Optimized arrangement and curve estimation method based on IMU (Inertial Measurement Unit) nodes

Abstract

The invention discloses an optimal arrangement and curve estimation method based on IMU nodes, and belongs to the technical field of three-dimensional curve measurement and restoration. According to the method, initial nodes are generated through uniform sampling on a curve, and the sensitivity value of each node to a curve recovery error is calculated by using local attitude information collected by an IMU sensor, so that a local error distribution function is constructed. And aiming at the error sensitivities of different nodes, a density adaptive sampling strategy is adopted to realize the optimization of the overall node distribution. According to the method, the validity of the optimization point distribution strategy is quantitatively verified by combining an error evaluation index. Through the combination of density adaptive point distribution and spherical interpolation restoration, the method can significantly reduce curve reconstruction errors, improves the space efficiency of IMU arrangement, and is suitable for flexible cables and other three-dimensional curve accurate measurement scenes. The method has the advantages of being simple in algorithm, capable of achieving automatic point distribution, high in error control precision and the like, and an efficient and reliable technical scheme is provided for three-dimensional curve measurement and restoration.

Description

TECHNICAL FIELD

[0001] The present application relates to three-dimensional curve measurement and restoration technology, and in particular to an IMU node-based optimized arrangement and curve estimation method. BACKGROUND

[0002] In the application of three-dimensional curve measurement and restoration such as flexible cable, accurately obtaining the shape of the curve in space is of great significance for path planning, attitude control and state monitoring. The existing technology usually adopts the method of uniform sampling or fixed interval point arrangement, obtains the discrete nodes of the curve through the sensor or visual system, and then uses interpolation, fitting or filtering method for curve reconstruction. However, this kind of method has the following obvious shortcomings: Firstly, the uniform point arrangement strategy fails to consider the local complexity of the curve in space. The nodes in the high-curvature or rapidly changing area are too sparse, resulting in large curve restoration error. The nodes in the flat or slowly changing area are too dense, causing sensor redundancy, increased data processing amount and waste of computing resources. Secondly, when using the acceleration, angular velocity or direction information collected by the IMU or other inertial measurement units, the traditional method usually only performs simple interpolation or filtering processing, without fully considering the contribution difference of different nodes to the overall restoration error, so that the node arrangement cannot be optimized for local error.

[0003] In addition, the existing technology lacks an adaptive node arrangement strategy based on error sensitivity, which makes it difficult to achieve high-precision restoration in complex three-dimensional curve scenarios, and it is also difficult to balance the efficiency of sensor arrangement and the rational use of space resources. Therefore, there is an urgent need for a three-dimensional curve restoration method that can analyze the local error sensitivity of nodes based on the IMU measurement data, and perform non-uniform adaptive point arrangement, so as to realize dense node arrangement in high error sensitive areas and sparse node arrangement in low error sensitive areas, thereby improving the curve restoration accuracy, reducing the number of redundant nodes, and being applicable to complex three-dimensional curve scenarios such as flexible cable. SUMMARY

[0004] To solve the above problems, the present application provides a three-dimensional curve error adaptive optimized point arrangement and curve restoration method based on IMU node, which can realize dense node arrangement in high error sensitive areas and sparse node arrangement in low error sensitive areas, and combine spherical interpolation technology for high-precision curve restoration, thereby significantly improving the curve measurement and restoration accuracy, optimizing the efficiency of sensor arrangement, and being applicable to complex three-dimensional curve measurement scenarios.

[0005] The technical scheme adopted by the present application is as follows: S1, generating a high-resolution sampling grid in the normalized arc length interval of the three-dimensional curve to be measured; S2, collecting IMU measurement data of each node through the IMU sensor, and calculating the local sensitivity value of each node to the curve recovery error; S3, smoothing the sensitivity values of the nodes to form an error sensitivity function, and constructing a density function according to the function, for describing the node distribution density of each section of the curve; S4, obtaining a cumulative distribution function by cumulatively summing the density function, generating target sampling points on the density cumulative distribution function according to equal probability intervals, and obtaining corresponding arc length positions through interpolation reverse lookup; S5, converting the arc length positions into curve parameters through inverse function mapping to obtain a non-uniformly arranged node parameter set; S6, continuously interpolating and reconstructing along the curve by using the node tangent vector and the spherical interpolation method to obtain the recovered curve; S7, evaluating the recovery method under the node distribution strategy according to the root mean square error and the maximum error between the recovered curve and the true curve; if the error is less than a set value, repeating the above steps for curve reconstruction; until the error of the reconstructed curve is less than or equal to the set value, outputting the reconstructed curve.

[0006] Further, the step S1 specifically includes the following sub-steps: S1.1, establishing a continuous parameter function P(u) through three-dimensional cubic spline interpolation of the sampled discrete points, wherein u∈[0,1] is the curve parameter, corresponding to the actual curve arc length s; S1.2, unifying curves of different lengths or shapes through arc length normalization and parameter mapping method.

[0007] Further, in the step S2, the local sensitivity value step specifically includes: S2.1, performing unitization processing on the tangent vector of each node, and applying a small rotational disturbance in the direction perpendicular to the tangent vector; S2.2, respectively performing spherical interpolation curve recovery by using the tangent vectors after positive and negative disturbances, and calculating the error change between the curve and the true node under each condition; S2.3, obtaining the node error sensitivity value by dividing the positive and negative error difference by the disturbance angle: ; Wherein, δ represents the angle of the small disturbance, are the curve reconstruction errors calculated under the conditions of positive and negative disturbances, respectively.

[0008] Further, in the step S4, a minimum threshold is added to the node error sensitivity value during the construction of the density function, to prevent negative or zero values.

[0009] Furthermore, in step S5, the method for obtaining the set of non-uniformly arranged node parameters is as follows: First, a density function is constructed using the node error sensitivity, and a cumulative distribution function is calculated using the density function; then, sampling points are generated on the cumulative distribution function at equal probability intervals, and the corresponding curve arc length positions are obtained by reverse lookup of the sampling points; finally, the arc length positions are mapped to curve parameters to obtain the set of non-uniformly arranged node points.

[0010] Furthermore, in step S6, the spherical interpolation method specifically includes: first, normalizing the tangent vectors of adjacent nodes and performing linear interpolation along the sphere containing the two vectors to obtain continuously changing direction vectors; then, calculating the displacement increment by accumulating along each segment of the curve according to a preset subdivision step size, and accumulating sequentially from the starting node to obtain the coordinates of each reconstructed node.

[0011] Furthermore, in step S2, the IMU measurement data collected by the IMU sensor includes three-dimensional acceleration, angular velocity or direction information at the node, thereby obtaining the tangent vector information at the node; at the same time, the known spatial coordinates of the starting node and the tangent vector of the starting node are used as initial conditions, and curve reconstruction is performed in combination with the IMU measurement information of each node.

[0012] Furthermore, at least one set of three-axis accelerometers, three-axis gyroscopes, and three-axis magnetometers are provided at the node.

[0013] The beneficial effects of this invention: The three-dimensional curve error adaptive optimization point layout and curve restoration method based on IMU nodes proposed in this invention can significantly improve the accuracy of curve restoration. By calculating the local error sensitivity of each initial sampling node and combining it with a non-uniform adaptive point layout strategy, the node density is increased in high error-sensitive areas and reduced in low error-sensitive areas, achieving accurate capture of local curve features. This effectively reduces the root mean square error (RMS) and maximum error, improving the overall accuracy of curve restoration. Compared with the traditional uniform point layout method, this invention can highlight the contribution of the features of complex regions of the curve to the restoration accuracy, making the restoration result closer to the true curve.

[0014] Furthermore, this invention optimizes sensor layout and spatial resource utilization through density-adaptive node placement. While ensuring reconstruction accuracy, it reduces the number of redundant nodes, lowers the cost and power consumption of sensors such as IMUs, and achieves efficient node placement and resource optimization. This not only reduces hardware investment but also improves system operating efficiency, enabling more economical and reliable acquisition of curve morphology information in engineering applications. This invention utilizes IMU measurement data acquired along the curve, combined with node local error sensitivity analysis, to construct a non-uniform node placement strategy. This achieves dense node placement in high-error-sensitive areas and sparse node placement in low-error-sensitive areas, thereby improving the accuracy and reliability of 3D curve reconstruction in flexible cable scenarios.

[0015] This invention possesses strong universality and robustness. By statistically analyzing the error sensitivity of multiple representative curves and synthesizing the point placement strategy, the universal error sensitivity function and non-uniform point placement strategy generated by this invention can adapt to curve scenarios with different curvatures, stresses, or environmental conditions, and are particularly suitable for engineering applications involving underwater towed cables and other complex three-dimensional curve measurement and restoration. Through this strategy, even under complex curve shapes or variable stress environments, high restoration accuracy and stability can be maintained.

[0016] The method of this invention is simple and easy to implement. The algorithm is based on spherical interpolation (SLERP) and error sensitivity analysis, and the calculation process is clear. It can be directly implemented in a computer or embedded system. By combining a non-uniform point distribution strategy and IMU data application, it can reduce the number of redundant nodes and engineering costs while ensuring accuracy. At the same time, it can improve the reliability of flexible curve or cable morphology measurement, providing an efficient and reliable technical solution for path planning, attitude control, and condition monitoring. Attached Figure Description

[0017] To illustrate the technical solution more clearly, the following figure will briefly introduce the accompanying drawings used in the existing description.

[0018] Figure 1 This is a flowchart illustrating the steps involved in restoring the shape of a space curve.

[0019] Figure 2 These are comparison images of the spatial curve restoration effects. Detailed Implementation

[0020] The method for optimizing the layout and curve estimation of IMU nodes includes the following steps: S1. Curve Sampling and Parameterization: Within the normalized arc length range of the 3D curve to be measured, a high-resolution sampling grid is first generated to quickly acquire the spatial coordinates and tangent vector information of the curve at any arc length position. The density of the sampling grid can be adjusted according to the target accuracy requirements, and the number of sampling points can be flexibly configured according to the total length of the curve and the local curvature to ensure that there are enough sampling points in the high curvature region.

[0021] S1.1 Establishes a continuous parametric function from the sampled discrete points using three-dimensional cubic spline interpolation. ,in These are curve parameters, corresponding to the actual curve arc length. This parameter function allows for the direct acquisition of curve coordinates and tangent vectors at any arc length, enabling a continuous description of the curve.

[0022] S1.2 This invention enables curves of different lengths or shapes to be processed uniformly through arc length normalization and parameter mapping methods, providing a general and repeatable basis for subsequent node sensitivity analysis, density function construction, and non-uniform point placement.

[0023] S2. Node Error Sensitivity Analysis: This invention calculates the local sensitivity of each initial sampling node to the overall curve restoration error, as follows: S2.1 Normalization of tangent vectors: Normalize the tangent vectors of each node to ensure consistent length during spherical interpolation; S2.2 Small rotational perturbation: Apply a small rotational perturbation in a direction perpendicular to the tangent vector. The perturbation angle can be set according to the accuracy requirements. S2.3 Positive and Negative Perturbation Restoration: The tangent vectors after positive and negative perturbations are restored using spherical interpolation (SLERP) to obtain the perturbation curve; S2.4 Error Calculation: Calculate the error between the positive and negative disturbance curves and the actual nodes (root mean square error RMS and maximum error). S2.5 Sensitivity Calculation: The error sensitivity value of each node is obtained by dividing the difference between positive and negative errors by the disturbance angle. , where δ represents the angle of a small perturbation, and the quantization node affects the overall restoration error; Furthermore, through the above steps, the present invention can identify high-error-sensitive areas and low-error-sensitive areas, thereby providing a basis for non-uniform point distribution.

[0024] S3. Density Function Construction and Non-uniform Node Placement: Based on the node error sensitivity value, this invention further constructs a non-uniform node placement strategy: S3.1 Sensitivity Smoothing Processing: The node sensitivity values ​​are smoothed to form a continuous error sensitivity function, and a minimum threshold is added to prevent zero or negative values, ensuring that the cumulative distribution function is strictly monotonically increasing; S3.2 Density Function Generation: The error sensitivity function is used as the node density function. The error sensitivity function is large in high-sensitivity regions and small in low-sensitivity regions. S3.3 Cumulative Distribution Function Calculation: The cumulative distribution function is obtained by summing (integrating) the density function, which is used to generate target nodes through non-uniform sampling; S3.4 Non-uniform node generation: Sampling points are generated at equal probability intervals on the cumulative distribution function, and the corresponding arc length positions are obtained through inverse function mapping. Then, the curve parameters are mapped through parameter function to obtain a set of non-uniformly distributed nodes. S4. Curve Restoration Method: Utilizing non-uniformly arranged nodes and their tangent vectors, this invention achieves continuous curve restoration through spherical interpolation (SLERP). The specific steps are as follows: S4.1 Tangent Vector Interpolation: Normalize the tangent vectors of adjacent nodes and perform linear interpolation along the sphere containing the two vectors to obtain a continuously changing direction vector; S4.2 Displacement Accumulation: The displacement increment is calculated by accumulating along each segment of the curve according to the preset subdivision step size. The increment is accumulated sequentially from the starting node to obtain the coordinates of each reconstructed node. S4.3 Error Assessment: Compare the restored curve with the true curve, calculate the root mean square error (RMS) and the maximum error, and use them to evaluate the accuracy of the point placement strategy and restoration method.

[0025] Furthermore, repeat the above operation on the points that have been re-placed, place the points again, and reconstruct the curve. When the error of the reconstructed curve is less than the set value, it can be considered that the point placement method has a good curve restoration effect.

[0026] Furthermore, error sensitivity functions are calculated for multiple representative curves, and corresponding non-uniform point placement strategies are generated based on the nodal local error sensitivity of each curve. This invention obtains a comprehensive error sensitivity function under different curvatures and bending regions by statistically analyzing, normalizing, and weighting the error sensitivity functions of these representative curves, thus forming a universally applicable error sensitivity function. Moreover, this comprehensive function can reflect the overall error distribution characteristics of underwater towed cables under various stress, environmental, and bending conditions, providing a scientific basis for practical engineering point placement.

[0027] Based on this universal error sensitivity function, this invention further generates a unified non-uniform node layout strategy. This involves using a density-adaptive method to increase sampling nodes in high-error-sensitive regions and decrease nodes in low-error-sensitive regions, thereby optimizing the overall node layout. By generating target sampling points at equal probability intervals on the cumulative distribution function and mapping these sampling points to curve arc length positions, a set of node layouts that can be directly applied to practical engineering projects is finally obtained. Example 1

[0028] During implementation, the flexible cable or other 3D curve to be tested is first sampled at high resolution. One end of the curve is a fixed endpoint, and the other end is a moving point. The coordinates of the moving point are determined by parametric equations, which are as follows:

[0029] Several fixed points (four in this embodiment) are selected along a straight line from the fixed end to the mobile end. A sinusoidal perturbation is then applied to these points to obtain a mathematical model of the cable used for seabed operations. The total curve length is approximately 200. Over 100,000 sampling points are generated at normalized arc length intervals of 0.001 to ensure accurate spatial coordinates and tangent vector information at any arc length. Subsequently, continuous parametric functions are established using three-dimensional cubic spline interpolation on the sampling points. ,in This represents curve parameters mapped to actual arc length. This parameter function allows for the rapid calculation of curve coordinates and tangent vectors at any location, providing a continuous basis for subsequent error sensitivity analysis and non-uniform point distribution.

[0030] For the initially uniformly arranged nodes, this embodiment selects 30 nodes as representatives. For the tangent vector of each node, it is first normalized, and then a small rotational perturbation of ±1° is applied in the direction perpendicular to the tangent vector. The curves after positive and negative perturbations are reconstructed using the Spherical Interpolation (SLERP) method, and the root mean square error (RMS) and maximum error between the curves and the actual nodes are calculated. Then, the formula is used... Calculate the error sensitivity value for each node, where This represents the radian value of the perturbation angle. After completing the sensitivity calculation for all nodes, a set of discrete sensitivity data is obtained. This reflects the degree to which each position of the curve affects the restoration error.

[0031] Subsequently, the discrete sensitivity values ​​are smoothed, and a minimum threshold is added to the function to prevent zero or negative values, ensuring that the cumulative distribution function is strictly monotonically increasing. This sensitivity function is used as the node density function, and non-uniform sampling nodes are generated at equal probability intervals in the [0,1] interval using the cumulative distribution function (CDF). The corresponding actual normalized arc length position is obtained by interpolation and mapped back to the curve parameters, finally obtaining the set of non-uniformly arranged node parameters. The above operation is repeated on the obtained non-uniformly arranged nodes to obtain the node arrangement method with the minimum reconstruction curve error within a certain number of iterations.

[0032] To further improve universality, this embodiment introduces multi-curve statistical analysis and point placement function optimization methods. Specifically, in practical engineering applications, the error sensitivity function of a single curve may be limited by specific shapes, local curvature distributions, or IMU measurement errors. To overcome this problem, this embodiment selects multiple representative curve samples, covering different lengths, curvatures, and spatial layout characteristics of curves under the expected application scenarios. For each representative curve, the nodal local error sensitivity function is calculated according to the aforementioned method, and a corresponding non-uniform point placement strategy is generated.

[0033] Subsequently, statistical analysis was performed on the error sensitivity functions of multiple curves, including the mean function and the weighted average function, to obtain a comprehensive error sensitivity function applicable to the entire curve family. Based on this comprehensive function, a comprehensive node density function was constructed, forming a universal non-uniform sampling reference model. When generating non-uniform sampling nodes, the node set obtained through this universal sampling function can simultaneously adapt to the local characteristics of different curves, ensuring sufficient node density in high-error-sensitive regions while reducing redundant nodes in low-sensitivity regions, thereby improving the applicability and robustness of the overall sampling strategy in practical engineering environments.

[0034] After the node layout is determined, this embodiment uses IMU measurement data at each node to obtain tangent vector information. The IMU includes a set of triaxial accelerometers, triaxial gyroscopes, and an optional triaxial magnetometer, transmitting measurement data to the calculation module in real time via a data acquisition interface. Curve reconstruction begins with the known spatial coordinates and tangent vector of the starting node, and continuously accumulates displacement increments along each segment of the curve using spherical interpolation (SLERP) to obtain the reconstructed curve. To verify the reconstruction accuracy, this embodiment compares the reconstructed curve with the true curve, calculating the root mean square error (RMS) and the maximum error.

[0035] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for optimizing the layout and curve estimation of IMU nodes, characterized in that, Includes the following steps: S1. Initial nodes are generated by uniformly sampling on the curve, with an arc length interval between adjacent nodes; a high-resolution sampling grid is generated within the normalized arc length interval of the three-dimensional curve to be measured. S2. Collect IMU measurement data of each initial node through IMU sensor, and calculate the local sensitivity value of each initial node to curve restoration error; S3. The sensitivity values ​​of each node are smoothed to form an error sensitivity function, and a density function is constructed based on this function to describe the node density of each segment of the curve. S4. The density function is summed to obtain the cumulative distribution function. Target sampling points are generated on the density cumulative distribution function at equal probability intervals, and the corresponding arc length position is obtained by interpolation. S5. Convert the arc length position into curve parameters through inverse function mapping to obtain a set of non-uniformly arranged node parameters; S6. Using the tangent vector of the target sampling point and the spherical interpolation method, continuous interpolation reconstruction is performed along the curve to obtain the restored curve; S7. Evaluate the restoration method under the node placement strategy based on the root mean square error and maximum error between the restored curve and the true curve. If the error is less than the set value, repeat the above steps to reconstruct the curve. Output the reconstructed curve when the error of the reconstructed curve is less than or equal to the set value.

2. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 1, characterized in that, Step S1 specifically includes the following sub-steps: S1.1 The initial node is used to establish a continuous parametric function P(u) through three-dimensional cubic spline interpolation, where u∈[0,1] is the curve parameter, corresponding to the actual curve arc length s; S1.2 unifies curves of different lengths or shapes by using arc length normalization and parameter mapping methods.

3. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 2, characterized in that, In step S2, the local sensitivity value step specifically involves: S2.1 Normalizes the tangent vector of each node and applies a small rotational perturbation in the direction perpendicular to the tangent vector; S2.2 Restore the spherical interpolation curve using the tangent vectors after positive and negative perturbations respectively, and calculate the error change between the curve and the real node in each case; S2.3 The nodal error sensitivity value is obtained by dividing the difference between positive and negative errors by the disturbance angle: ; Where δ represents the angle of the small perturbation. These represent the curve reconstruction errors calculated under positive and negative perturbation conditions, respectively.

4. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 3, characterized in that, In step S4, during the construction of the density function, a minimum threshold is added to the node error sensitivity value to prevent negative or zero values ​​from appearing.

5. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 4, characterized in that, In step S5, the method for obtaining the set of non-uniformly arranged node parameters is as follows: First, a density function is constructed using the node error sensitivity, and a cumulative distribution function is calculated using the density function; then, sampling points are generated on the cumulative distribution function at equal probability intervals, and the corresponding curve arc length positions are obtained by looking up the sampling points; finally, the arc length positions are mapped to curve parameters to obtain the set of non-uniformly arranged node points.

6. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 5, characterized in that, In step S6, the spherical interpolation method specifically includes: first, normalizing the tangent vectors of adjacent nodes and performing linear interpolation along the sphere containing the two vectors to obtain continuously changing direction vectors; then, calculating the displacement increment by accumulating along each segment of the curve according to a preset subdivision step size, and accumulating sequentially from the starting node to obtain the coordinates of each reconstructed node.

7. The method for optimizing the layout and curve estimation of IMU nodes according to any one of claims 1-6, characterized in that: The IMU measurement data acquired by the IMU sensor includes three-dimensional acceleration, angular velocity or direction information at the initial node, and then the tangent vector information at the node is obtained; at the same time, the curve is reconstructed by combining the known spatial coordinates of the starting node and the tangent vector of the starting node with the IMU measurement information of each node.

8. The method for optimizing the layout and curve estimation based on IMU nodes according to claim 7, characterized in that, At least one set of three-axis accelerometers, three-axis gyroscopes and three-axis magnetometers are installed at the node.

Images