Adaptive node optimization track fine adjustment method based on B spline curve

Abstract

The invention discloses a B spline curve-based adaptive node optimization track fine adjustment method, which comprises the following steps of: S1, acquiring track measurement data, and determining a track lifting and lining quantity limit value; s2, adopting a Douglas-Peucker algorithm to insert an initial node, and obtaining an initial node vector; s3, carrying out adaptive node insertion on a B spline curve based on residual analysis to obtain an optimization variable and a node vector which are solved through constraints; and forming a B spline fitting curve according to the solved node vector and optimization variable, and obtaining the adjustment amount of each measurement point in the deviation point set of the track section to be finely adjusted. According to the method, B spline nodes are adaptively inserted through residual analysis, track fine adjustment automation is achieved, the efficiency is higher than that of a cubic spline, and abrupt change is avoided; the smoothness of track fine adjustment is optimized by controlling the minimum node spacing; the node number is greatly compressed, so that manual editing is facilitated; the method is suitable for the fields of various rail transit, data fitting, path planning and the like and is high in universality.

Description

Technical Field

[0001] This invention relates to the field of railway operation and maintenance, and in particular to an adaptive node optimization track fine-tuning method based on B-spline curves. Background Technology

[0002] The smoothness of railway tracks is crucial to the safety and stability of train operation. To maintain the safe and smooth operation of railway trains, railway bureaus invest significant manpower and resources annually in large-scale track fine-tuning to ensure track smoothness. Before fine-tuning, designers need to develop a track fine-tuning plan based on track measurement data (deviation or 10m chord, 20m chord indices). The design process must consider the impact of factors such as the catenary conductor pull-out value, bridge eccentricity, and clearance on the track alignment limits.

[0003] The current scheme design mainly relies on manual simulation and adjustment, which is inefficient and the quality depends mainly on human experience, resulting in significant differences in schemes designed by different people. Some scholars have proposed a track fine-tuning method based on 10th-order polynomial fitting, which can automatically generate track fine-tuning schemes. However, the 10th-order polynomial only fits the deviation curve of the entire section without piecewise fitting, which can easily lead to the "Runge phenomenon" when fitting deviation curves with large fluctuations. Other scholars have proposed a track fine-tuning algorithm that estimates track deviation characteristics. Based on cubic interpolation spline curves, it inserts piecewise nodes at the peaks and troughs to achieve an approximate fit to the deviation curve, effectively reducing the absolute value of the residuals. However, the residuals cannot be strictly controlled within the specified range according to design requirements, especially for ballasted tracks where the track lifting volume is non-negative. Still others have proposed a track fine-tuning algorithm based on quadratic programming. Using cubic interpolation spline curves as a reference and combining quadratic programming algorithms, it can achieve automated track fine-tuning schemes under constraints. However, this method, which inserts cubic spline curves between adjacent measurement points, results in an excessively large iterative calculation matrix, making fine-tuning calculations slow for long tracks. Furthermore, the large number of boundary points makes subsequent editing impossible. Summary of the Invention

[0004] To address the problems existing in the background technology, this invention provides an adaptive node optimization trajectory fine-tuning method based on B-spline curves. While achieving fine-tuning, the number of inserted nodes is significantly reduced compared to the amount of measurement data, and the subsequent editability of the fine-tuning scheme is preserved.

[0005] Therefore, the present invention adopts the following technical solution:

[0006] An adaptive node optimization trajectory fine-tuning method based on B-spline curves includes:

[0007] S1, obtain the set of deviation points of the track section to be finely adjusted, form the deviation value curve, and clarify the upper and lower limits of the adjustable starting and shifting amount of each measurement point in the deviation point set.

[0008] S2, Insert initial nodes on the deviation curve and obtain the initial node vector. ;

[0009] S3, Adaptive node insertion of B-spline curves based on residual analysis, yields the optimization variables obtained through constraint solving. and node vectors This includes the following steps:

[0010] S31, based on the initial node vector Construct the B-spline basis function matrix ;

[0011] S32, construct a quadratic convex optimization problem under constraints;

[0012] S33, Solve the quadratic convex optimization problem under the constraints constructed in S32:

[0013] The quadratic convex optimization problem under the given constraints is solved using the general interior-point method. If the solution is successful, the node vectors that constitute the final B-spline curve obtained through the constraint solution are obtained. and optimization variables If the solution fails, proceed to S4; if the solution fails, proceed to S34.

[0014] S34, Obtain the new node vector Repeat steps S31-S33 until a solution is found, then execute step S4. If a solution is not found after reaching the maximum number of iterations, it proves that the problem has no solution and the constraints need to be modified, the upper and lower limits need to be adjusted, and step S2 should be executed.

[0015] S4, based on the node vector and optimization variables A B-spline fitting curve is formed to obtain the adjustment amount of each measurement point in the deviation point set of the track segment to be fine-tuned.

[0016] Each measurement point in the deviation point set mentioned in step S1 above is... ;in, The mileage value of the measurement point. The deviation value of the measurement point. , This refers to the quantity of measured track data.

[0017] The upper and lower limits of the adjustable track lifting amount for each measuring point in the deviation point set are obtained based on the catenary guide height pull-out value, bridge eccentricity, and clearance value. For the set of deviation value points The upper limit of the value, For the set of deviation value points The lower limit of the value.

[0018] Step S2 above includes the following steps:

[0019] S21, based on the set of deviation points and the upper and lower limits of the adjustable starting track amount, set the minimum insertion spacing of the nodes. and tolerance The limit difference Based on the adjustable upper and lower limits of the starting track quantity and Determined by the absolute value of the mean;

[0020] S22, the node sequence is formed by the start and end points of the deviation value curve. , , ;

[0021] S23, the Douglas-Peucker algorithm is used to insert the initial node to obtain the node sequence. ;

[0022] S24, for the node sequence Perform filtering to obtain the filtered node sequence. ;

[0023] S25, the node sequence is processed using a normalization algorithm. Transform into initial node vector .

[0024] Step S24 above includes the following steps:

[0025] Calculate the node sequence Distance between adjacent node values ,in, , For node sequence The number of nodes in;

[0026] if Less than Then delete from the node sequence. Obtain the filtered node sequence ;in, To finally construct the initial node vector Dimensions Let B be the degree of the B-spline curve. .

[0027] In step S25 above, a normalization algorithm is used to normalize the node sequence. Transform into initial node vector The first and last vector values ​​are repeated. Next, take The normalization algorithm for node vector values ​​is as follows:

[0028] (1)

[0029] in, The track length needs to be adjusted.

[0030] Step S31 above includes:

[0031] The B-spline basis functions are constructed using the DeBoor recursive algorithm.

[0032] (2)

[0033] Among them, take , ;

[0034] Constructing the B-spline basis function matrix :

[0035] (3).

[0036] The specific steps of step S32 above are as follows:

[0037] 1) Set optimization variables The following constrained optimization problem is established:

[0038] (4)

[0039] in, This is the measured track deviation vector;

[0040] Expanding the objective function in the above equation into a quadratic form:

[0041] (5)

[0042] In the formula It is a symmetric positive definite matrix;

[0043] 2) Introduce nonnegative slack variables Transform the above inequality constraints into standard form:

[0044] (6)

[0045] in, for identity matrix , ;

[0046] 3) Introduce Lagrange multipliers and By constructing the Lagrangian function, we obtain a quadratic convex optimization problem under constraints:

[0047] (7).

[0048] In step S34 above, a new node vector is obtained. The method is as follows:

[0049] The optimization variables are solved using the unconstrained least squares method:

[0050] (8)

[0051] Calculate the residual value sequence of the measurement points:

[0052] (9)

[0053] Calculate the largest Mileage value of the corresponding measurement point The mileage value For the position where a new node needs to be inserted;

[0054] right After normalization according to formula (1), the new node vector is obtained. .

[0055] The adjustment formula for each measurement point in the deviation point set of the track segment to be fine-tuned in step S4 above is:

[0056] (10).

[0057] Compared with the prior art, the present invention has the following beneficial effects:

[0058] 1. This invention achieves adaptive insertion of B-spline curve fitting nodes through residual analysis, realizing automated design of track fine-tuning schemes under constraints. It is more efficient than cubic interpolation spline algorithm and does not cause abrupt changes in segments with high node density.

[0059] 2. The method of the present invention can optimize the smoothness of long, medium and short waves of the fine-tuning scheme by controlling the minimum node spacing, which greatly improves the smoothness of track fine-tuning.

[0060] 3. The method of the present invention generates a significantly reduced number of nodes for describing B-spline curves compared to the number of measurement points, and subsequent manual editing can be achieved by adjusting the node positions, which has great flexibility.

[0061] 4. The method of the present invention can be applied not only to the design of track fine-tuning schemes for various rail transit systems such as conventional railways, urban rail transit, high-speed railways, and heavy-haul railways, but also to the fitting of constrained B-spline curves for discrete data, and can also be used in fields such as train route planning, and has strong versatility and applicability. Attached Figure Description

[0062] Figure 1 This is a flowchart of the orbit fine-tuning method of the present invention;

[0063] Figure 2 This is a schematic diagram illustrating the insertion of initial nodes using the Douglas-Peucker algorithm in the method of this invention;

[0064] Figure 3 This is a schematic diagram of adaptive node insertion of B-spline curves based on residual analysis in the method of the present invention;

[0065] Figure 4 This is a schematic diagram comparing the deviation before and after optimization in Embodiment 1 of the present invention;

[0066] Figure 5 This is a schematic diagram of the optimized adjustment amount in Embodiment 1 of the present invention;

[0067] Figure 6 This is a schematic diagram comparing the 10m chord before and after optimization in Embodiment 1 of the present invention;

[0068] Figure 7 This is a schematic diagram comparing the deviation before and after optimization in Embodiment 2 of the present invention;

[0069] Figure 8 This is a schematic diagram of the optimized adjustment amount in Embodiment 2 of the present invention;

[0070] Figure 9 This is a schematic diagram comparing the difference in long-wavelength 300m vector distance before and after optimization in Embodiment 2 of the present invention. Detailed Implementation

[0071] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0072] like Figure 1 As shown, the adaptive node optimization trajectory fine-tuning method based on B-spline curves of the present invention includes the following steps:

[0073] S1. Obtain track measurement data and determine the starting and shifting limits, including the following steps:

[0074] S11, acquire the measurement data of the track segment to be fine-tuned, i.e., the set of deviation value points, and form a deviation value curve (a curve composed of the set of deviation value points of the segment to be fine-tuned), where the measurement points... , The mileage value of the measurement point. The deviation value of the measurement point. , This represents the quantity of measured track data.

[0075] S12, based on the overhead contact line pull-out value, bridge eccentricity, and clearance value, determine the upper and lower limits of the adjustable track lifting amount for each measurement point in the deviation value set obtained in S11. For the set of deviation value points The upper limit of the value, For the set of deviation value points The lower limit of the value.

[0076] S2, use the Douglas-Peucker algorithm to insert the initial node and obtain the initial node vector. This includes the following steps:

[0077] S21, Set the minimum insertion spacing for nodes. and tolerance .

[0078] The minimum node insertion spacing is used to control the distance between insertion points, enabling control over the smoothness of long, medium, and short waves in different application scenarios; the track smoothness of conventional railways generally only needs to meet certain requirements. The smoothness index of the midpoint chord is generally... A value greater than 15 can meet the smoothness requirements; the smoothness of high-speed railway tracks mainly meets the following requirements. Long-wave smoothness index of moment difference, generally Value greater than It can meet the ride comfort requirements; under difficult conditions The value can be appropriately reduced to improve the convergence speed of the model, but the smoothness index needs to be checked after solving; limit error according to and It is determined by the absolute value of the mean.

[0079] S22, the node sequence is formed by the start and end points of the deviation value curve. , , .

[0080] S23, with Figure 2 Taking the deviation curve shown as an example, the Douglas-Peucker algorithm is used to insert initial nodes, and the node sequence is obtained through the following steps. :

[0081] S231, Connect the first and last points A and G of the deviation curve with a straight line, calculate the distance between all points and the straight line, and find the maximum distance value. The maximum value in this embodiment The corresponding point is point C;

[0082] S232, using With limit In comparison, if If so, then proceed to step S233; if If the iteration ends, then the iteration ends.

[0083] S233, the maximum distance value The mileage value on the X-axis corresponding to point C As the node position, Insert a sequence of nodes and divide the curve into two parts, with point C as the boundary.

[0084] S234. Following the methods in S231 to S233, process curves ABC and CDEFG respectively.

[0085] conduct After several iterations, the final node sequence is obtained. .

[0086] S24, Calculate the node sequence Distance between adjacent node values , , For node sequence The number of nodes in, if Less than Then delete from the node sequence. Obtain the filtered node sequence .in, To finally construct the initial node vector Dimensions Let be the degree of the B-spline curve. Since only the curvature continuity (i.e., the continuity of the second derivative) needs to be ensured after trajectory adjustment, according to the characteristics of the B-spline curve, only the degree of curvature needs to be ensured. That's fine, but on the other hand, in order to reduce the impact range of a single node, so The value should be as small as possible, so in this example, we take... .

[0087] S25, the node sequence is processed using a normalization algorithm. Transform into initial node vector The first and last vector values ​​are repeated. Four times. The normalization algorithm for node vector values ​​is as follows:

[0088] (1)

[0089] in, The track length needs to be adjusted.

[0090] S3, based on residual analysis, implements adaptive node insertion of B-spline curves, obtaining the optimization variables through constraint solving. and node vectors ,refer to Figure 3 The specific steps are as follows:

[0091] S31, based on the initial node vector obtained in S2 Construct the B-spline basis function matrix .

[0092] The B-spline basis functions are constructed using the DeBoor recursive algorithm.

[0093] (2)

[0094] Among them, take , .

[0095] Constructing the B-spline basis function matrix :

[0096] (3)

[0097] S32, construct a quadratic convex optimization problem under constraints.

[0098] 1) Set optimization variables Establish the following constrained optimization problem:

[0099] (4)

[0100] in, The measured track deviation vector; the upper and lower limits of the adjustable track shifting amount. and As a constraint, ensure that the adjustment amount at each measurement point is within the adjustable range.

[0101] The objective function in the above equation is a typical least-squares problem, which can be expanded into a quadratic form:

[0102] (5)

[0103] In the formula The matrix is ​​a symmetric positive definite matrix, which guarantees that the optimization problem is a convex problem.

[0104] 2) Introduce nonnegative slack variables Transform the above inequality constraints into standard form:

[0105] (6)

[0106] in, for identity matrix , .

[0107] 3) Introduce Lagrange multipliers and Constructing the Lagrangian function results in a quadratic convex optimization problem under constraints:

[0108] (7)

[0109] S33, Solve the quadratic convex optimization problem under the constraints constructed in S32:

[0110] The above problem can be solved using the general interior-point method. If the solution is successful, the final node vectors constraining the B-spline curve are obtained. and optimization variables If the solution fails, proceed to S4; if the solution fails, proceed to S34.

[0111] S34, Obtain the new node vector And repeat steps S31-S33. Specifically, obtain the new node vector. The method is as follows:

[0112] The optimization variables are solved using the unconstrained least squares method:

[0113] (8)

[0114] Calculate the residual value sequence of the measurement points:

[0115] (9)

[0116] Calculate the largest Mileage value of the corresponding measurement point This position is where the new node needs to be inserted, achieving adaptive insertion of B-spline curve fitting nodes. After normalization according to formula (1), the new node vector is obtained. Repeat steps S31-S33 until a solution is found, then proceed to S4. If the problem remains unsolved after reaching the maximum number of iterations, it indicates that the problem is unsolvable and the constraints need to be modified, with adjustments made to the upper and lower limits. and Execute S2.

[0117] S4, the node vector obtained from S3 and optimization variables The formula for calculating any point on the B-spline curve. This allows us to obtain the adjustment amount for each measurement point in the deviation point set of the track segment to be fine-tuned:

[0118] (10).

[0119] This invention uses B-spline curves (represented by a small number of nodes) to fit deviation values ​​(a large number of discrete point data) to obtain an optimized trajectory deviation curve. This optimized curve is represented by node vectors, which can be manually edited by adjusting the node vectors. In other words, the B-spline curve expression method enables the fitting result to be manually adjustable.

[0120] Example 1

[0121] Taking a ballasted track section of a conventional railway as an example, this section is 2.5km long with a measurement point spacing of 1m, totaling 2500 measurement points. The entire line has significant undulations, with a maximum horizontal deviation of 61.2mm and a minimum of -65.4mm. The horizontal optimization setting for track shifting ranges from -20mm to 20mm. Since conventional railways only consider the 10m midpoint chord index, the minimum node insertion spacing is set based on experience. , tolerance .

[0122] Optimization of deviation and node distribution before and after optimization, as follows Figure 4 As shown in the figure, a total of 32 nodes were inserted after optimization, with a minimum node spacing of 41.3m. The node distribution in the figure demonstrates that the adaptive node optimization algorithm can reduce residuals by densifying nodes in areas with significant fluctuations. The adjustment amounts are as follows: Figure 5 As shown, the maximum guide distance is 20mm, and the minimum is -15.1mm, strictly controlled within the range of [-20mm, 20mm], meeting the guide distance limit requirements. Comparison of 10m chord performance before and after optimization: Figure 6 As shown, the maximum peak value before optimization was 7.1 mm, and after optimization it was 0.8 mm, significantly reducing shortwave irregularities. Compared with cubic interpolation splines, the convergence speed was improved by 82%, and the number of insertion nodes was reduced by 16%.

[0123] Example 2

[0124] Taking the fine-tuning of ballastless track on a certain operating high-speed railway line as an example, this section of the line is 2.5km long, with a measurement point spacing of 0.625m and a total of 4000 measurement points. The elevation deviation of the line is optimized, and the elevation adjustment amount for the operation is obtained. The maximum elevation deviation in the measurement data is -2.7mm, and the minimum is -19.5mm. Since this is fine-tuning of ballastless track, the adjustment limit is strictly controlled by the fastener adjustment amount. During maintenance, the adjustment amount should generally not be too large, and the elevation adjustment amount is controlled within [-2mm, 8mm]. To ensure long-wave smoothness (… (vector distance difference), minimum node insertion spacing is , tolerance .

[0125] The elevation deviation and node distribution before and after optimization are as follows: Figure 7 As shown, after optimization, a total of 8 nodes were inserted, with a minimum node spacing of 208.3m; the elevation adjustment is as follows. Figure 8 As shown, the maximum is 5.1mm and the minimum is -2mm, which meets the adjustment limit requirements; the optimized 10m chord (elevation) index and 30m vector distance difference index are almost 0 and are not shown again. The 300m vector distance difference index before and after optimization is as follows: Figure 9 As shown, the peak value was reduced from 6 mm before optimization to 1.3 mm after optimization, and long-wavelength irregularities were significantly reduced. Compared with cubic interpolation splines, the convergence efficiency was improved by 84%, and the number of inserted nodes was reduced by 12%.

Claims

1. A B-spline curve based adaptive node optimization orbit fine-tuning method, characterized in that, Comprise: S1, obtain the deviation point set of the track section to be fine-tuned, form a deviation value curve, and determine the upper and lower limit values of the adjustable track setting amount of each measurement point in the deviation point set; S2, inserting an initial node on the deviation value curve, obtaining an initial node vector ; S3, B-spline curve adaptive knot insertion based on residual analysis, obtaining the optimization variables solved by constraints and node vectors comprising the steps of: S31, constructing a B-spline basis function matrix according to the initial node vector , constructing a B-spline basis function matrix ; S32, construct a quadratic convex optimization problem under the constraint condition; S33, solve the quadratic convex optimization problem under the constraint condition constructed in S32: The quadratic convex optimization problem under the constraint is solved by using a general interior point method, and if the solving is successful, a node vector finally constituting the B-spline curve is obtained by the constraint solving and the optimization variable , S4 is executed; if the solving fails, S34 is executed S34, obtaining a new node vector and repeating S31-S33 until a solution is found, performing S4; if a solution is not found after a maximum number of iterations, it is proved that the problem has no solution and the constraints need to be modified, adjusting the upper and lower limits, performing S2; S4, according to the node vector and optimization variables , forming a B-spline fitting curve, obtaining the adjustment amount of each measurement point in the deviation point set of the orbit segment to be fine-tuned.

2. The adaptive node optimization orbit fine-tuning method of claim 1, wherein, The deviation point set in S1 is ; wherein, is the mileage value of the measurement point, is the deviation value of the measurement point, , is the number of track measured data; The upper and lower limit values of the adjustable amount of the catenary pulling-in at each measurement point in the deviation point set are obtained according to the catenary height pulling-out value, the bridge eccentricity and the limit value, The upper limit value of the value in the deviation point set The lower limit value of the value in the deviation point set The upper limit value of the value in the deviation point set The lower limit value of the value in the deviation point set 3. The adaptive node optimization orbit fine-tuning method of claim 2, wherein, S2 comprises the following steps: S21, setting a node minimum insertion interval according to the deviation point set and the upper and lower limit values of the adjustable starting track volume and a limit difference , the limit difference is determined according to the absolute value of the upper and lower limit values of the adjustable starting track volume and ; S22, a node sequence is constituted by the start and end points of the deviation value curve , , ; S23, insert initial nodes using Douglas-Peucker algorithm to obtain node sequence ; S24, screening the node sequence to obtain a screened node sequence ; S25, using a normalization algorithm to convert the node sequence into an initial node vector .

4. The adaptive node optimization orbit fine-tuning method of claim 3, wherein, S24 comprises the following steps: computing the sequence of nodes wherein the distance between adjacent nodes values is wherein, , is the number of nodes in the sequence of nodes is the number of nodes in the sequence of nodes If is less than , then delete from the node sequence ; wherein is the dimension of the final initial node vector , is the degree of the B-spline curve, .

5. The adaptive node optimization orbit fine-tuning method of claim 4, wherein, In step S25: The node sequence is converted into an initial node vector using a normalization algorithm The initial node vector is converted into a final node vector using a normalization algorithm The final node vector is converted into a final node vector using a normalization algorithm The final node vector is converted into a final node vector using a normalization algorithm The normalization algorithm for the node vector values is shown below: (1) wherein, is the length of the track to be adjusted.

6. The adaptive node optimization orbit fine-tuning method of claim 5, wherein, Step S31 comprises: The B-spline basis function is constructed by using the DeBoor recursive algorithm, (2) wherein, taking , ; Constructing b-spline basis function matrices : (3)。 7. The adaptive node optimization orbit fine-tuning method of claim 6, wherein, The specific steps of S32 are as follows: 1) Set optimization variables , establish the following constrained optimization problem: (4) wherein, is the measured orbit error vector; The objective function in the above formula is expanded into a quadratic form: (5) In the formula is a symmetric positive definite matrix; 2) Introduce non-negative slack variables Transform the inequality constraints into standard form: (6) wherein is identity matrix, , ; 3) Introduce Lagrange multipliers and Construct the Lagrangian function, resulting in a quadratic convex optimization problem with constraints: (7)。 8. The adaptive node optimization orbit fine-tuning method of claim 7, wherein, S34 obtaining a new node vector The method is as follows: The optimization variable is solved by using the unconstrained least square method: (8) The residual value sequence of the measurement point is calculated: (9) calculating the maximum odometer value of the corresponding measuring point , the odometer value a new node position needs to be inserted; To The new node vector is obtained after normalization according to formula (1) .

9. The adaptive node optimization orbit fine-tuning method of claim 8, wherein, The adjustment amount formula of each measurement point in the deviation point set of the track section to be fine-tuned in S4 is: (10)。

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