are input into the target state estimation algorithm to perform azimuth-only target motion analysis and underwater target tracking, or to display the motion status of the underwater target relative to the sensor chain.

[0014] According to a second aspect of this disclosure, a sensor chain dynamic spatial positioning system for a ship towing chain is provided, the system comprising: The modeling module is used to establish models of the tow chain and sensor chain in a two-dimensional inertial reference frame. The tow chain is discretized along its length into several rigid tow cable segments that are hinged to each other. The parameters of each tow cable segment include the segment length, mass, and equivalent diameter. The upstream end of the first tow cable segment is defined as the traction point, and the adjacent connection points between each tow cable segment are defined as nodes. The sensor chain is hinged to the downstream end of the downstream tow cable segment and records the position, velocity, and acceleration of the ship at each moment as the motion input of the traction point. The static initialization module is used to perform static balance calculations on the towing chain and the sensor chain at a given traction speed, to obtain the initial pitch angle, initial depth and tension distribution, and to use them as the initial conditions for dynamic solution. The velocity field calculation module is used to obtain the local velocity of each position point within any tow cable segment based on the translational and rotational states of its upstream node, and to decompose the local velocity relative to the axis of the tow cable segment into normal velocity components and tangential velocity components. The load calculation module is used to determine the distributed hydrodynamic loads acting on each of the tow cable segments and the sensor chain based on the normal velocity component, the tangential velocity component, fluid density, equivalent diameter, and resistance parameters. The torque and equation establishment module is used to establish torque balance equations including the distributed hydrodynamic load and downstream reaction force, with the upstream nodes of each tow cable segment and sensor chain as reference points, and to write the torque balance equations as a set of second-order ordinary differential equations with the pitch angle of each segment as unknowns. The numerical solution module is used to synchronously solve the second-order ordinary differential equations at a uniform time step to obtain the pitch angle and angular velocity of each tow cable segment and the sensor chain as a function of time. The kinematic recursion module is used to calculate the position and velocity of each node according to the pitch angle and length of each tow cable segment in a geometric recursive manner, and output the position of the sensor chain relative to the inertial reference frame from the centroid coordinates of the sensor chain.

[0015] The technical solution disclosed herein has the following beneficial effects: This disclosure discretizes the towing chain into several rigidly hinged segments, introduces physical quantities such as chain mass parameters and equivalent diameter, uses the ship's motion as the traction boundary, and combines static equilibrium to obtain a reasonable initial attitude and depth. Based on this, it considers the normal and tangential hydrodynamic effects of each segment in the fluid, establishes a set of dynamic equations with the pitch angle of each segment as the state variable, and performs simultaneous numerical solutions. This allows for a unified description of the dynamic response of the towing chain and sensor chain throughout the entire time history. Through this dynamic spatial positioning method, the bending morphology and attitude evolution of the towing chain and sensor chain in water can be more realistically reflected under varying ship speeds and maneuvering conditions, thereby significantly improving the estimation accuracy of the sensor chain's center of mass position and reducing the target motion analysis error introduced by the uncertainty of the chain's attitude. Attached Figure Description

[0016] Figure 1 This is a flowchart of a sensor chain dynamic spatial positioning method for a ship towing chain according to an embodiment of this specification; Figure 2 This is a structural block diagram of a sensor chain dynamic spatial positioning system for a ship towing chain, as described in an embodiment of this specification. Detailed Implementation

[0017] Example embodiments will now be described more fully with reference to the accompanying drawings. However, example embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided to make this disclosure more comprehensive and complete, and to fully convey the concept of the example embodiments to those skilled in the art. The described features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a full understanding of embodiments of this disclosure. However, those skilled in the art will recognize that the technical solutions of this disclosure can be practiced with one or more of the specific details omitted, or other methods, components, apparatus, steps, etc., can be employed. In other instances, well-known technical solutions are not shown or described in detail to avoid obscuring various aspects of this disclosure.

[0018] Furthermore, the accompanying drawings are merely illustrative of this disclosure. The same reference numerals in the drawings denote the same or similar parts, and therefore repeated descriptions of them will be omitted. Some block diagrams shown in the drawings are functional entities and do not necessarily correspond to physically or logically independent entities. These functional entities may be implemented in software, in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0019] This invention provides a method for dynamic spatial positioning of a ship's towing chain using sensors. (Refer to...) Figure 1 The diagram shown is a flowchart illustrating a dynamic spatial positioning method for a ship towing chain using sensors, according to an embodiment of the present invention. This method can be applied to electronic devices such as personal computers, servers, and controllers. The method can be executed by a device, which can be implemented in software and / or hardware. Specifically, the method may include the following steps S101-S107.

[0020] In step S101, a model of the towing chain and sensor chain is established in a two-dimensional inertial reference frame. The towing chain is discretized along its length into several rigid towing cable segments that are hinged to each other. The parameters of each towing cable segment include the segment length, mass, and equivalent diameter. The upstream end of the first towing cable segment is defined as the traction point, and the adjacent connection points between each towing cable segment are defined as nodes. The sensor chain is hinged to the downstream end of the downstream towing cable segment and records the position, velocity, and acceleration of the ship at each moment as the motion input of the traction point.

[0021] First, a fixed two-dimensional inertial reference frame is selected within a plane encompassing the ship's direction of travel and the water depth. This reference frame remains stationary relative to the seawater and seabed, facilitating a unified description of the ship's, tow chain's, and sensor chain's motion within the same coordinate system. The inertial reference frame can be set with a point on the still water surface as its origin, the x-axis as a horizontal projection along the ship's direction of travel, and the y-axis along the vertical direction. Thus, the motion of the entire tow chain and sensor chain is constrained within this two-dimensional plane. Based on this, the actual continuous flexible tow chain is discretized using the lumped mass concept, treating it as a series of interconnected rigid tow cable segments. These rigid tow cable segments are kinematically connected through hinge points, allowing the calculation to approximate the macroscopic shape of the entire flexible cable by tracking only the translational and rotational states of a finite number of rigid bodies. For each tow cable segment, its physical parameters, such as length, mass, and equivalent diameter, are pre-defined. The segment length can be determined based on the actual cable length and discretization accuracy requirements. The mass can be considered as the lumped mass of the cable segment in the direction of gravity. The equivalent diameter is used to characterize the equivalent diameter of the segment under fluid forces in water. These parameters can be derived from the cable structure design and hydrodynamic characteristic calibration. Through the above processing, the originally continuous tow cable is abstracted into a finite number of rigid units with definite mass, geometric dimensions, and equivalent shape parameters, laying the foundation for establishing the force and moment balance relationship in the subsequent process.

[0022] Geometrically, the upstream end of the first tow cable segment of the tow chain is defined as the traction point. This traction point is connected to the ship's hull via a trolley or tow cable pulley, and is the unique dynamic boundary between the tow chain and the ship. The connection point between any two tow cable segments is defined as a node. Each node has time-varying coordinates in a two-dimensional inertial reference frame, used to describe the relative attitude and positional constraints between the tow cable segments. The sensor chain is positioned at the very end of the tow chain, with its upstream end hinged to the downstream end of the downstream tow cable segment. This hinge point is also defined as a node, allowing the sensor chain to rotate relative to the tow cable end in a two-dimensional plane, thus reflecting the attitude changes of the sensor chain relative to the tow cable end under the influence of gravity, buoyancy, and hydrodynamics. In an exemplary approach for ease of explanation, the tow chain can be discretized along its length into two rigid tow cable segments: a first tow cable segment and a second tow cable segment. The upstream end of the first tow cable segment is the traction point A, which is rigidly connected to the vessel. The connection node between the first and second tow cable segments is node C, representing the hinge position of the two cable segments. The downstream end of the second tow cable segment is hinged to node E, the upstream end of the sensor chain. Node E is both the end node of the second tow cable segment and the connection node between the sensor chain and the tow cable system. This combination of two tow cable segments and a sensor chain segment allows for the capture of relative attitude changes between the upstream and downstream segments of the tow cable and between the sensor chain while maintaining model simplicity.

[0023] When describing the motion of the nodes, the translational state of the ship at each moment is used as the boundary input of the traction point. That is, throughout the simulation or calculation process, the changes in the ship's position, velocity, and acceleration in the two-dimensional inertial reference frame over time are recorded. The fixed geometric relationship of the ship's towing cable traction point in the ship's hull coordinate system is transformed into the inertial reference frame, thereby obtaining the coordinates of traction point A at each discrete moment. ,speed and acceleration In the dynamic solution, the calculation process is discretized into equally spaced time steps on the time axis, and the time corresponding to the k-th time step is denoted as . The position of the traction point in the inertial reference frame is denoted as... Speed ​​is denoted as Acceleration is denoted as The equations describe the dynamics of the tow chain and sensor chain, where position has the dimension of length, velocity has the dimension of length / time, and acceleration has the dimension of length / time squared. Each of these physical quantities is used as a known boundary condition at each time step and substituted into the dynamic equations of the tow chain and sensor chain to solve for the pitch angle and angular velocity at the corresponding time. During numerical integration, the time step is used to discretize the solution process, and the known position, velocity, and acceleration of the traction point are used as initial conditions for synchronous solution. Since traction point A is rigidly connected to the ship, the macroscopic motion of the ship can be directly equivalent to the translational boundary conditions of traction point A during modeling. This allows subsequent mechanical analysis to be performed only within the tow chain and sensor chain system, without explicitly modeling the hull structure in the same set of dynamic equations.

[0024] After completing the geometric and kinematic modeling described above, in order to further establish the dynamic equations needed in subsequent steps based on this discrete model, the torque balance relationship with reference points can be considered at traction point A, node C, and node E respectively. Specifically, for the system consisting of two rigid tow cable segments and the sensor chain, at each node, the rigid body connected to that node can be regarded as a whole, and the external forces and inertial torques acting on it can be summarized, and the equilibrium equation for rotation can be written at that node. Taking traction point A as an example, the rotation of the first tow cable segment about traction point A can be regarded as a rigid body rotation problem. Considering the effects of the weight and buoyancy of the first tow cable segment at its center of mass, the equivalent effect of the hydrodynamic force on the cable segment in the water, the internal forces and internal torques from the direction of node C, and the constraint reaction forces between the traction point and the ship, the torques of these forces relative to traction point A are vector-summed. The resulting total external torque is balanced with the rate of change of angular momentum of the first tow cable segment about traction point A, thus establishing a torque balance equation at traction point A. Similarly, at node C, the first and second tow cable segments are extracted separately, and their respective force and torque contributions to node C are analyzed, resulting in the corresponding torque balance relationships. At node E, the second tow cable segment and the sensor chain are considered together, and a torque balance equation for node E is established. In this way, in the simplest demonstration case of two rigid tow cable segments plus a sensor chain segment, three torque balance equations with pitch angles or equivalent attitude variables of each segment as unknowns can be obtained at the traction point A, node C, and node E. These three equations are solved simultaneously in the same time step, thus providing a unified and consistent modeling foundation for further deriving the dynamic equations and spatial attitude of the entire tow chain and sensor chain system in subsequent steps.

[0025] In step S102, at a given traction speed, static balance calculations are performed on the towing chain and the sensor chain to obtain the initial pitch angle, initial depth, and tension distribution, which are then used as the initial conditions for the dynamic solution.

[0026] Specifically, given a traction speed, we first assume the vessel is traveling at a stable, constant speed in calm water. At this point, the tow chain and sensor chain in a two-dimensional inertial reference frame will gradually converge to a set of equilibrium attitudes and spatial shapes that do not change with time, i.e., the so-called steady-state equilibrium configuration. To obtain this steady-state configuration, this embodiment performs static equilibrium calculations on the tow chain and sensor chain as a whole: From an overall perspective, the tow chain is considered to be composed of several rigid tow cable segments connected in series, with the sensor chain hinged to the last tow cable segment, the lower end being a free end, and the upper traction point connected to the vessel via a cable car or pulley. After the traction speed is given, the fluid forces such as resistance and lift on the cable segments and sensor chain in the water tend to stabilize. At this point, the attitude and forces at each node satisfy the balance relationship of force and torque, and there are no longer any acceleration terms that change with time. To simplify the dynamic solution, this embodiment first solves the steady-state static equilibrium problem. The initial pitch angle, depth, and tension distribution of the drag chain and sensor chain are obtained through static equilibrium calculation, and these are used as the initial conditions for the dynamic equation. This allows the time-domain numerical integration to start from the physically reasonable configuration that satisfies force equilibrium, thereby improving the stability of the numerical solution and reducing meaningless transient oscillations.

[0027] The static equilibrium calculation specifically includes: in the vertical direction, making the sum of the vertical components of the gravity, buoyancy, and traction tension of the towing chain and the sensor chain zero to determine the initial pitch angle; in the horizontal direction, balancing the total towing force of all towing cable segments and the sensor chain with the horizontal component of the traction tension to determine the initial depth and the tension distribution along the towing chain; the obtained initial pitch angle, initial depth, and tension distribution serve as the initial conditions for the second-order ordinary differential equation system.

[0028] In vertical static equilibrium, the vertical components of gravity, buoyancy, and traction tension are considered for the entire tow chain and sensor chain. The gravity of each rigid cable segment of the tow chain can be expressed as the product of the segment's mass and gravitational acceleration. The gravity of the entire sensor chain can be expressed as the mass multiplied by gravitational acceleration. The buoyancy is calculated based on the displacement volume and fluid density. Under steady-state equilibrium, the overall pitch attitude of the towing system near the traction point is represented by angles. The angle is described as the average tilt angle of the traction chain relative to the vertical direction in a two-dimensional inertial reference frame; the direction of the tension on the sensor chain from the cable end is represented by the angle. The characterization is defined as the angle between the cable end tension and the vertical direction. By decomposing the forces acting on the sensor chain, the static equilibrium relationship in the vertical direction can be obtained: ; in, The magnitude of the tension applied by the end of the tow cable to the sensor chain. Let represent the total buoyancy acting on the sensor chain. The above formula shows that the difference between the weight and buoyancy of the sensor chain is balanced by the vertical component of the cable tension. For each segment of the tow chain, a similar vertical force balance can be used: the difference between the weight and buoyancy of each tow cable segment, as well as the vertical component of the tension transmitted from the downstream segment or the sensor chain, are balanced by the vertical component of the tension transmitted from the upstream segment or the traction point. By performing vertical force balance segment by segment along the tow cable from bottom to top, a set of tension component relationships related to the pitch angle can be obtained, and based on this, the pitch attitude of the entire tow chain and sensor chain in steady state, i.e., the initial pitch angle, can be determined.

[0029] In the horizontal static equilibrium, the main consideration is the balance between the total horizontal towing force and the horizontal component of the traction tension experienced by the rigid towing cable segments of the towing chain and the sensor chain body during navigation in water. For the sensor chain body, it experiences resistance along the course of travel at a constant traction speed. This resistance can be calculated based on the fluid density, the equivalent force-bearing area of ​​the sensor chain, and the resistance coefficient, and is balanced with the horizontal component of the cable end tension in steady state. This balance relationship can be expressed as: ; Therefore, the total horizontal drag force of the sensor chain is equal to the horizontal component of the tension applied at its hinge points. For each segment of the drag chain, at a given traction speed, each segment will also experience a horizontal drag force related to its geometry, equivalent diameter, pitch angle, and local flow velocity. These drag forces are transmitted upwards along the tow cable and balanced by the horizontal components of tension at the upstream and downstream nodes of each segment. Overall, at the traction point, the horizontal component of the traction tension should be equal to the sum of the drag forces along the entire tow cable and sensor chain. By establishing the balance relationship between the drag force and the horizontal components of tension in the horizontal direction, and combining it with the force balance in the vertical direction, the pitch angle of each segment, the position of each node, and the magnitude and direction of the tension along the tow chain from top to bottom can be solved, thus obtaining the initial depth and tension distribution.

[0030] In static equilibrium calculations, the given traction speed directly determines the magnitude of the hydrodynamic forces exerted by the fluid on each segment of the tow chain and the sensor chain. It is typically assumed that the fluid is uniform and stationary, and the relative velocity between the tow cable and the sensor chain is approximately equal to the projection of the ship's velocity in the inertial reference frame onto each segment. Therefore, after a given traction speed, there is a coupling between the horizontal towing force of each segment and its orientation (pitch angle). Simultaneously, the difference between gravity and buoyancy in the vertical direction is also reflected in the pitch attitude through the direction and magnitude of the tension. The static equilibrium solution seeks a set of pitch angles and tension distributions that satisfy force balance in both the vertical and horizontal directions under this coupling relationship, ensuring that each node of the tow cable and the sensor chain forms a self-consistent spatial configuration in the two-dimensional inertial reference frame. The initial pitch angle obtained from the solution describes the tilt attitude of the tow chain and sensor chain relative to the vertical direction in steady state, the initial depth is given by the coordinates of each node in the vertical direction, and the tension distribution reflects the force transmission along each segment of the tow chain from the traction point to the sensor chain.

[0031] Finally, the initial pitch angle, initial depth, and tension distribution along the tow chain obtained through the above static equilibrium calculation are used as the initial conditions for the subsequent second-order ordinary differential equation system with the pitch angle of each segment as the unknown. This is directly used in the time domain numerical integration, so that the dynamic solution starts from the physically reasonable configuration that satisfies the force balance, thereby improving the stability and accuracy of the dynamic spatial positioning results of the tow chain and sensor chain.

[0032] In step S103, for any of the tow cable segments, the local velocity of each position point within the segment is obtained based on the translational and rotational states of its upstream node, and the local velocity relative to the axis of the tow cable segment is decomposed into a normal velocity component and a tangential velocity component.

[0033] Wherein, for any infinitesimal element within any of the tow cable segments with an arc length r from its upstream node, its local velocity is the sum of the translational velocity and angular velocity of the upstream node and the vector product of the position vector pointing from the upstream node to the infinitesimal element.

[0034] For any rigid tow cable segment, it is necessary to determine the local velocity distribution at any point within the segment, based on the translational and rotational states of the upstream node of the tow cable segment, within the aforementioned two-dimensional inertial reference frame. Furthermore, the local velocity is decomposed relative to the axis of the tow cable segment. For this purpose, the tow cable segment is considered a rigid body, its upstream node is traction point A, and the velocity of traction point A in the two-dimensional inertial reference frame is denoted as... The velocity components in the corresponding coordinate directions are respectively and ,Right now The tow cable segment rotates in a two-dimensional plane about an axis perpendicular to that plane; its angular velocity vector is denoted as... ,in It is a unit vector pointing out of the paper. Let be the angular velocity scalar of the tow cable segment in the plane. Taking the upstream node of the tow cable segment as a reference, consider any point X within the segment. The arc length of this point from the upstream node along the axis of the tow cable segment is r. The position vector from the traction point A to point X can then be written as: ,in Let X be the attitude angle of the tow cable segment axis relative to the x-axis of the inertial reference frame. According to the principles of rigid body kinematics, the local velocity at point X is... The local velocity is the sum of the translational velocity and rotational velocity of the upstream node, i.e., the sum of the translational velocity and angular velocity of the upstream node and the vector product of the position vector pointing from the upstream node to the infinitesimal element. This can be expressed as: ; in Expanding the above vectors, we get: ; Furthermore: ; After calculating the cross product, the components of the local velocity along the coordinate axes are as follows: ; Therefore, for any infinitesimal element within a tow cable segment with an arc length r from its upstream node, its local velocity is precisely obtained by superimposing the translational velocity and angular velocity of the upstream node with the vector product of the position vector pointing from the upstream node to the infinitesimal element, fully demonstrating the velocity field distribution characteristics of the rigid tow cable segment under the combined action of translation and rotation.

[0035] After obtaining the vector expression for the local velocity, to facilitate subsequent processing, it is necessary to further determine the magnitude of the local velocity and its pointing angle relative to the coordinate axes of the inertial reference frame. Local velocity vector The modulus can be directly calculated from the above components, and the result is as follows: ; The above expression reflects that the magnitude of the local velocity depends not only on the translational velocity of the upstream node, but also on the angular velocity of the tow cable segment. Micro-element position r and attitude angle of tow cable segment Related. The direction of local velocity can be represented by its angle relative to the x-axis. This is indicated by the ratio of the velocity components: ; Under this definition, the local velocity vector The direction in a plane is determined by angle. The only key point is that, in subsequent decomposition relative to the tow cable segment axis, this angle and the tow cable segment attitude angle can be directly used. The difference is used to determine the components of the local velocity in the axial and normal directions.

[0036] Considering that the axial direction of the tow cable segment in the two-dimensional plane is determined by the attitude angle To decompose the local velocity relative to the axis of the tow cable segment into normal and tangential components, the axis of the tow cable segment can be considered as a line passing through the upstream node and making an angle of θ with the horizontal direction. The straight line. Thus, the local velocity vector... The angle between the cable section axis and the tow cable axis is Under this geometric relationship, the component of the local velocity along the axis of the towing cable segment is the tangential velocity component, and the component perpendicular to the axis of the towing cable segment is the normal velocity component. The magnitude of the normal component of the local velocity can be directly obtained through the sine relationship between the magnitude of the local velocity and the included angle, specifically: ; in The magnitude of the local velocity vector. This represents the component of the local velocity in the normal direction of the towing cable segment. This normal velocity component describes the lateral motion of the infinitesimal element relative to the axis of the towing cable segment and is a key quantity determining the normal hydrodynamic force (i.e., the fluid action that generates bending moment on the towing cable segment). Correspondingly, the local velocity in the same direction as the axis of the towing cable segment constitutes a tangential velocity component. The tangential velocity component is distributed along the axis of the towing cable segment and is mainly related to the parallel flow along the length of the cable, usually associated with the tangential action that generates axial drag. In this step, only the decomposition process of the local velocity is explained, that is, through the direction angle of the local velocity vector. and the attitude angle of the towing cable section The projections of the local velocity onto the two orthogonal directions of the towing cable segment, namely the normal and tangential directions, are determined, thereby obtaining the corresponding normal velocity component and tangential velocity component at each arc length position r. This lays the foundation for subsequent calculations of hydrodynamic loads such as normal towing and tangential towing based on these two velocity components.

[0037] In step S104, the distributed hydrodynamic loads acting on each of the tow cable segments and the sensor chain are determined based on the normal velocity component, the tangential velocity component, and the fluid density, equivalent diameter, and resistance parameters. The distributed hydrodynamic load includes normal towing and tangential towing, wherein: the normal towing acts along the normal direction, and its element strength is calculated by the fluid density, normal drag coefficient, equivalent diameter of the towing cable segment, element length, and the product of normal velocity component and velocity magnitude; the tangential towing acts along the axial direction of the towing cable segment, and its element strength is calculated by the fluid density, tangential drag coefficient, equivalent force-bearing area of ​​the element, and half the product of tangential velocity component and velocity magnitude; the directions of both the normal towing and the tangential towing are opposite to the corresponding local velocity direction.

[0038] In this embodiment, based on the local velocity vectors obtained at any position within each tow cable segment according to the translational and rotational states of the upstream node, it is necessary to further combine the normal and tangential velocity components of the local velocity with fluid density, equivalent diameter, and drag parameters to determine the distributed hydrodynamic loads acting on each tow cable segment and sensor chain. This embodiment considers the tow cable segment and sensor chain as slender bodies immersed in a uniform fluid, and their local velocity in the two-dimensional inertial reference frame is given in the previous step. For any infinitesimal element within a tow cable segment with an arc length r from the upstream node, this infinitesimal element can be considered as a small line segment of length dr, subjected to pressure drag (normal drag) and shear friction drag (tangential drag) generated by the fluid flow. Due to the local velocity vector... Generally, it does not coincide with the axis of the towing cable segment. Therefore, it can be decomposed relative to the axis of the towing cable segment into a tangential velocity component along the axis and a normal velocity component perpendicular to the axis. The normal velocity component reflects the degree of lateral flow through the infinitesimal element relative to the axis of the towing cable segment and mainly determines the magnitude of the normal drag. The tangential velocity component reflects the degree of relative slippage of the infinitesimal element along the axis and is closely related to shear friction drag.

[0039] To accurately represent distributed hydrodynamic loads, this embodiment uses a vector expression that aligns the hydrodynamic forces on the infinitesimal element with the direction of the local velocity vector. For any tow cable segment, with its upstream node as point A, the local velocity vector... The components in the inertial reference frame have been clearly given in the previous implementation step: ; The local velocity magnitude is: ; in, and These are the velocity components of the upstream traction point A in the x and y directions of the inertial reference frame, respectively. Let be the angular velocity of the tow cable segment. The attitude angle of the tow cable segment relative to the horizontal axis. Let be the unit basis vector of the inertial reference frame. Based on this local velocity vector and its magnitude, in a fluid density of , The equivalent diameter D of the towing cable section and the normal resistance coefficient are: The tangential drag coefficient is Under the given conditions, a distributed model is performed on the normal and tangential drag forces acting on the infinitesimal elements on the cable segment.

[0040] For normal drag, this embodiment considers it as resistance formed by pressure distribution around the cross-section of the towing cable segment. This resistance acts locally along the normal direction of the infinitesimal element and is proportional to the square of the normal velocity component. By substituting the local velocity component, the normal drag coefficient, and the equivalent diameter into the resistance calculation formula, the normal drag force vector acting on the infinitesimal element at arc length position r can be obtained, expressed as: ; In the above formula, The term corresponds to the component of the local velocity in the normal direction of the tow cable segment, and its square reflects the increase of normal drag with the square of the normal velocity component; the preceding... The term gives the vector direction of the local velocity in the inertial reference frame, such that the normal drag is constructed along the opposite direction of the local velocity vector. Coefficient This incorporates the effects of fluid density, normal drag coefficient, and equivalent diameter of the towing cable segment on the normal drag strength. The infinitesimal element length dr indicates that the force is in the form of linear density after integrating over a unit arc length. The negative sign indicates that the direction of the normal drag is opposite to the direction of the local velocity vector, thus hindering local motion.

[0041] For tangential towing, this embodiment considers it as resistance caused by shear friction between the fluid and the surface of the towing cable segment. Tangential towing acts along the axial direction of the towing cable segment and is mainly related to the tangential component of the local velocity. Considering that the circumferential force-bearing area of ​​the towing cable segment about its axis is related to its diameter, the equivalent force-bearing area of ​​the infinitesimal element can be written as... When the fluid density is The tangential drag coefficient is Under the given conditions, the tangential drag force vector of the infinitesimal element can be written as: ; in, The term corresponds to the tangential component of the local velocity along the axis of the tow cable segment, and its square reflects the change of tangential drag with the square of the tangential velocity component. This reflects the area of ​​the infinitesimal element subjected to force along the circumference. This is the scaling factor for tangential drag. Similar to normal drag, the velocity direction term preceding the tangential drag vector ensures that the overall direction of tangential drag is opposite to the direction of the local velocity vector, thus correctly representing its effect of hindering axial slippage.

[0042] Through the two vector expressions described above, this embodiment continuously presents the distribution of normal and tangential drag on the arc-length coordinate r. That is, for any tow cable segment, the distributed hydrodynamic load along the segment can be obtained based on parameters such as local velocity, fluid density, equivalent diameter, and normal and tangential drag coefficients over its entire length from the upstream node to the downstream node. In specific implementation, the sensor chain can also be regarded as a slender body composed of several infinitesimal elements. Following the same form as the tow cable segment, its equivalent diameter, stressed length, and corresponding hydrodynamic drag coefficient are used to calculate the normal and tangential drag at any arc-length position on the sensor chain, thereby obtaining the distributed hydrodynamic load of the entire tow chain and sensor chain within a unified framework. It is important to emphasize that in this step, only the distribution and magnitude of the two types of hydrodynamic loads, normal drag and tangential drag, are determined on each segment. Their directions are all opposite to the direction of the corresponding local velocity, which is used to characterize the resistance effect of the fluid on the moving tow cable segment and sensor chain. These distributed hydrodynamic loads will be further incorporated into the torque balance and dynamic equations in subsequent steps.

[0043] In step S105, with the upstream nodes of each tow cable segment and sensor chain as reference points, a moment balance equation including the distributed hydrodynamic load and downstream reaction force is established, and the moment balance equation is written as a set of second-order ordinary differential equations with the pitch angle of each segment as unknown. When establishing the torque balance equation, the method further includes: taking the upstream node of each tow cable segment as the moment center, performing torque integration on the normal towing along the length direction of the tow cable segment to obtain the hydrodynamic torque of the tow cable segment about the upstream node, and tangential towing does not generate torque about the upstream node; the reaction force and reaction torque transmitted from the downstream adjacent tow cable segment and the sensor chain to the upstream node of the tow cable segment through the connecting node are included in the external torque of the tow cable segment; the torque integration caused by the normal towing along the length direction of the tow cable segment is numerically approximated using the Gauss-Legendre five-point quadrature method to obtain the total external torque of each tow cable segment and the sensor chain about its upstream node.

[0044] Furthermore, for each of the aforementioned tow cable segments and sensor chains, an angular momentum balance relationship is established according to the angular momentum theorem, with its upstream node as the reference point. That is, the external torque of the tow cable segment or the sensor chain is expressed as the sum of the inertial torque term about its center of mass and the torque term caused by the translational acceleration of the center of mass relative to the upstream node. Let the external torque be equal to the derivative of the angular momentum with respect to time, and a second-order ordinary differential equation is obtained with the pitch angle of the tow cable segment or the sensor chain as the unknown.

[0045] The boundary conditions of the second-order ordinary differential equation system are set as follows: the displacement, velocity, and acceleration of the traction point of the towing chain at any time are the same as the displacement, velocity, and acceleration of the ship; the downstream end of the sensor chain is a free end, and its reaction force and reaction torque are zero.

[0046] In order to incorporate the aforementioned distributed hydrodynamic loads and the reaction forces and moments transmitted by the downstream section and sensor chain through the connecting nodes into the dynamic equations, this embodiment establishes a moment balance relationship for each tow cable segment and sensor chain with its upstream node as the reference point. Using the angular momentum theorem, this moment balance relationship is written as a second-order ordinary differential equation with the pitch angle of each segment as the unknown quantity. Specifically, taking any tow cable segment as a rigid body AB, its upstream traction point A is connected to the ship, and its downstream node B is hinged to the downstream tow cable segment or sensor chain. In a two-dimensional inertial reference frame, the angular momentum of this rigid body about point A is... This can be written as the sum of the angular momentum at the center of mass G and the translational momentum of the center of mass with respect to point A: ; in, Let G be the angular momentum of the rigid body about its center of mass G. Let G be the vector pointing from the upstream traction point A to the centroid G. Let G be the linear momentum acting on the center of mass G. The resultant force of external forces acting on this rigid body, including gravity, buoyancy, drag, and downstream reaction force, is denoted as . Its torque about G is According to the angular momentum theorem, the sum of the external torques at point A should be equal to the sum of the time derivative of the angular momentum and the term of the translational acceleration of the center of mass, that is: ; in, Let I be the moment of inertia about the center of mass, and let I be the moment of inertia of the cable segment about the center of mass. Let be the angular acceleration of that pitch angle segment. Let be the translational acceleration of the upstream traction point A. Substituting the values, we obtain the torque balance equation with the traction point A as the moment center: ; in, This represents the total external torque acting on the rigid body AB (in this scenario, it is mainly the equivalent force of hydrodynamic drag and downstream reaction forces) relative to the traction point A. It can be seen that this external torque is clearly decomposed into two parts: one part is the inertial torque about the center of mass. The other part is the additional torque caused by the translational acceleration of the center of mass relative to the upstream node. .

[0047] After clarifying the expression for the moment of inertia of the entire rigid body segment, in order to introduce the distributed hydrodynamic load obtained in the previous step into the moment balance equation, it is necessary to perform moment integration with respect to the normal towing along the length direction, taking the upstream node of each tow cable segment as the moment center, to obtain the hydrodynamic moment of the tow cable segment about the upstream node. Consider a tow cable segment AB of length L, and arbitrarily take a infinitesimal element X on it with an arc length r from the traction point A. The area of ​​this infinitesimal element is... The local velocity is The angle between the x-axis and the inertial reference frame is Its normal velocity component is The corresponding normal drag force is: ; in, For fluid density, Let be the normal drag coefficient, and D be the equivalent diameter of the towing cable segment. This normal drag acts in the opposite direction to the local velocity, and because its line of action has a non-zero normal distance from the rigid body axis, it generates a torque about the traction point A. The infinitesimal torque generated by this infinitesimal force about the traction point A is: ; in Let X be the vector pointing from traction point A to infinitesimal element X. The expression for the local velocity vector obtained in the previous step is: ; And the corresponding speed: ; And velocity direction angle: ; Substituting into the normal drag formula, we can obtain the normal drag vector form on the infinitesimal element: ; The drag torque of the infinitesimal element is applied to the entire length. By integrating, the hydrodynamic torque of the entire tow cable segment at traction point A can be obtained: ; In the above formula and Substituting and rearranging the specific expression, we can obtain the scalar form of the hydrodynamic torque: ; From the above derivation, it can be seen that only the normal pressure drag will produce a non-zero torque at the traction point A, while the tangential drag along the cable segment axis, whose line of action passes through the axis of rotation, does not produce a torque about the upstream node; that is, tangential drag does not produce a torque about the upstream node. Therefore, when establishing the torque balance equation, it is only necessary to integrate the torque term of the normal drag. External torque can be taken into account. Tangential drag only affects translational balance and axial tension, and does not affect rotational balance with respect to the upstream node.

[0048] Since the above-mentioned normal drag integral term is difficult to obtain analytically under normal circumstances, and the integral contains a nonlinear combination of pitch angle, pitch velocity, and upstream node translational velocity, this embodiment uses the Gauss-Legendre five-point quadrature method along the length of the tow cable segment for specific numerical calculations. Numerical approximation is performed. The arc length interval [0, L] is mapped to a standard interval, and five Gaussian points are selected within this interval. The integral is approximated by a weighted sum of the integrals at these five sampling points, thus obtaining the hydrodynamic torque of each tow cable segment and sensor chain about its upstream node at each time step. This approach ensures both integration accuracy and computational efficiency, making it suitable for repeated calls within subsequent time steps.

[0049] After considering the torque contribution of distributed hydrodynamic loads on the nodes, it is also necessary to include the reaction forces and reaction moments transmitted from the downstream adjacent tow cable segments and sensor chains to the upstream nodes of the tow cable segment through the connecting nodes into the external torque of the tow cable segment. Taking the example of two tow cable segments and one sensor chain, a torque balance equation is established for the first tow cable segment at the traction point A. At the discrete k-th time step, the pitch angle of the first segment is denoted as... The acceleration of traction point A is The mass and length of this segment are respectively The moment of inertia is Then, the angular momentum balance equation with reference point A can be written as: ; in The total external torque acting on segment 1 at the k-th time step consists of two parts: one part is the total torque of the first segment's self-distributed normal drag relative to the traction point A. The other part is the reaction torque generated by the second segment and the sensor chain on segment 1 through its connecting node B. Therefore, it can be written as: ; Among them, the reaction torque This can be represented by the reaction force component at the downstream node B of the first segment. Let the component of the reaction force acting on node B in the inertial reference frame at the k-th time step be... Then its torque about the traction point A is: ; Here Essentially, it is an equivalent reaction force formed by the combined inertial force, normal drag force, and tangential drag force of the second tow cable segment and sensor chain, as well as the external force at the end of the array segment. Specifically, it can be expressed as follows: ; in The inertial force component of the center of mass of the second tow cable segment. and These are the normal and tangential towing components acting on the second tow cable segment (obtained by integrating the distributed hydrodynamic loads as described above). This represents the additional external force component of the sensor chain at this node. By incorporating these downstream reaction forces and their torques into the torque balance equation at traction point A, dynamic coupling between the upstream and downstream tow cable segments and the sensor chain is achieved.

[0050] Similarly, for the second tow cable segment and the sensor chain itself, with their respective upstream nodes as reference points, the torque balance equations are established according to the above method: on the one hand, the normal tow along the length direction is converted into hydrodynamic torque about the upstream node through integration or the Gauss-Legendre five-point quadrature method; on the other hand, the reaction force and reaction torque transmitted from the downstream components are included in the external torque of that node, and this external torque is expressed as the sum of the inertial torque term of the corresponding rigid body and the torque term caused by the translational acceleration of the center of mass. Finally, for each tow cable segment and sensor chain, its external torque is set to be equal to the derivative of the angular momentum with respect to time, that is, the torque balance equations of each segment are satisfied. This yields second-order ordinary differential equations with pitch angles of each segment as unknowns. By simulating all such second-order ordinary differential equations in node order, a system of overall second-order ordinary differential equations is formed, with pitch angles of each segment as state variables and including distributed hydrodynamic loads and upstream-downstream coupled reaction forces. This system of equations is then solved by time integration in subsequent steps to obtain the dynamic attitude and spatial position of the tow chain and sensor chain throughout the entire motion process.

[0051] In step S106, the second-order ordinary differential equations are solved synchronously with a uniform time step to obtain the pitch angle and angular velocity of each tow cable segment and the sensor chain as a function of time.

[0052] Based on the calculations in the preceding steps, the rigid body segment AB (length L, mass m, equivalent diameter D) is in rotational equilibrium about the upstream node A. The pitch angle of the first segment is used as an example. For example, its second-order ordinary differential equation can be written as: ; After establishing the second-order ordinary differential equations with the pitch angles of each segment as unknowns, these equations are simultaneously numerically integrated at a unified time step to obtain the pitch angles and angular velocities of each tow cable segment and sensor chain evolving over time. Specifically, for each rigid tow cable segment and sensor chain discretized from the tow cable, the torque balance relationship is established at its upstream node, ultimately yielding second-order ordinary differential equations. Each equation can be viewed as a rotation equation with the pitch angle of that segment as the unknown, with nonlinear terms such as local hydrodynamics, downstream reaction force, and translational acceleration of the upstream node on the right-hand side. In these equations, the pitch angle of each segment is not only related to the angular velocity and angular acceleration of that segment, but also coupled with adjacent segments and the sensor chain through the reaction force and reaction torque at the connecting nodes. Therefore, the equations of all segments must be treated as a whole system of second-order ordinary differential equations, rather than solved segment by segment in isolation. On the time axis, the simulation interval is divided into equally spaced discrete moments. A fixed time step is selected uniformly. Within each time step, using the pitch angle and angular velocity of the previous moment as initial values, the pitch acceleration at that time step is calculated, thus advancing the pitch angle and angular velocity to the next moment. In this way, over the entire time interval, the pitch angle of each segment... With angular velocity All of them evolve synchronously on the same time grid, ensuring that the dynamic coupling relationship between the tow cable segments and between the tow cable segments and the sensor chain is consistent in time.

[0053] In terms of specific numerical solution methods, the second-order ordinary differential equation system is first implemented by performing a conventional state expansion, and then each equation is... Rewrite the second-order equation for the unknowns as two first-order equations, with pitch angle as the equation. With angular velocity As paired state variables for this segment, the pitch angles and angular velocities of all tow cable segments and sensor chains are aggregated into a unified state vector, with a unified time step. The following numerical integration is performed in the form of a system of ordinary differential equations. The corresponding numerical integration employs the fourth-order Runge-Kutta method, synchronously updating the pitch angle and angular velocity of each segment within the same time step. That is, at a certain time step... First, the local velocity distribution, normal drag, and tangential drag on each segment are calculated using the current pitch angle and angular velocity. Then, the total external torque of each segment about the upstream node is calculated according to the torque balance equation given in step S105, and the pitch angular acceleration at that moment is obtained by substituting it into the torque balance equation. Subsequently, the fourth-order Runge-Kutta method is used to calculate the pitch angular acceleration at that moment within the time interval. Perform a complete update on the state vector to ensure that all tow cable segments and sensor chains are in the state vector. The pitch angle and angular velocity at each time step are simultaneously updated. This is essentially a synchronous solution: at each time step, the state evolution of all segments proceeds uniformly, without one segment updating first and another lagging behind, thus achieving synchronous numerical solution of the second-order ordinary differential equations at a unified time step.

[0054] In the numerical solution process, since the right-hand side of the second-order ordinary differential equation contains terms such as hydrodynamic torque and towing force obtained by integration along the cable segment length, it is necessary to numerically approximate these definite integrals about the arc length at each time step. To ensure computational accuracy and maintain consistency with the aforementioned torque modeling, the Gauss-Legendre five-point quadrature method is used for integration. The integral along the arc length [0,L] of the towing cable segment is numerically approximated using the Gauss-Legendre quadrature method. Specifically, several Gaussian points are selected in the standard interval [0,L]. and their corresponding weights And through linear mapping The sampling points are converted to the arc length interval [0,L], and the weighted summation is used to approximate the arc length integral.

[0055] The continuous integral can be approximated as a finite weighted sum. For example, the integral of the drag torque over a certain segment can be written as: ; in At the arc length position The local moment density value is calculated using the pitch angle and angular velocity at the current time step. Similarly, a fast and stable numerical estimate can be performed on all definite integral terms appearing in each time step, and these estimates are substituted into the right-hand side of the corresponding second-order ordinary differential equation to calculate the pitch acceleration at the current time step. Subsequently, these accelerations, along with the current pitch angle and angular velocity, are advanced within the same time step using the fourth-order Runge–Kutta method to obtain the pitch angle and angular velocity for the next time step.

[0056] Regarding initial conditions, solving the second-order ordinary differential equations requires providing the initial pitch angle and initial angular velocity for each segment. This embodiment uses the steady-state configuration obtained from static equilibrium calculations as the starting attitude for the dynamic solution, i.e., the initial pitch angle of each tow cable segment and sensor chain when they reach steady state at a given traction speed. To reflect the initial state of the towline-sensor system being aligned with the ship's course, the initial angular velocity is typically set to zero, i.e. This indicates that there is no inertial effect of angular acceleration history at the initial time of calculation, and the attitude change is driven only by the ship's subsequent maneuvers and hydrodynamic effects. Using such initial conditions, under a unified time step, the fourth-order Runge-Kutta method synchronously advances all pitch angles and angular velocities, which can obtain the time-domain response of pitch attitude and angular velocity of the tow chain and sensor chain throughout the entire process of ship straight-line navigation, turning maneuvers, and re-stabilization. This provides complete dynamic input for subsequent geometric recursion based on pitch angles and segment lengths, and calculation of the trajectories of each node and sensor chain's center of mass in the two-dimensional inertial reference frame.

[0057] In step S107, the position and velocity of each node are calculated using a geometric recursive method based on the pitch angle and length of each tow cable segment, and the position of the sensor chain relative to the two-dimensional inertial reference frame is output from the centroid coordinates of the sensor chain.

[0058] Among them, in a unified time step The pitch angles of each tow cable segment and the sensor chain have been obtained. and its angular velocity Then, using geometric recursion, the position and velocity of each node are calculated in a two-dimensional inertial reference frame. Pitch angle Let be the angle between this segment's axis and the horizontal x-axis, and let Li be the arc length from the upstream node to the downstream node; with the upstream node position... ,speed This represents the input for this segment, with the unit axis vector taken as... The geometric position of the downstream node is then directly given by first-order recursion: ; This formula accumulates segment by segment from top to bottom, yielding the coordinates of all nodes. The corresponding velocity recursion is obtained from rigid body kinematics: the velocity of the downstream node equals the translational velocity of the upstream node plus the velocity increment caused by the rotation of this segment around the upstream node. In the two-dimensional case, this can be written as: ; Where k is a unit vector perpendicular to the two-dimensional plane. Since the segment length is constant, the above equation gives the form of a linear superposition of the nodal velocity with respect to the upstream nodal velocity and the current segment angular velocity.

[0059] Applying the above general formula to a typical arrangement of two tow cable segments plus a sensor chain, it can be recursively deduced in the order of "traction point A, intermediate node C, upstream end of sensor E, sensor center of mass G": given the coordinates and velocity of traction point A. The length of the first tow cable section and the pitch angle are ,but: ; ; ; ; The length of the second tow cable section and the pitch angle are: ,get: ; ; ; ; The sensor chain is considered as a slender body with its axis pitch angle being... The upstream end is at point E, and the geometric distance of the centroid along its axis from point E is denoted as . (when uniformly distributed) ,in (where is the length of the chain). Then the centroid of the sensor chain... The position and velocity are given by the following formula: ; If the mass or buoyancy distribution inside the chain is uneven, then... Let E be the actual centroid distance of the chain about the upstream end E; the above form remains unchanged. For the general case of multiple tow cables, simply recursively calculate the positions. With speed recursion Apply sequentially to all segments, and finally press on the sensor chain body. By performing an offset and isomorphic velocity correction along the axis, the time values ​​of the target node and the centroid can be obtained.

[0060] In one embodiment, the method further includes: The centroid position of the output sensor chain and the azimuth measurement obtained by the passive detector are input into the target state estimation algorithm to perform azimuth-only target motion analysis and underwater target tracking, or to display the motion status of the underwater target relative to the sensor chain.

[0061] After obtaining the centroid positions of the sensor chain at each time point through the aforementioned steps, these centroid positions are used as the spatial coordinates of the observation platform in a two-dimensional inertial reference frame. These coordinates, along with the azimuth measurement values ​​output by the passive detector at the same time point, are input into the target state estimation algorithm to perform azimuth-only target motion analysis and underwater target tracking. Specifically, at discrete time points... In this embodiment, the coordinates of the center of mass of the sensor chain in the two-dimensional inertial reference frame have been obtained through recursive derivation of towing cable dynamics and geometry. The azimuth observation given by a passive detector to a certain underwater target is denoted as This orientation is the polar coordinate angle between the target and the center of mass of the sensor chain. In the target state estimation algorithm, let the target state vector be: ; in Let the target's position coordinates be in a two-dimensional inertial reference frame. Let x and y be the velocity components of the target. The target motion model can adopt a uniform linear model, with a sampling time interval of 1. When the discrete state transition relation is expressed as: ; ; in This represents process noise, used to characterize the deviation between the target's actual motion and the ideal uniform velocity model. The observation equation reflects only the measurement characteristics of azimuth. In this embodiment, the azimuth observation is written as follows, using the geometric relationship between the center of mass of the sensor chain and the target position: ; in For passive detectors at time The measured target location, To measure noise, The arctangent function (in implementation, the four-quadrant arctangent function can be used). To ensure that the azimuth angle is arrive (Continuous within the range). Therefore, the coordinates of the centroid of the sensor chain are... The position parameter of the observation platform is directly entered into the measurement function. By combining the platform's true spatial position obtained from the aforementioned tow cable dynamics with the passive detector's orientation measurement within the same target state estimation algorithm framework, target motion analysis under orientation-only conditions can be achieved.

[0062] In the specific process of solving the target state, nonlinear filtering algorithms such as Extended Kalman Filter (EPF), Unscented Kalman Filter (UKF), or Particle Filter (PFF) can be used to recursively estimate the aforementioned state-space model. Taking EPF as an example, at each time step, the target position and velocity are first predicted using the state transition equation. Then, the observation equation is constructed using the current azimuth observation and the position of the center of mass of the sensor chain to correct the predicted state. Since the observation only has one dimension—azimuth—and the measurement function is in a nonlinear arctangent form, the filtering algorithm needs to adjust the measurement function. The target state is linearized or its statistical characteristics are approximated by sampling points. During this process, the position of the sensor chain's center of mass is considered a known input, and its time variation is given by the aforementioned tow cable dynamics and geometric recursion. As the time step progresses, the algorithm continuously absorbs information from the platform position and azimuth measurements, gradually converging to the target's trajectory and velocity estimates, thereby achieving target motion analysis and underwater target tracking under azimuth-only conditions. This implementation can also input the estimated target position sequence and the sensor chain's center of mass position sequence into the display or tactical decision module to display the target's motion relative to the sensor chain in curve form, such as plotting the target trajectory, nearest point, relative distance, and relative azimuth change curves, allowing operators to intuitively observe the underwater target's motion relationship relative to the towed sensor chain. Throughout the process, the position of the sensor chain's center of mass in the two-dimensional inertial reference frame is always used as the platform position input, and the azimuth measurement obtained by the passive detector is used as the sole observation, achieving continuous estimation and situational awareness of the target's motion state without relying on target distance measurements.

[0063] Based on the same line of thought, such as Figure 2 The diagram shown is a structural block diagram of a sensor chain dynamic spatial positioning system for a ship towing chain according to an embodiment of the present invention. The system includes: Modeling module 201 is used to establish a model of the towing chain and sensor chain in a two-dimensional inertial reference frame. The towing chain is discretized along its length into several rigid towing cable segments that are hinged to each other. The parameters of each towing cable segment include the segment length, mass and equivalent diameter. The upstream end of the first towing cable segment is defined as the traction point, and the adjacent connection points between each towing cable segment are defined as nodes. The sensor chain is hinged to the downstream end of the downstream towing cable segment and records the position, velocity and acceleration of the ship at each moment as the motion input of the traction point. The static initialization module 202 is used to perform static balance calculations on the towing chain and the sensor chain at a given traction speed, to obtain the initial pitch angle, initial depth and tension distribution, and to use them as the initial conditions for dynamic solution. The velocity field calculation module 203 is used to obtain the local velocity of each position point in any of the tow cable segments based on the translational and rotational states of its upstream nodes, and to decompose the local velocity relative to the axis of the tow cable segment into normal velocity components and tangential velocity components. The load calculation module 204 is used to determine the distributed hydrodynamic loads acting on each of the tow cable segments and the sensor chain based on the normal velocity component, the tangential velocity component, and fluid density, equivalent diameter, and resistance parameters. The torque and equation establishment module 205 is used to establish a torque balance equation including the distributed hydrodynamic load and downstream reaction force, with the upstream node of each tow cable segment and sensor chain as the reference point, and write the torque balance equation as a set of second-order ordinary differential equations with the pitch angle of each segment as the unknown. The numerical solution module 206 is used to synchronously solve the second-order ordinary differential equations at a uniform time step to obtain the pitch angle and angular velocity of each tow cable segment and the sensor chain as a function of time. The kinematic recursion module 207 is used to calculate the position and velocity of each node in a geometric recursive manner based on the pitch angle and length of each tow cable segment, and output the position of the sensor chain relative to the inertial reference frame from the centroid coordinates of the sensor chain.

[0064] The specific details of the above system have been described in detail in the method section of the implementation plan. For any undisclosed details, please refer to the implementation plan of the method section, and therefore will not be repeated here.

[0065] This system discretizes the towing chain into several rigidly hinged segments, introduces physical quantities such as chain mass parameters and equivalent diameter, uses the ship's motion as the traction boundary, and combines static equilibrium to obtain a reasonable initial attitude and depth. Based on this, it considers the normal and tangential hydrodynamic effects of each segment in the fluid, establishes a set of dynamic equations with the pitch angle of each segment as the state variable, and performs synchronous numerical solutions, thus providing a unified description of the dynamic response of the towing chain and sensor chain throughout the entire time history. Through this dynamic spatial positioning method, the bending morphology and attitude evolution of the towing chain and sensor chain in water can be more realistically reflected under varying ship speeds and maneuvering conditions, significantly improving the estimation accuracy of the sensor chain's center of mass position and reducing the target motion analysis error introduced by the uncertainty of the chain's attitude.

[0066] The accompanying drawings are merely illustrative of the processes included in the methods according to exemplary embodiments of this disclosure and are not intended to be limiting. It is readily understood that the processes shown in the drawings do not indicate or limit the temporal order of these processes. Furthermore, it is readily understood that these processes may be executed synchronously or asynchronously, for example, in multiple modules.

[0067] It should be noted that although several modules or units of the system have been mentioned in the detailed description above, this division is not mandatory. In fact, according to exemplary embodiments of this disclosure, the features and functions of two or more modules or units described above can be embodied in one module or unit. Conversely, the features and functions of one module or unit described above can be further divided and embodied by multiple modules or units.

[0068] Other embodiments of this disclosure will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this disclosure that follow the general principles of this disclosure and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of this disclosure are indicated by the claims.

[0069] It should be understood that this disclosure is not limited to the precise structures described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this disclosure is limited only by the appended claims.

Claims

1. A method of dynamic spatial positioning of a sensor chain of a marine tow chain, characterized in that, The method comprises: a model of a towed chain and a sensor chain body is established in a two-dimensional inertial reference frame, the towed chain is discretized into a plurality of rigid towline segments which are hingedly connected along the length, parameters of each towline segment include segment length, mass and segment equivalent diameter, an upstream end of a first towline segment is defined as a traction point, adjacent connecting points between each towline segment are defined as nodes, the sensor chain body is hingedly connected to a downstream end of a most downstream towline segment, a position, a speed and an acceleration of the ship at each time are recorded as motion input of the traction point; under a given traction speed, static force balance calculation is performed on the towed chain and the sensor chain body to obtain initial pitch angle, initial depth and tension distribution, and the initial conditions are taken as initial conditions for dynamics solving; for any towline segment, local speed of each position point in the segment is obtained according to the translation and rotation state of the upstream node, and the local speed is decomposed into a normal velocity component and a tangential velocity component relative to the axis of the towline segment; distributed hydrodynamic loads acting on each towline segment and the sensor chain body are determined according to the normal velocity component, the tangential velocity component and fluid density, equivalent diameter and resistance parameters; torque balance equations including the distributed hydrodynamic loads and downstream reaction forces are established with the upstream nodes of each towline segment and the sensor chain body as reference points, the torque balance equations are written into a second-order ordinary differential equation group with the pitch angle of each segment as an unknown quantity; the second-order ordinary differential equation group is synchronously numerically solved under a unified time step to obtain the pitch angle and angular velocity of each towline segment and the sensor chain body varying with time; the position and speed of each node are calculated in a geometric recursive manner according to the pitch angle of each towline segment and the segment length, and the position of the sensor chain body relative to the two-dimensional inertial reference frame is output from the mass center coordinates of the sensor chain body.

2. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, the towed chain is discretized into two rigid towline segments, a first towline segment and a second towline segment, along the length direction, an upstream end of the first towline segment is a traction point A connected with the ship, a connecting node between the first towline segment and the second towline segment is a node C, a downstream end of the second towline segment is hingedly connected with an upstream end node E of the sensor chain body, torque balance equations are respectively established at the traction point A, the node C and the node E, and the three torque balance equations are solved together.

3. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, the calculation of the local speed and the distributed hydrodynamic load satisfies the following rules: for a microelement in any towline segment at an arc length r from the upstream node, the local speed of the microelement is the sum of the translation speed and the angular velocity of the upstream node and the vector product of the position vector from the upstream node to the microelement; the distributed hydrodynamic load includes normal drag and tangential drag, wherein: the normal drag acts along the normal direction, and the microelement strength is calculated by the product of the fluid density, the normal resistance coefficient, the segment equivalent diameter of the towline segment, the microelement length, the normal velocity component and the speed; the tangential drag acts along the axial direction of the towline segment, and the microelement strength is calculated by half of the product of the fluid density, the tangential resistance coefficient, the equivalent force area of the microelement, the tangential velocity component and the speed. The directions of the normal drag and the tangential drag are opposite to the corresponding local velocity directions.

4. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, In establishing the moment balance equation, further comprising: Moment integration of the normal drag along the length direction of each of the towline segments is performed with the upstream node of the towline segment as the center to obtain the hydrodynamic moment of the towline segment about the upstream node, and the tangential drag does not generate a moment about the upstream node; The reaction force and reaction moment transferred by the downstream adjacent towline segment and the sensor chain to the upstream node of the towline segment are added to the external moment of the towline segment; The moment integration of the normal drag along the length direction of the towline segment is performed by using Gauss-Legendre five-point quadrature method for numerical approximation to obtain the total external moment of each of the towline segments and the sensor chain about its upstream node.

5. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, In establishing the moment balance equation, further comprising: for each of the towline segments and the sensor chain, respectively taking its upstream node as the reference point, and establishing the angular momentum balance relationship according to the angular momentum theorem, namely: The external moment of the towline segment or the sensor chain is expressed as the sum of the inertial moment item about its center of mass and the moment item caused by the translational acceleration of the center of mass relative to the upstream node; Letting the external moment equal the derivative of the angular momentum with respect to time, a second-order ordinary differential equation is obtained with the pitch angle of the towline segment or the sensor chain as the unknown.

6. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, The static balance calculation comprises: In the vertical direction, the sum of the vertical components of the gravity, buoyancy and traction tension of the tow drag chain and the sensor chain is made to be zero to determine the initial pitch angle; In the horizontal direction, the total drag force and the horizontal components of the traction tension of all the towline segments and the sensor chain are balanced to determine the initial depth and the tension distribution along the tow drag chain; The obtained initial pitch angle, initial depth and tension distribution are taken as the initial conditions of the second-order ordinary differential equation set.

7. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, The boundary conditions of the second-order ordinary differential equation set are set as: The displacement, velocity and acceleration of the traction point of the tow drag chain at any time are the same as those of the ship; The downstream end of the sensor chain is a free end, and the reaction force and reaction moment thereof are taken as zero; And in solving the second-order ordinary differential equation set, comprising: The second-order ordinary differential equation set is numerically integrated with a uniform time step, and the fourth-order Runge-Kutta method is used to update the pitch angle and angular velocity of all the towline segments and the sensor chain at each time step.

8. The method of dynamic spatial positioning of a sensor chain of a marine towing chain according to claim 1, characterized in that, Further comprising: The center of mass position of the output sensor chain is input into a target state estimation algorithm together with the bearing measurement obtained by the passive detector to perform target motion analysis and underwater target tracking in bearing only, or used to display the motion situation of the underwater target relative to the sensor chain.

9. A sensor chain dynamic spatial positioning system for a marine towing chain, characterized in that Comprising: a modeling module for modeling the towed chain and the sensor chain body in a two-dimensional inertial reference frame, discretizing the towed chain along its length into a plurality of rigid cable segments connected to each other at their ends, each of the cable segments having parameters including segment length, mass and equivalent diameter, defining an upstream end of a first of the cable segments as a tow point, and defining adjacent connection points between the cable segments as nodes, the sensor chain body being connected to a downstream end of a last of the cable segments, and recording a position, a velocity and an acceleration of the vessel at each time as motion input of the tow point; a static initialization module for performing static force balance calculation on the towed chain and the sensor chain body at a given tow velocity to obtain initial pitch angle, initial depth and tension distribution, and using the initial conditions as initial conditions for dynamic solution; a velocity field calculation module for obtaining local velocities at each position in a cable segment according to a translational and rotational state of an upstream node of the cable segment, and decomposing the local velocities into normal and tangential velocity components with respect to an axis of the cable segment; a load calculation module for determining distributed hydrodynamic loads on each of the cable segments and the sensor chain body according to the normal and tangential velocity components, fluid density, equivalent diameter and drag parameters; a moment and equation establishment module for establishing moment balance equations of each of the cable segments and the sensor chain body with respect to upstream nodes of the cable segments and the sensor chain body, the moment balance equations including the distributed hydrodynamic loads and downstream reaction forces, and writing the moment balance equations as a second-order ordinary differential equation group with pitch angles of the cable segments as unknowns; a numerical solution module for synchronously numerically solving the second-order ordinary differential equation group at a uniform time step to obtain pitch angles and angular velocities of each of the cable segments and the sensor chain body varying with time; a kinematics recursion module for calculating positions and velocities of the nodes according to the pitch angles of the cable segments and segment lengths in a geometric recursion manner, and outputting a position of the sensor chain body with respect to the inertial reference frame from a mass center coordinate of the sensor chain body.

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