A method and system for dynamic spatial positioning of a sensor chain of a marine tow chain

Abstract

The present application relates to underwater positioning technology field of ship, specifically relates to a kind of sensor chain body dynamic space positioning method and system of ship tow chain, method includes: in two-dimensional inertial reference system, tow chain is discretized into several articulated rigid segments, sensor chain body is connected to terminal, and with the ship movement as traction point boundary;According to given traction speed, initial pitch angle, depth and tension distribution are obtained by static equilibrium;According to the translation and rotation state of each segment upstream node, local velocity in segment is calculated, and distributed hydrodynamic load is obtained in combination with fluid parameters, and moment balance equation is established at each upstream node, forming second-order ordinary differential equation group with pitch angle as unknown quantity;Numerical solution is solved under unified time step, and geometric recursion is carried out according to each node and sensor chain body centroid position, to realize its dynamic positioning in inertial reference system.The present application can improve the precision and reliability of underwater target motion analysis and target tracking.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of underwater positioning of ships, and particularly relates to a sensor chain body dynamic space positioning method and system for a ship towing chain. BACKGROUND

[0002] In the prior art, a ship-mounted towing chain realizes continuous observation of the upper ocean layer through the coupling of the towing cable pitch angle and the sailing speed based on the dynamic profile scanning principle of the ship towing multi-node sensor chain body. Specifically, the ship-mounted towing chain releases an electromagnetic coupling towing chain through a ship-mounted winch, and a sensor array chain body is carried on the towing cable. The chain body can carry a temperature-salinity-depth sensor (CTD), an ecological sensor (chlorophyll, CDOM, turbidity, etc.), and the like. The chain body moves within a specific water layer range. The multiple sensors observation nodes distributed at intervals on the chain body can measure the hydrodynamic data of the position in real time according to the carried sensors.

[0003] In order to accurately estimate the target position and motion state from the azimuth information of the hydrodynamic data, the real position of each sensor unit in the inertial space needs to be known in advance. However, the towing chain is affected by gravity, buoyancy and water dynamics varying with the ship speed during sailing, and its attitude and spatial shape changes over time, especially under strong dynamic conditions such as ship acceleration, deceleration and turning. The spatial position of the sensor chain body and the ship trajectory are not a simple geometric relationship. In the prior art, the chain body shape is often estimated by simplifying it into a static catenary, an empirical curve or relying on a small number of depth / attitude sensors for interpolation calculation. Such methods are mostly based on the steady state or weak dynamic assumption, and can easily produce large deviations when the ship is in strong motion, the speed changes greatly or the water dynamic effect is significant. SUMMARY

[0004] In view of the above technical problems, the present application provides a sensor chain body dynamic space positioning method and system for a ship towing chain. Under the known ship motion conditions, the dynamic relationship between the towing chain and the sensor chain body is established to realize high-precision and continuous calculation of the spatial position of the sensor chain body in the inertial reference frame over time, thereby improving the accuracy and reliability of underwater target motion analysis and target tracking.

[0005] Other characteristics and advantages of the present disclosure will become apparent from the following detailed description, or will be learned by practice of the present disclosure.

[0006] According to an aspect of the present application, a sensor chain body dynamic space positioning method for a ship towing chain is provided, which comprises:

[0007] A model of the towed chain and the sensor chain body is established in a two-dimensional inertial reference frame, the towed chain is discretized into several rigid towline segments which are hinged to each other along the length, the parameters of each towline segment include segment length, mass and segment equivalent diameter, the upstream end of the first towline segment is defined as the towing point, the adjacent connection points between each towline segment are defined as nodes, the sensor chain body is hinged to the downstream end of the most downstream towline segment, the position, velocity and acceleration of the towing vessel at each time are recorded as the motion input of the towing point;

[0008] Under a given towing speed, static force balance calculation is performed on the towed chain and the sensor chain body to obtain the initial pitch angle, initial depth and tension distribution, which are used as the initial conditions for dynamic solving;

[0009] For any towline segment, the local velocity of each position point in the segment is obtained according to the translation and rotation state of the upstream node, and the local velocity is decomposed into a normal velocity component and a tangential velocity component relative to the axis of the towline segment;

[0010] According to the normal velocity component, the tangential velocity component, and the fluid density, equivalent diameter and resistance parameters, the distributed hydrodynamic load acting on each towline segment and the sensor chain body is determined;

[0011] Taking the upstream nodes of each towline segment and the sensor chain body as reference points, a moment balance equation is established, which includes the distributed hydrodynamic load and the downstream reaction force, the moment balance equation is written as a second-order ordinary differential equation group with the pitch angle of each segment as the unknown quantity;

[0012] The second-order ordinary differential equation group is numerically solved synchronously 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;

[0013] According to the pitch angle of each towline segment and the segment length, the position and velocity of each node are calculated in a geometric recursive manner, and the position of the sensor chain body relative to the two-dimensional inertial reference frame is output from the centroid coordinates of the sensor chain body.

[0014] Further, the towed chain is discretized into two rigid towline segments along the length direction, which are the first towline segment and the second towline segment, the upstream end of the first towline segment is the towing point A connected to the towing vessel, the connection node between the first towline segment and the second towline segment is node C, the downstream end of the second towline segment is hinged to the upstream end node E of the sensor chain body; the moment balance equations are established at the towing point A, node C and node E respectively, and the three moment balance equations are solved simultaneously.

[0015] Further, the calculation of the local velocity and the distributed hydrodynamic load satisfies the following rules:

[0016] 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.

[0017] The distributed hydrodynamic loads include normal towing and tangential towing, wherein:

[0018] 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.

[0019] 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.

[0020] The directions of both the normal drag and the tangential drag are opposite to the corresponding local velocity directions.

[0021] Furthermore, the process of establishing the torque balance equation also includes:

[0022] 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.

[0023] 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.

[0024] 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.

[0025] 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:

[0026] 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;

[0027] 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.

[0028] Furthermore, the static equilibrium calculation includes:

[0029] in vertical direction, sum of gravity, buoyancy and vertical component of tension of the towline and the sensor chain is zero to determine the initial pitch angle;

[0030] in horizontal direction, total drag and horizontal component of tension of all the towline segments and the sensor chain are balanced to determine the initial depth and tension distribution along the towline;

[0031] the initial pitch angle, the initial depth and the tension distribution are taken as initial conditions of the second order ordinary differential equations.

[0032] Further, boundary conditions of the second order ordinary differential equations are set as:

[0033] displacement, velocity and acceleration of the towline's tow point are same as those of the host ship at any time;

[0034] downstream end of the sensor chain is free end, and its reaction force and reaction torque are taken as zero;

[0035] and, in solving the second order ordinary differential equations, including:

[0036] numerical integration of the second order ordinary differential equations is performed with uniform time step, and fourth order Runge-Kutta method is used to update pitch angle and angular velocity of all the towline segments and the sensor chain at each time step.

[0037] Further, the method further includes:

[0038] output of the sensor chain's mass center position is input into target state estimation algorithm together with bearing measurements obtained by passive sonar to perform target motion analysis and underwater target tracking in bearing only, or used to display motion situation of underwater target relative to the sensor chain.

[0039] According to a second aspect of the disclosure, a sensor chain dynamic space positioning system of a ship's towline is provided, the system includes:

[0040] a modeling module for establishing a model of the towline and the sensor chain in a two-dimensional inertial reference frame, discretizing the towline along length into a plurality of mutually hinged rigid towline segments, parameters of each towline segment including segment length, mass and segment equivalent diameter, defining upstream end of a first towline segment as a tow point, and adjacent connecting points between each towline segment as nodes, the sensor chain being hinged to downstream end of the most downstream towline segment, recording position, velocity and acceleration of the host ship at each time as motion input of the tow point;

[0041] a static initialization module for calculating static equilibrium of the towed chain and the sensor chain body under a given towing speed to obtain initial pitch angle, initial depth and tension distribution, and taking them as initial conditions for dynamic solving;

[0042] a velocity field calculation module for obtaining local velocity of each position point in a segment of the towed cable according to the translation and rotation state of the upstream node thereof, and decomposing the local velocity into normal velocity component and tangential velocity component relative to the axis of the towed cable segment;

[0043] a load calculation module for determining distributed hydrodynamic load acting on each towed cable segment and the sensor chain body according to the normal velocity component, the tangential velocity component, and fluid density, equivalent diameter and drag parameter;

[0044] a moment and equation establishment module for taking the upstream node of each towed cable segment and sensor chain body as reference point to establish moment balance equation containing the distributed hydrodynamic load and downstream reaction force, and writing the moment balance equation into second-order ordinary differential equation group with pitch angle of each segment as unknown quantity;

[0045] a numerical solving module for synchronously numerically solving the second-order ordinary differential equation group under unified time step to obtain pitch angle and angular velocity of each towed cable segment and the sensor chain body varying with time;

[0046] a kinematics recursion module for calculating position and velocity of each node according to pitch angle of each towed cable segment and segment length in geometric recursion manner, and outputting position of the sensor chain body relative to the inertial reference system from the mass center coordinates of the sensor chain body.

[0047] The technical solution of the present disclosure has the following beneficial effects:

[0048] The present disclosure discretizes the towed chain into several mutually hinged rigid towed cable segments, introduces physical quantities such as chain body mass parameter and equivalent diameter, takes ship motion as towing boundary, obtains reasonable initial attitude and depth in combination with static equilibrium, then considers normal and tangential hydrodynamic action of each segment in fluid, establishes dynamic equation group with pitch angle of each segment as state quantity and performs synchronous numerical solving, so that dynamic response of the towed chain and the sensor chain body is uniformly described in the whole time history. Through the above dynamic spatial positioning method, the bending shape and attitude evolution of the towed chain and the sensor chain body in water can be more truly reflected under ship speed change and maneuvering conditions, so as to significantly improve the estimation accuracy of the mass center position of the sensor chain body and reduce the target motion analysis error introduced by chain body attitude uncertainty. BRIEF DESCRIPTION OF DRAWINGS

[0049] Figure 1A flow chart of a sensor chain dynamic spatial positioning method of a ship tow chain according to an embodiment of the present specification;

[0050] Figure 2 A structural block diagram of a sensor chain dynamic spatial positioning system of a ship tow chain according to an embodiment of the present specification. DETAILED DESCRIPTION

[0051] Example implementations will now be described more fully with reference to the accompanying drawings. Example implementations may, however, be implemented in many different forms and should not be construed as limited to the implementations set forth herein; rather, these implementations are provided so that this disclosure will be thorough and complete, and will fully convey the inventive aspects to those skilled in the art. The described features, structures, or characteristics can be combined in one or more implementations. In the following description, numerous specific details are provided to give a thorough understanding of implementations of the disclosure. One skilled in the relevant art will recognize, however, that the implementations of the disclosure can be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures have not been described in detail to avoid obscuring aspects of the disclosure.

[0052] Furthermore, the accompanying drawings are only schematic and are non-limiting. Like references signs denote like parts, for the sake of clarity and for brevity of description it will be appreciated that not all of the components of the figures will be described in detail herein. Some of the blocks in the diagrams are functional blocks that can be implemented by software, hardware, or a combination of both. The functional blocks can be implemented in one or more hardware modules or integrated circuits, or in different network and / or processor devices and / or microcontroller devices.

[0053] The present disclosure provides a sensor chain dynamic spatial positioning method of a ship tow chain. Referring to Figure 1 A flow chart of a sensor chain dynamic spatial positioning method of a ship tow chain according to an embodiment of the present specification;

[0054] In step S101, a model of the towed chain and the sensor chain body is established in a two-dimensional inertial reference frame, the towed chain is discretized along the length into a plurality of rigid towline segments that are hingedly connected to each other, parameters of each towline segment include segment length, mass and segment equivalent diameter, an upstream end of the first towline segment is defined as a traction point, adjacent connection points between the towline segments are defined as nodes, the sensor chain body is hingedly connected to a downstream end of the most downstream towline segment, and the position, velocity and acceleration of the ship at each time are recorded as the motion input of the traction point.

[0055] First, a fixed two-dimensional inertial reference frame is selected in a plane containing the ship sailing direction and the water depth direction, the reference frame does not move with time relative to the seawater and seabed, facilitating the unified description of the motion of the ship and the towed chain and the sensor chain body in the same coordinate system. The inertial reference frame can be set with a point on the calm water surface as the origin, the x-axis along the horizontal projection of the ship's forward direction, and the y-axis along the vertical direction, so that the motion of the entire towed chain and the sensor chain body is constrained in the two-dimensional plane. On this basis, the actual continuous flexible towed chain is discretized using the lumped mass idea, and it is regarded as being composed of a plurality of rigid towline segments connected to each other, each rigid towline segment is kinematically connected through the hinge point, so that only the translational and rotational states of a limited number of rigid bodies need to be tracked to approximately represent the macroscopic form of the entire flexible cable body during calculation. For each towline segment, the physical parameters such as segment length, mass and segment equivalent diameter are pre-defined, the segment length can be determined according to the actual cable length and the discretization accuracy requirement, the mass can be regarded as the lumped mass of the segment cable body in the gravity direction, and the equivalent diameter is used to represent the equivalent diameter of the segment in the water subjected to fluid action. These parameters can be derived from cable structure design and hydrodynamic characteristic calibration. Through the above processing, the originally continuous towline is abstracted into a limited number of rigid units with clear mass, geometric size and equivalent shape parameters, laying a foundation for subsequent establishment of force and torque balance relationship.

[0056] In the geometric relationship, the upstream end of the first segment of the tow chain is defined as the traction point, which is connected to the ship body through a winch or a towline pulley and is the only dynamic boundary between the tow chain and the ship. The connection between any two segments of the towline is defined as a node, and each node has a time-varying coordinate in the two-dimensional inertial reference frame, which is used to describe the relative attitude and position constraint between the towline segments. The sensor chain is arranged at the end of the tow chain, and its upstream end is connected to the downstream end of the most downstream towline segment through a hinge. The hinge point is also defined as a node, so that the sensor chain can rotate relative to the end of the towline in the two-dimensional plane, thereby reflecting the attitude change of the sensor chain relative to the end of the towline under the action of gravity, buoyancy and hydrodynamic force. In an exemplary manner for ease of description, the tow chain can be discretized into only two rigid towline segments along the length direction, which are the first towline segment and the second towline segment. The upstream end of the first towline segment is the traction point A, which is rigidly connected to the ship. The connection node between the first towline segment and the second towline segment is node C, which represents the hinge position of the two cable bodies. The downstream end of the second towline segment is hinged to the upstream end node E of the sensor chain, and node E is both the end node of the second towline segment and the connection node between the sensor chain and the towline system. Through the combination of the two towline segments and the sensor chain, the relative attitude change between the front and rear segments of the towline and the sensor chain can be captured while keeping the model simple.

[0057] When describing the motion of the node, the translational state of the ship at each time is taken as the boundary input of the traction point, that is, during the entire simulation or calculation process, the position, velocity and acceleration of the ship in the two-dimensional inertial reference frame are recorded as a function of time. The fixed geometric relationship of the ship towline traction point in the ship coordinate system is converted to the inertial reference frame, so that the coordinates , velocity and acceleration of the traction point A at each discrete time are obtained. In the dynamics solution, the calculation process is discretized into equal time steps on the time axis, and the time corresponding to the kth time step is denoted as The position of the traction point in the inertial reference frame is denoted as , and the velocity is denoted as where the position quantity has the dimension of length, the velocity quantity has the dimension of length / time, and the acceleration quantity has the dimension of length / time2. The above-mentioned physical quantities at each time step are substituted into the dynamic equations of the towed chain and the sensor chain body as known boundary conditions for solving the pitch angle and angular velocity of each section at the corresponding time. In the numerical integration process, the time step is used to discretize the solving process, and the known position, velocity and acceleration of the towing point are used as initial conditions for synchronous solving. Since the towing point A is fixed to the ship through a rigid connection, the macro motion of the ship can be directly equivalent to the translational boundary condition of the towing point A in modeling, so that the subsequent mechanical analysis can be carried out only in the towed chain and sensor chain body system, without the need to explicitly model the ship structure in the same dynamic equation set.

[0058] After the above-mentioned geometric and kinematic modeling is completed, in order to further establish the dynamic equations needed in the subsequent steps on the basis of the discrete model, the torque balance relationship at the node can be considered at the towing point A, the node C and the node E, respectively. Specifically, for the system composed of two rigid towline sections and a sensor chain body, the rigid body connected to the node can be regarded as a whole at each node, and the external force and inertial torque acting on it are summarized, and the balance equation about rotation is written at the node. Taking the towing point A as an example, the rotation of the first towline section around the towing point A is regarded as a rigid body rotation problem, the gravity and buoyancy of the first towline section at its center of mass, the equivalent action of the hydrodynamic force on the cable section in water, the internal force and internal torque from the direction of node C, and the constraint reaction force between the towing point and the ship are considered, the torque of these forces relative to the towing point A is vector summed, and the total external torque is balanced with the angular momentum change rate of the first towline section about the towing point A, thereby establishing a torque balance equation at the towing point A. Similarly, at the node C, the first towline section and the second towline section are cut off respectively, the force and torque contributions of each to the node C are analyzed, and the corresponding torque balance relationship is written; at the node E, the second towline section and the sensor chain body are considered, and the torque balance equation about the node E is established. In this way, in the simplest demonstration case of two rigid towline sections plus a sensor chain body, three torque balance equations with the pitch angle or equivalent attitude variable of each section as the unknown quantity can be obtained at the towing point A, the node C and the node E respectively, and the above-mentioned three equations are solved simultaneously at the same time step, thereby providing a unified and consistent modeling basis for further deriving the dynamic equation set and spatial attitude of the entire towed chain and sensor chain body system in the subsequent steps.

[0059] In step S102, under a given towing speed, static force balance calculation is performed on the towed chain and the sensor chain body to obtain the initial pitch angle, initial depth and tension distribution, which are used as initial conditions for dynamic solving.

[0060] Specifically, under the condition of given towing speed, it is firstly considered that the ship sails at a stable constant speed in calm water, at this time the tow chain and the sensor chain body gradually converge to a set of time-invariant equilibrium posture and spatial shape, i.e. the so-called steady-state equilibrium configuration, in the two-dimensional inertial reference frame. In order to obtain the steady-state configuration, the embodiment carries out static force balance calculation on the whole tow chain and sensor chain body: in the overall perspective, the tow chain is regarded as a series of rigid tow cable segments, the sensor chain body is hinged to the last tow cable segment, and the lower end is a free end, and the upper end towing point is connected with the ship through a cable car or a pulley. After the towing speed is given, the fluid action such as resistance and lift of the cable segments and the sensor chain body in the water tends to be stable, at this time the posture and force of each node satisfy the balance relationship of force and torque, and there is no time-varying acceleration term. In order to simplify the dynamics solving, in the embodiment, the steady-state static force balance problem is solved first, the initial pitch angle, depth and tension distribution of the tow chain and the sensor chain body are obtained through static force balance calculation, and are taken as the initial conditions of the dynamic equation, so that the time domain numerical integration starts from the physically reasonable configuration satisfying the force balance, thereby improving the stability of numerical solving and reducing meaningless transient oscillation.

[0061] The static force balance calculation specifically includes: in the vertical direction, the sum of the vertical components of the gravity, buoyancy and towing tension of the tow chain and the sensor chain body is zero, so as to determine the initial pitch angle; in the horizontal direction, the total tow force and the horizontal components of the towing tension of all the tow cable segments and the sensor chain body are balanced, so as to determine the initial depth and the tension distribution along the tow chain; and the obtained initial pitch angle, initial depth and tension distribution are taken as the initial conditions of the second-order ordinary differential equation set.

[0062] In the static force balance in the vertical direction, the vertical components of the gravity, buoyancy and towing tension of the tow chain and the sensor chain body are considered as a whole. The gravity of each rigid tow cable segment of the tow chain can be represented by the product of the segment mass and the gravitational acceleration, and the gravity of the whole sensor chain body can be represented by the product of the mass and the gravitational acceleration ; the buoyancy is converted according to the displacement volume and the fluid density. In the steady-state equilibrium configuration, the overall pitch attitude of the tow system near the towing point is denoted by an angle , which is the average inclination angle of the tow chain in the two-dimensional inertial reference frame relative to the vertical direction; the direction of the cable end tension of the sensor chain body is denoted by an angle , which is defined as the included angle of the cable end tension relative to the vertical direction. In the sensor chain body part, the force is decomposed to obtain the static force balance relationship in the vertical direction:

[0063] ;

[0064] wherein, Tension applied to the sensor chain by the cable end, Total buoyancy acting on the sensor chain. The above equation shows that the difference between the weight and the buoyancy of the sensor chain is balanced by the vertical component of the cable end tension. For each segment of the tow chain, a similar vertical force balance can be applied: the difference between the weight and the buoyancy of each segment of the tow cable, and 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 towing point. By applying the vertical force balance to each segment of the tow cable from bottom to top, a set of tension component relationships related to the pitch angle can be obtained, and based on which the pitch attitude of the entire tow chain and the sensor chain in the steady state, i.e. the initial pitch angle, can be determined.

[0065] In the horizontal static force balance, the balance between the total horizontal tow force and the horizontal component of the towing tension is mainly considered when the rigid tow cable segments of the tow chain and the sensor chain are sailing in water. For the sensor chain part, it receives a resistance along the sailing direction at a constant towing speed, which can be calculated according to the fluid density, the equivalent force area of the sensor chain and the drag coefficient, and is balanced with the horizontal component of the cable end tension in the steady state, which can be expressed as:

[0066] ;

[0067] As can be seen, the total horizontal tow force of the sensor chain is equal to the horizontal component of the tension applied at its hinge point. For each segment of the tow chain, each segment of the tow cable will also receive a horizontal tow force related to its geometric shape, equivalent diameter, own pitch angle and local flow rate, etc., which is transmitted from bottom to top along the tow cable and is balanced by the horizontal component of the tension at the upstream and downstream nodes of each segment. Overall, at the towing point, the horizontal component of the towing tension should be equal to the sum of the tow forces along the entire tow cable and the sensor chain. By simultaneously balancing the tow force and the horizontal component of the tension in the horizontal direction, and combining the vertical force balance, the pitch angle of each segment, the position of each node and the size and direction of the tension along the tow chain from top to bottom can be solved, so as to obtain the initial depth and tension distribution.

[0068] In the static force balance calculation process, the given towing speed directly determines the hydrodynamic force of the fluid on each segment of the towed chain and the sensor chain body. It is usually assumed that the fluid is uniform and stationary, and the relative speed of the towed chain and the sensor chain body is approximately equal to the projection of the ship's speed in the inertial reference frame in the direction of each segment. Therefore, after the given towing speed, there is a coupling between the horizontal towed force of each segment and its orientation (pitch angle). At the same time, the difference between the gravity and the buoyancy in the vertical direction is also reflected in the pitch attitude through the tension direction and size. The static force balance solution is to find a set of pitch angles and tension distributions that satisfy the force balance in the vertical and horizontal directions under this coupling relationship, so that each node of the towed chain and the sensor chain body forms a self-consistent spatial configuration in the two-dimensional inertial reference frame. The initial pitch angle obtained by solving describes the inclination attitude of the towed chain and the sensor chain body relative to the vertical direction in the steady state, and the initial depth is given by the coordinates of each node in the vertical direction, while the tension distribution reflects the force transmission along each segment of the towed chain from the towing point to the sensor chain body.

[0069] Finally, the initial pitch angle, initial depth and tension distribution along the towed chain obtained by the above static force balance calculation are used as the initial conditions of the subsequent second-order ordinary differential equation with the pitch angle of each segment as the unknown, which is directly used in time domain numerical integration, so that the dynamic solution starts from a physically reasonable configuration that satisfies the force balance, thereby improving the stability and accuracy of the dynamic spatial positioning results of the towed chain and the sensor chain body.

[0070] In step S103, for any towed chain segment, the local velocity of each position point in the segment is obtained according to the translational and rotational state of the upstream node, and the local velocity is decomposed into a normal velocity component and a tangential velocity component relative to the axis of the towed chain segment.

[0071] wherein, for any microelement in the towed chain segment with an arc length of r from the upstream node, the local velocity of the microelement is the sum of the translational velocity and the angular velocity of the upstream node and the vector product of the position vector from the upstream node to the microelement.

[0072] For any rigid towed chain segment, it is necessary to obtain the local velocity distribution of any position point in the segment based on the translational and rotational state of the upstream node of the towed chain segment in the aforementioned two-dimensional inertial reference frame, and to decompose the local velocity relative to the axis of the towed chain segment on this basis. To this end, the towed chain segment is regarded as a rigid body, and the upstream node of the towed chain segment is the towing point A. The velocity of the towing point A in the two-dimensional inertial reference frame is denoted as The corresponding coordinate direction velocity components are and that is, The towed chain segment rotates in the two-dimensional plane around the axis perpendicular to the plane, and the angular velocity vector is denoted as wherein is a unit vector pointing out of the paper, is the angular velocity of the streamer segment in the plane. Taking any point X in the segment as reference, where the arc length of the point X from the upstream node along the axis of the streamer segment is r, the position vector from the towing point A to the point X can be written as where is the attitude angle of the axis of the streamer segment relative to the x-axis of the inertial reference frame. According to the principle of rigid body kinematics, the local velocity of the point X is which is the superposition of the translational velocity of the upstream node and the rotational velocity around the upstream node, i.e. the local velocity is the sum of the translational velocity and the angular velocity of the upstream node and the vector product of the position vector from the upstream node to the microelement. It can be expressed as:

[0073] ;

[0074] where . Expanding the above vectors, we get:

[0075] ;

[0076] Further, we have:

[0077] ;

[0078] After calculating the cross product, the component form of the local velocity in the coordinate axis direction is:

[0079] ;

[0080] It can be seen that for any microelement in the streamer segment with an arc length of r from the upstream node, its local velocity is exactly obtained by superimposing the translational velocity and angular velocity of the upstream node and the vector product of the position vector from the upstream node to the microelement, which fully reflects the velocity field distribution characteristics of the rigid streamer segment under the joint action of translation and rotation.

[0081] After obtaining the vector expression of the local velocity, in order to facilitate subsequent processing, it is necessary to further obtain the magnitude of the local velocity and the pointing angle relative to the coordinate axis of the inertial reference frame. The magnitude of the local velocity vector can be directly calculated from the above components, and the result is:

[0082] ;

[0083] The above expression reflects that the magnitude of the local velocity is not only related to the translational velocity of the upstream node, but also related to the angular velocity of the streamer segment, the position r of the microelement, and the attitude angle of the streamer segment. The direction of the local velocity can be represented by its included angle relative to the x-axis, which is given by the ratio of the velocity components:

[0084] ;

[0085] Under this definition, the local velocity vector The direction in the plane is determined by the angle The only decision, in the subsequent decomposition relative to the axis of the streamer section, can directly use the angle and the streamer section attitude angle The difference between the local velocity in the axial and normal components.

[0086] Considering that the axis direction of the streamer section in the two-dimensional plane is determined by the attitude angle , in order to decompose the local velocity relative to the axis of the streamer section into two parts of normal and tangential, the axis of the streamer section can be regarded as a straight line passing through the upstream node and making an angle of with the horizontal direction. In this way, the angle between the local velocity vector and the axis of the streamer section is . Under this geometric relationship, the component of the local velocity along the axis direction of the streamer section is the tangential velocity component, and the component along the direction perpendicular to the axis direction of the streamer section is the normal velocity component. Among them, the normal component of the local velocity can be directly obtained by the sine relationship between the local velocity and the angle, which is:

[0087] ;

[0088] Where represents the length of the local velocity vector, is the component of the local velocity in the normal direction of the streamer section. This normal velocity component describes the lateral motion component of the infinitesimal relative to the axis of the streamer section, which is the key quantity to determine the normal hydrodynamic force (i.e. the part of the fluid action that produces bending moment on the streamer section). Correspondingly, the local velocity in the same direction as the axis of the streamer section constitutes the tangential velocity component, which is distributed along the axis direction of the streamer section, mainly related to the parallel flow along the length direction of the cable body, and usually related to the tangential action that produces axial drag. In this step, only the decomposition process of the local velocity is described, that is, the projection of the local velocity in the normal and tangential directions of the streamer section is determined by the direction angle of the local velocity vector and the attitude angle of the streamer section, so as to obtain the corresponding normal velocity component and tangential velocity component at each arc length position r, thereby laying a foundation for subsequent calculation of normal drag and tangential drag and other hydrodynamic loads based on these two velocity components.

[0089] In step S104, the distributed hydrodynamic load acting on each of the streamer sections and the sensor chain body is determined according to the normal velocity component, the tangential velocity component, and the fluid density, the equivalent diameter and the resistance parameter;

[0090] The distributed hydrodynamic load includes normal drag and tangential drag, wherein: the normal drag acts along the normal direction, and the micro-element strength of the normal drag is calculated by the product of the fluid density, the normal resistance coefficient, the segment equivalent diameter of the towline segment, the micro-element length, the normal velocity component and the velocity magnitude; the tangential drag acts along the axial direction of the towline segment, and the micro-element strength of the tangential drag is calculated by the product of the fluid density, the tangential resistance coefficient, the equivalent force area of the micro-element, half of the product of the tangential velocity component and the velocity magnitude; the directions of the normal drag and the tangential drag are opposite to the corresponding local velocity direction.

[0091] Wherein, on the basis of obtaining the local velocity vector at any position in each towline segment according to the translation and rotation states of the upstream node, the distributed hydrodynamic load acting on each towline segment and the sensor chain body needs to be determined by further combining the normal velocity component and the tangential velocity component of the local velocity, and the fluid density, the equivalent diameter and the resistance parameters. In this embodiment, the towline segment and the sensor chain body are regarded as an elongated body immersed in a uniform fluid, and the local motion velocity of the elongated body in the two-dimensional inertial reference frame is given by the previous step. For a micro-element with an arc length of r in any towline segment, it can be considered as a small linear segment with a length of dr, which is subjected to the pressure resistance (normal drag) and shear friction resistance (tangential drag) generated when the fluid flows through. Since the local velocity vector Generally, it does not coincide with the axis of the towline segment, so it can be decomposed into a tangential velocity component along the axis direction and a normal velocity component perpendicular to the axis direction. The normal velocity component reflects the degree of lateral flow of the micro-element relative to the axis of the towline segment, and mainly determines the size of the normal drag; the tangential velocity component reflects the degree of relative slip of the micro-element along the axis direction, and is closely related to the shear friction drag.

[0092] In order to accurately represent the distributed hydrodynamic load, this embodiment adopts the way of writing the hydrodynamic force on the micro-element as a vector expression consistent with the direction of the local velocity vector. For any towline segment, taking its upstream node as point A, the local velocity vector The components in the inertial reference frame have been explicitly given in the previous implementation step:

[0093] ;

[0094] The local velocity magnitude is:

[0095] ;

[0096] Wherein, and are the velocity components of the upstream traction point A in the x and y directions of the inertial reference frame, respectively, is the angular velocity of the towline segment, an attitude angle of the streamer segment with respect to a horizontal axis, is a unit basis vector of the inertial frame. Based on this local velocity vector and the velocity magnitude, the normal drag force and the tangential drag force on a differential element of the streamer segment are modeled in a distributed manner, under the conditions that the fluid density is , the equivalent diameter of the streamer segment is D, the normal drag coefficient is , and the tangential drag coefficient is .

[0097] For the normal drag, the present embodiment regards it as a resistance formed by the pressure distribution around the cross section of the streamer segment, which acts locally in the normal direction of the differential 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 relationship, the normal drag differential force vector acting on the differential element at the arc length position r can be obtained, which is expressed as:

[0098] ;

[0099] In the above formula, the term corresponds to the component of the local velocity in the normal direction of the streamer segment, and its square reflects the increase of the normal drag with the square of the normal velocity component; the preceding term gives the vector direction of the local velocity in the inertial reference frame, so that the normal drag is constructed to act in the opposite direction of the local velocity vector. The coefficient then integrates the effects of the fluid density, the normal drag coefficient, and the equivalent diameter of the streamer segment on the strength of the normal drag, and the differential element length dr indicates that the force is in the form of linear density after integration on the unit arc length. The negative sign indicates that the direction of the normal drag is opposite to that of the local velocity vector, i.e., it plays a role of hindering the local motion.

[0100] For the tangential drag, the present embodiment regards it as a resistance caused by the shear friction between the fluid and the surface of the streamer segment, which acts in the direction of the axis of the streamer segment and is mainly related to the tangential component of the local velocity. Considering that the circumferential force bearing area of the streamer segment around its axis is related to the diameter, the equivalent force bearing area of the differential element can be written as , under the conditions that the fluid density is and the tangential drag coefficient is , the tangential drag differential force vector of the differential element can be written as:

[0101] ;

[0102] wherein, the term corresponds to the tangential component of the local velocity in the direction of the axis of the streamer segment, and its square reflects the change of the tangential drag with the square of the tangential velocity component; reflects the force bearing area of the differential element in the circumferential direction, is the proportionality coefficient of tangential drag. Similar to the normal drag, the velocity direction term in front of the tangential drag vector ensures that the overall direction of the tangential drag is opposite to the direction of the local velocity vector, so as to correctly represent the role of hindering the axial slip.

[0103] Through the above two vector expressions, the embodiment continuously gives the distribution forms of the normal drag and the tangential drag on the arc length coordinate r, that is, for any towline segment, the distributed hydrodynamic load along the segment can be obtained according to the local velocity, fluid density, equivalent diameter, and normal and tangential drag coefficients, etc. In the specific implementation, for the sensor chain body part, it can also be regarded as an elongated body composed of a plurality of microelements, and the normal drag and the tangential drag at any arc length position on the sensor chain body are calculated in the same form as the towline segment, using the equivalent diameter, the force length and the corresponding hydrodynamic drag coefficient thereof, so as to obtain the distributed hydrodynamic load of the entire tow drag chain and the sensor chain body in a unified framework. It needs to be emphasized that in this step, only the distribution forms and sizes of the normal drag and the tangential drag on each segment are determined, and the directions of them are opposite to the directions of the corresponding local velocities, which are used to represent the resistance effect of the fluid on the moving towline segment and the sensor chain body. These distributed hydrodynamic loads will be further brought into the moment balance and the dynamics equation in the subsequent steps.

[0104] In step S105, the moment balance equation containing the distributed hydrodynamic load and the downstream reaction force is established with the upstream node of each towline segment and sensor chain body as the reference point, and the moment balance equation is written as a second-order ordinary differential equation group with the pitch angle of each segment as the unknown quantity;

[0105] In establishing the moment balance equation, it also includes: taking the upstream node of each towline segment as the center of moment, the normal drag along the length direction of the towline segment is subjected to moment integration to obtain the hydrodynamic moment of the towline segment about the upstream node, and the tangential drag does not generate moment about the upstream node; the reaction force and reaction moment transmitted to the upstream node of the towline segment by the downstream adjacent towline segment and the sensor chain body through the connecting node are counted into the external moment of the towline segment; the moment integral of the normal drag along the length direction of the towline segment is subjected to numerical approximation by Gauss-Legendre five-point quadrature method to obtain the total external moment of each towline segment and the sensor chain body about its upstream node.

[0106] and, for each of the tow cable segments and the sensor chain body, respectively taking its upstream node as the reference point, an angular momentum balance relationship is established according to the angular momentum theorem, that is, the external moment of the tow cable segment or the sensor chain body is expressed as the sum of the inertial moment item about the center of mass and the moment item caused by the translational acceleration of the center of mass relative to the upstream node; the external moment is equal to the derivative of the angular momentum with respect to time, and a second-order ordinary differential equation with the pitch angle of the tow cable segment or the sensor chain body as the unknown quantity is obtained.

[0107] The boundary conditions of the second-order ordinary differential equation group are set as follows: the displacement, velocity and acceleration of the towing point of the tow chain at any time are the same as the displacement, velocity and acceleration of the ship; the downstream end of the sensor chain body is a free end, and the reaction force and the reaction moment are taken as zero.

[0108] In order to unify the foregoing distributed hydrodynamic load and the reaction force and the reaction moment transmitted by the connecting node between the downstream segment and the sensor chain body into the dynamic equation, in the embodiment, for each of the tow cable segments and the sensor chain body, the moment balance relationship is established by taking its upstream node as the reference point, and the moment balance relationship is written into the second-order ordinary differential equation with the pitch angle of each segment as the unknown quantity by means of the angular momentum theorem. Specifically, taking any tow cable segment as a rigid body AB, the upstream towing point A of the rigid body is connected with the ship, and the downstream node B is hinged with the downstream tow cable segment or the sensor chain body. In the two-dimensional inertial reference frame, the angular momentum of the rigid body about point A can be written as the sum of the angular momentum at the center of mass G and the additional item of the translational momentum of the center of mass about point A:

[0109] ;

[0110] wherein, is the angular momentum of the rigid body about its center of mass G, is a vector from the upstream towing point A to the center of mass G, is the linear momentum acting on the center of mass G. The resultant force of the external forces such as gravity, buoyancy, tow force, downstream reaction force acting on the rigid body is denoted as , and the moment about G is . According to the angular momentum theorem, the total external moment at point A should be equal to the sum of the derivative of the angular momentum with respect to time and the translational acceleration item, that is:

[0111] ;

[0112] wherein, is the inertial moment item about the center of mass, I is the rotational inertia of the tow cable segment about the center of mass, is the angular acceleration of the pitch angle of the segment, is the translational acceleration of the upstream towing point A. Substituting , the moment balance equation with the towing point A as the moment center can be obtained:

[0113] ;

[0114] where, is the total external moment of the external force acting on the rigid body AB (in this scenario, the equivalent force of the water drag and downstream reaction force) with respect to the towing point A. It can be seen that this external moment is explicitly decomposed into two parts: one part is the inertial moment of the inertial force about the center of mass , and the other part is the additional moment due to the translational acceleration of the center of mass with respect to the upstream node .

[0115] After the expression of the inertial moment of the whole rigid body is obtained, in order to introduce the distributed water drag load obtained in the previous step into the moment balance equation, the normal drag along the length direction needs to be integrated with respect to the upstream node of each segment of the towline as the center of moment, to obtain the water drag moment of this towline segment about the upstream node. Consider a towline segment AB with length L, and take a microelement X with arc length r from the towing point A on it. The microelement area of this microelement is , the local velocity is , the angle between the x-axis of the inertial reference frame and the local velocity is , and the normal velocity component is . The corresponding normal drag microelement force is:

[0116] ;

[0117] where, is the fluid density, is the normal drag coefficient, and D is the equivalent diameter of this segment of the towline. This normal drag acts in the opposite direction of the local velocity, and since its action line has a non-zero normal distance from the rigid body axis, it will generate a moment about the towing point A. The microelement moment generated by this microelement force on the towing point A is:

[0118] ;

[0119] where is the vector from the towing point A to the microelement X. The local velocity vector expression obtained in the previous step is:

[0120] ;

[0121] and the corresponding velocity magnitude is:

[0122] ;

[0123] and the velocity direction angle is:

[0124] ;

[0125] Substitute the normal drag formula, and the normal drag vector form on the micro-element can be obtained:

[0126] ;

[0127] Integrate the micro-element drag torque on the whole length , and the hydrodynamic torque of the whole section at the towing point A can be obtained:

[0128] ;

[0129] Substitute the specific expressions of and in the above formula and organize them, and the scalar form of the hydrodynamic torque can be obtained:

[0130] ;

[0131] From the above derivation, it can be seen that only the pressure drag along the normal direction will produce a non-zero torque about the towing point A, and the tangential drag along the cable segment axis direction, whose action line passes through the rotation axis, does not produce torque about the upstream node, i.e., the tangential drag does not produce torque about the upstream node. Therefore, when establishing the torque balance equation, only the integral torque term of the normal drag needs to be taken into account in the external torque, and the tangential drag only affects the translational balance and the axial tension, and does not enter the rotational balance about the upstream node.

[0132] Since the above normal drag integral term is difficult to obtain an analytical solution in general cases, and the integrand contains nonlinear combinations of pitch angle, pitch angle velocity, and upstream node translational velocity, in the specific numerical calculation, Gauss-Legendre five-point quadrature method is used to numerically approximate along the length direction of the streamer segment. That is, the arc length interval [0, L] is mapped to the standard interval, and five Gauss points are selected in the interval, and the weighted sum of the integrand at these five sampling points is approximated instead of integration, so that the hydrodynamic torque of each streamer segment and the sensor chain body about its upstream node at each time step is obtained. This processing can not only ensure the integration accuracy, but also take into account the calculation efficiency, which is suitable for subsequent multiple calls in time stepping.

[0133] After considering the torque contribution of the distributed hydrodynamic load to the node, the reaction force and reaction torque of the downstream adjacent streamer segment and the sensor chain body transmitted to the upstream node of the streamer segment through the connecting node also need to be taken into account in the external torque of the streamer segment. Taking the example of two streamer segments plus a sensor chain body, the torque balance equation of the first streamer segment at the towing point A is established. At the discrete kth time step, the pitch angle of the first streamer segment is denoted as , the acceleration of the towing point A is , and the mass and length of the segment are , the moment of inertia is . The angular momentum balance equation with respect to the reference point A can be written as:

[0134] ;

[0135] where is the total external moment acting on segment 1 at the kth time step, which consists of two parts: one is the total moment of the first segment's distributed normal drag with respect to the reference point A , and the other is the reaction moment of the second segment and the sensor chain through the connecting node B on segment 1 . Therefore, it can be written as:

[0136] ;

[0137] where the reaction moment can be expressed 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 kth time step be , then its moment about the reference point A is:

[0138] ;

[0139] Here is essentially the equivalent reaction force formed by the inertial force, normal drag force and tangential drag force of the second segment of the towline and the sensor chain, as well as the external force at the end of the array segment, which is specifically expressed as:

[0140] ;

[0141] where is the inertial force component of the second segment of the towline, and are the normal and tangential drag components (obtained by integrating the aforementioned distributed hydrodynamic load) acting on the second segment of the towline, and is the additional external force component of the sensor chain on the node. By including these downstream reaction forces and their moments into the moment balance equation of the reference point A, the coupling of the upstream and downstream towline segments and the sensor chain in dynamics is realized.

[0142] Similarly, for the second segment of the streamer and the sensor chain body itself, the same torque balance equation is established with the respective upstream node as the reference point, according to the above method: on the one hand, the normal drag along the length direction is converted into the hydrodynamic torque about the upstream node by integral or Gauss-Legendre five-point quadrature method; on the other hand, the reaction force and reaction torque from the more downstream components are taken into the external torque of the node, and the external torque is expressed as the sum of the inertial torque term of the corresponding rigid body and the torque term caused by the center of mass translational acceleration. Finally, for each segment of the streamer and the sensor chain body, the external torque is equal to the derivative of the angular momentum with respect to time, that is, the torque balance equation of each segment node satisfies the form of , so as to obtain the second-order ordinary differential equations with the pitch angle of each segment as the unknown quantity respectively. By combining all the second-order ordinary differential equations of the segments in the order of the nodes, a whole second-order ordinary differential equation group with the pitch angles of the segments as the state variables, including distributed hydrodynamic loads and upstream and downstream coupled reaction forces, is formed, which is uniformly solved by time integration in the subsequent steps, and is used to obtain the dynamic attitude and spatial position of the streamer chain and the sensor chain body in the whole motion process.

[0143] In step S106, the second-order ordinary differential equation group is synchronously solved at a unified time step to obtain the pitch angle and angular velocity of each streamer segment and the sensor chain body changing with time.

[0144] Based on the calculation of the foregoing steps, the rotation balance of the rigid segment AB (length L, mass m, equivalent diameter D) about the upstream node A. Taking the pitch angle of the first segment as an example, the second-order ordinary differential equation thereof can be written as:

[0145] ;

[0146] After the second-order ordinary differential equations with the pitch angles of the segments as the unknown quantities are established, these equations are synchronously numerically integrated at a unified time step to obtain the pitch angle and angular velocity of each streamer segment and the sensor chain body evolving with time. Specifically, for each rigid streamer segment and the sensor chain body obtained by discretizing the streamer, the torque balance relationship established at the upstream node thereof is finally obtained, and the second-order ordinary differential equation is obtained, each of which can be regarded as a rotation equation with the pitch angle of the segment as the unknown quantity, and the right end includes nonlinear terms such as local hydrodynamic force, downstream reaction force and translational acceleration of the upstream node. In these equations, the pitch angle of each segment is not only related to the angular velocity and angular acceleration of the segment, but also coupled with the adjacent segment and the sensor chain body through the reaction force and reaction torque at the connecting node, so the equations of all the segments must be treated as a whole second-order ordinary differential equation group, and cannot be solved independently. On the time axis, the simulation interval is divided into equally spaced discrete time points , and a fixed time step is uniformly selected At each time step, the pitch angle and angular velocity at the previous time step are used as initial values to solve the pitch angle acceleration at the time step, so as to advance the pitch angle and angular velocity to the next time step. In this way, the pitch angle and angular velocity of each segment are synchronously evolved on the same time grid throughout the time interval, ensuring that the dynamic coupling relationship between the streamer segments and between the streamer segments and the sensor chain is consistent in time.

[0147] In the specific numerical solution method, the second-order ordinary differential equation is first expanded in the conventional manner in the implementation, and each second-order equation with as the unknown quantity is rewritten as two first-order equations, with the pitch angle and the angular velocity as the paired state quantities of the segment. The pitch angles and angular velocities of all streamer segments and sensor chains are combined into a unified state vector, which is numerically integrated in the form of an ordinary differential equation system at a unified time step . The corresponding numerical integration uses the fourth-order Runge-Kutta method to synchronously update the pitch angle and angular velocity of each segment within the same time step. That is, at a certain time step , the local velocity distribution, the normal drag, and the tangential drag on each segment are first calculated using the current pitch angle and angular velocity, and then the total external moment about the upstream node of each segment is calculated according to the moment balance equation given in step S105, and substituted into the moment balance to obtain the pitch angle acceleration at the time; subsequently, the state vector is updated once in the time interval using the fourth-order Runge-Kutta method, so that the pitch angle and angular velocity of all streamer segments and sensor chains at are simultaneously given new values. Essentially, this is a synchronous solution: at each time step, the state evolution of all segments is uniformly advanced, and there is no situation where one segment is updated and another segment lags behind, thereby achieving synchronous numerical solution of the second-order ordinary differential equation system at a unified time step.

[0148] In the numerical solution process, since the right side of the second-order ordinary differential equation contains terms such as the hydrodynamic moment and the drag force obtained by integrating along the length of the cable segment, numerical approximation of these integrals with respect to the arc length is also required at each time step. To ensure calculation accuracy and maintain consistency with the aforementioned moment modeling, the Gauss-Legendre five-point quadrature method is used for integration in the arc length direction [0, L] of the streamer segment. Specifically, a number of Gauss points and their corresponding weights are selected on the standard interval [0, L], and ​​The sampling points are converted to the arc length interval [0,L], and the weighted summation is used to approximate the arc length integral.

[0149] 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:

[0150] ;

[0151] 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.

[0152] 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.

[0153] 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.

[0154] 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 is the angle of the segment axis relative to the horizontal x-axis, and the segment length Li is the arc length from the upstream node to the downstream node; the upstream node position , velocity represents the input to the segment, and the unit axial vector takes . Then the geometric position of the downstream node is given directly by a first-order recursion as:

[0155] ;

[0156] This formula is accumulated from top to bottom segment by segment to obtain all node coordinates. The corresponding velocity recursion is obtained from the rigid body kinematics relationship: the velocity of the downstream node is equal to the translational velocity of the upstream node plus the velocity increment caused by the rotation of the segment around the upstream node, which can be written as:

[0157] ;

[0158] where k is the unit vector perpendicular to the two-dimensional plane. Since the segment length is constant, the above formula gives the linear superposition form of the node velocity to the upstream node velocity and the segment angular velocity.

[0159] Using the above general formula for the typical arrangement of two segments of tow cable plus a sensor chain body, the recursion can be performed in the order of "towing point A, intermediate node C, upstream end of sensor E, and centroid of sensor G": given the coordinates and velocity of the towing point A , the length and pitch angle of the first tow cable segment are , then:

[0160] ;

[0161] ;

[0162] ;

[0163] ;

[0164] The length and pitch angle of the second tow cable segment are , and the following is obtained:

[0165] ;

[0166] ;

[0167] ;

[0168] ;

[0169] The sensor chain body is considered as an elongated body, and the pitch angle of its axis is , and the geometric distance from the upstream end E to the centroid of the sensor chain body along its axis is denoted as (when uniformly distributed where is the chain length). The position and velocity of the sensor chain centroid are given by:

[0170] ;

[0171] If the mass or buoyancy distribution within the chain is not uniform, then is taken as the actual centroid distance of the chain from the upstream end E; the above form remains unchanged. For the general case of a multi-segment streamer, one simply recursively and recursively apply the above to all segments, and finally make a single axial offset and conformal velocity correction to the sensor chain at its to obtain the target node and centroid time quantities.

[0172] In one embodiment, the method further comprises:

[0173] inputting the outputted centroid position of the sensor chain together with the bearing measurements obtained by the passive detector into a target state estimation algorithm to perform bearing-only target motion analysis and underwater target tracking, or for displaying the motion situation of the underwater target relative to the sensor chain.

[0174] After obtaining the centroid position of the sensor chain at each time instant by completing the foregoing steps, the centroid position is inputted into a target state estimation algorithm as the spatial coordinates of the observation platform in the two-dimensional inertial reference frame together with the bearing measurements outputted by the passive detector at the same time instant, for performing bearing-only target motion analysis and underwater target tracking. Specifically, at discrete time , the embodiment has obtained the coordinates of the sensor chain centroid in the two-dimensional inertial reference frame , and the bearing observation given by the passive detector for a certain underwater target is denoted as , which is the polar coordinate angle of the target relative to the centroid of the sensor chain. In the target state estimation algorithm, the target state vector is denoted as:

[0175] ;

[0176] where is the position coordinates of the target in the two-dimensional inertial reference frame, is the velocity components of the target in the x and y directions. The target motion model can adopt a uniform straight line model, and when the sampling time interval is , the discrete state transition relationship is written as:

[0177] ;

[0178] ;

[0179] where is the process noise, which is used to characterize the deviation between the real motion of the target and the ideal uniform motion model. The measurement equation embodies the azimuth-only measurement characteristic. In this embodiment, the azimuth observation is written as the geometric relationship between the sensor chain mass center position and the target position by:

[0180] ;

[0181] where is the target azimuth measured by the passive detector at time , is the measurement noise, is the arctangent function (in the implementation, the four-quadrant arctangent function can be used to ensure the continuity of the azimuth angle in the range of to ). As can be seen, the sensor chain mass center coordinates enter the measurement function as the observation platform position parameters, and the real spatial position of the platform obtained by the aforementioned towed cable dynamics and the azimuth measurement of the passive detector are combined in the same target state estimation algorithm framework to realize the target motion analysis under the azimuth-only condition.

[0182] In the process of solving the target state, extended Kalman filtering, unscented Kalman filtering, or particle filtering, etc. nonlinear filtering algorithm can be used to recursively estimate the above state space model. Taking the extended Kalman filter as an example, at each time, firstly, the state transition equation is used to time predict the target position and velocity, and then the azimuth observation at the current time and the sensor chain mass center position are used to construct the observation equation to correct the predicted state. Since the observation has only one dimension of azimuth, and the measurement function is a nonlinear arctangent form, the filtering algorithm needs to linearize 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.

[0183] 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:

[0184] 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.

[0185] 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.

[0186] 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.

[0187] 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.

[0188] The torque and equation establishing module 205 is configured to establish torque balance equations including the distributed hydrodynamic loads and downstream reaction forces with each upstream node of the towed cable segments and the sensor chain body as a reference point, and write the torque balance equations as second-order ordinary differential equations with each segment pitch angle as an unknown quantity;

[0189] The numerical solution module 206 is configured to synchronously solve the second-order ordinary differential equations to obtain the pitch angle and angular velocity of each towed cable segment and the sensor chain body changing over time under a unified time step;

[0190] The kinematics recursive module 207 is configured to calculate the position and velocity of each node in a geometric recursive manner according to the pitch angle of each towed cable segment and the segment length, and output the position of the sensor chain body relative to the inertial reference system from the mass center coordinates of the sensor chain body.

[0191] The specific details in the above system have been described in detail in the method part, and the details not disclosed can be referred to the content of the method part, and thus will not be described again.

[0192] The system discretizes the towed chain into a plurality of rigid towed cable segments that are hingedly connected to each other, introduces physical quantities such as chain body mass parameters and equivalent diameters, takes the ship motion as a traction boundary, obtains a reasonable initial attitude and depth in combination with static force balance, and then considers the normal and tangential hydrodynamic forces of each segment in the fluid to establish a dynamics equation set with the pitch angle of each segment as a state quantity and perform synchronous numerical solution, so that the dynamic responses of the towed chain and the sensor chain body are uniformly described over the entire time history. Through the above dynamic spatial positioning method, the bending shape and attitude evolution of the towed chain and the sensor chain body in the water can be more truly reflected under the conditions of ship speed change and maneuvering, so as to significantly improve the estimation accuracy of the mass center position of the sensor chain body and reduce the target motion analysis error introduced by the attitude uncertainty of the chain body.

[0193] The above-described drawings are only schematic illustrations of the processes included in the method according to the exemplary embodiments of the present disclosure, and are not for limiting purposes. It is easily understood that the processes shown in the above-described drawings do not indicate or limit the time sequence of the processes. In addition, it is also easily understood that the processes can be executed synchronously or asynchronously, for example, in a plurality of modules.

[0194] It should be noted that although several modules or units of the system are mentioned in the above detailed description, such division is not mandatory. In fact, according to the exemplary embodiments of the present 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 into a plurality of modules or units.

[0195] Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the features disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the disclosure being indicated by the following claims.

[0196] It should be understood that the present disclosure is not limited to the precise structures herein described and illustrated in the drawings, and that various modifications and changes can be made without departing from its scope. The scope of the present disclosure is limited only by the claims that follow.

Claims

1. A method for dynamic spatial positioning of a ship towing chain using sensors, characterized in that, The method includes: 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.

2. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, 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. 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.

3. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, 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.

4. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, 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.

5. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, When 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.

6. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, 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.

7. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, The boundary conditions for 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.

8. The sensor chain dynamic spatial positioning method for a ship towing chain according to claim 1, characterized in that, 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.

9. A sensor chain dynamic spatial positioning system for a ship towing chain, characterized in that, include: 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.

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