JP6307211B2 - Method for processing measurement data such as measurement Masaya applying crossing method and system for executing the method - Google Patents

Description

  The present invention relates to a method for processing measurement data such as measurement arrow using the intersection method and a system for executing the method, in particular, a standing wave component extraction method for measurement data, a standing wave component extraction system for measurement data, and a standing wave decomposition for measurement data. The present invention relates to a movement amount calculation method, a movement amount calculation system, a movement amount standing wave component extraction method, a movement amount standing wave component extraction system, a movement amount standing wave decomposition method, and a movement amount error reduction method.

  Trajectory errors occur on railroad tracks. Track misalignment is the residual displacement that occurs on the track due to repeated train loads. Many of the tracks are premised on maintenance of track deviation within an allowable range, and inspections, judgments, and repairs are repeated to maintain and improve the quality control level. There are two types of trajectory error: absolute value and relative value. The absolute value of the trajectory error is directly measured by an instrument. The relative value of the trajectory error is measured between the start point, midpoint, and end point using a string. In the present specification, unless otherwise specified, “trajectory error” means a relative value using strings. Of the trajectory error, the vertical deviation of the trajectory shape is called “high / low error”, and the horizontal error is called “street error”. In the measurement of the trajectory error, a distance measured by setting the distance axis along the trajectory is used as the distance in the longitudinal direction of the trajectory. As shown in FIG. 26, a chord having a constant length L [m] is stretched on the survey line distance axis, and the distance (shortest distance) between the rail and the chord at the midpoint of the chord is 1/2 of the chord length L. Measured continuously in the direction of the survey line, with only overlapping. This measurement method is called the Masaya method. The separation between the rail and the string at the middle point of the string is called “Shinya”, the separation in the vertical direction is called “High and Low Masaya”, and the separation in the left and right direction is called “Dori Masaya”. The measured positive arrow value (hereinafter, “measurement positive arrow”, using symbol m) is compared with a predetermined positive arrow value (hereinafter, “design positive arrow”, using symbol d) determined from the design alignment. A part where the difference between the design positive arrow d [mm] and the measurement positive arrow m [mm] exceeds the management target value is extracted, and a repair plan is drawn up.

  In the straight line, the value of the design positive arrow d [mm] is basically 0 [mm] for both the high and low positive arrows and the street positive arrow regardless of the chord length L [m].

  For a plane curve or a vertical curve, the value of the design arrow d [mm] is calculated from the design curve radius (hereinafter R [m]) and the chord length L [m] to be used by the approximate expression (Formula 1). Is done. The vertical curve is used before and after the change in the vertical gradient in the trajectory.

  In particular, the plane curve used in railways includes a transition curve that gradually changes the curve radius R between a straight line (R = ∞) and a circular curve (R ≧ 160 [m]: constant value). The so-called linearity is set. The relaxation curve is provided in order to prevent a sudden change in centrifugal force generated in a vehicle traveling on the curve. The length of this relaxation curve (Transition Curve Length, hereinafter referred to as TCL [m]) is the train speed V [km / h] passing through the curve and the left and right rails set to reduce the centrifugal force on the curve. Is determined in relation to the inclination amount (hereinafter “cant”, using symbol C [mm]).

The design positive arrow value d _i at each point i in the plane curve gradually changes in the relaxation curve portion, becomes a constant value in the circular curve, and draws a trapezoidal shape as shown in FIG. This may be referred to as a “trapezoid Masaya diagram”. This figure is equivalent to a figure showing the steering wheel angle at each point when the vehicle runs in the curve. Changes and irregularities in the Masaya diagram are important factors that have a major impact on the riding comfort of railways. From a mathematical viewpoint, the design positive arrow value d _i corresponds to a design linear second-order difference value, that is, a curvature value.

  As is clear from the fact that the actual shape of the planar curve is a trapezoidal arrow diagram, the arrow value and the arrow waveform output by the arrow method are the linear shape of the actual trajectory (hereinafter “real” Shape ”).

  For example, as shown in FIG. 28, the positive arrow waveform (lower waveform) obtained by the 10 m string positive arrow method is different from the actual shape (upper waveform). In particular, at the locations indicated by double circles in the positive arrow waveform, an output called “pseudo-error” is generated, which is judged to be incorrect in the positive arrow waveform, even though the trajectory is not incorrect in the actual shape. ing.

When the actual shape is a sine wave of a wavelength Λ [m], the detection magnification h _Λ of the Lm chord Masaya in this sine wave is derived as follows using the addition theorem of trigonometric functions. .

FIG. 29 is a graph illustrating Formula 2. As shown in this figure, since the various wavelengths Λ included in the actual shape are amplified or attenuated by the detection magnification h _Λ inherent to the 10 m string by the 10 m string singer method, Is different from the actual shape.

Detection ratio h _lambda, based on the wavelength lambda [m] having the actual shape varies between 0-2. The amplitude of Λ = 10 [m] wavelength component is amplified twice as much in the case of 10 m string Masaya. When Λ> 10 [m], the detection magnification h _Λ decreases as the wavelength Λ increases, becomes 1/2 when Λ = 30 [m], falls below 1/20 when Λ = 100 [m], and the wavelength Components are hardly detected. When 0 <Λ <10 [m], the wavelength Λ at which detection is impossible (h _Λ = 0) is 1 / (2 × k) times the chord length L (k = 1, 2, 3...) Exists.

  However, trajectory deviations in the bands of wavelengths Λ = 5 [m], 2.5 [m] and 1.25 [m] are extremely rare considering the rigidity of the trajectory structure composed of rails and sleepers. Yes, it is not subject to management as a trajectory error.

  In this way, when the actual shape of the orbit is measured using the 10 m string Masaya method, the wavelength component of Λ = 10 [m] to 20 [m] existing in the actual shape is amplified more than the actual wavelength band, and the wavelength band beyond that Then, an exaggeration attenuated more than the actual is performed.

  This exaggeration is similar to the correction that combines the vibration characteristics of railway vehicles and human sensory characteristics in the speed range of conventional lines, and research by Dr. Yoshihiko Sato has revealed that "Limits of track height fluctuation from the standpoint of comfort", Railway Technical Research Report No. 549, August 1966). According to the study, the detection characteristics of 10m string Masaya (high and low) and the response magnification according to wavelength of the up-and-down motion of the train at a speed of 60 to 120 km / h are in good agreement. In other words, the 10 m string Masaya method can effectively extract a track error that causes a shake of a train traveling in the speed range.

  In the high-speed area (conventional line over 120 km / h, Shinkansen over 250 km / h), the long-wave orbital deviation that causes the shaking of the train can be achieved by using the Masaya method with a long chord length of L = 20 m and 40 m. (Hereinafter referred to as “long-wavelength trajectory error”) is known to be able to be extracted effectively (Hideyuki Takai, Masao Uchida, “Track Management Method for Conventional Lines of 160 km / h”, Journal of Japan Railway Facilities Association, 1991 (October P20-25) (Hideyuki Takai, Yoichi Sunaga, Kunio Takeshita “Track Management Method for High-Speed Shinkansen” Railway Research Institute Vol. 9, No-1, P13-18 1995). In the description of the present application, the “Shinkansen” means a Shinkansen railway defined in Article 2 of the National Shinkansen Development Law.

The positive arrow values of the chord lengths of 20 m and 40 m are calculated by the known “double chord formula” from the measurement value of the chord length of 10 m without extending the physical measurement chord length. If the measured value of the interval of p = 5 [m] by the 10 m string positive arrow method is m _i , the 20 m string positive arrow value m20 _i is calculated as m20 _i = m _i−1 + 2m _i + m _{i + 1} .

The points extracted in the inspection process and judgment process using the Masaya method are repaired by actually moving the track in the subsequent repair process. Hereinafter, this is described as “track maintenance work”. In track maintenance work, information on “where, how much, in which direction to move” (hereinafter referred to as “movement amount”, symbol g is used in order to set the design arrow value d _i as the target arrow value. mm]).

  One method of determining the amount of movement is the intersection method. The crossing method is a calculation method that converts the measured Masaya data crossed by half chord lengths into the amount of movement at each point using a geometric illustration method. The amount of movement by the crossing method has a lot of construction results on actual tracks.

  The geometric calculation principle of the intersection method will be described with reference to FIG.

Point i-1, i, the measurements versine at i + 1, respectively _{m i-1, m} i, and _{m i + 1.}

  A positive arrow that changes as the point moves is indicated by a symbol with a '.

When the point i is moved by + g _i , the positive arrow mi of the point _i is calculated by the following equation from FIG.

m ′ _i = m _i + g _i

Measurements versine _{m i-1, m i +} 1 at point i-1, i + 1, the end of each string, the effect of point i is + _{g i} moves from the midpoint connecting theorem, the following two equations (Equation 3, Equation The amount of positive arrows changes as in 4).

  At this time, it is assumed that the points i-1 and i + 1 are not moved.

  In this principle, angle approximation similar to the double chord formula is performed, but the chord length L = 10 [m] used for the crossing method and the plane curve radius R ≧ 160 [m] used for the railway curve. If considered, the effect of the approximation is a negligible number.

Next, the accumulation method will be described as one of the conventional methods for calculating the movement amount. Cumulative method, the measurement versine of each point _{m i,} the design versine and _{d i,} the versine difference at the point _i, when defined by _{f i = (d i -m i} ), the j-th (j The moving amount g _j of the point of ≧ 2) is a solution using the fact that it is given by the following equation using the positive arrow difference f _i up to the previous positive arrow measurement point (j−1).

The accumulation method is given this name because the positive arrow difference f _i is multiplied and accumulated as the station i increases.

Actually, for example, using a spreadsheet, a known measurement arrow m _i (i = 1, 2,..., N) is input to the calculation table, and a design arrow d _i (i = 1, 2,...) Is input. , N) is arbitrarily given, so that the movement amount g _i (i = 2,..., N + 1) is calculated in the target section. However, at the measurement point n which is the last measurement point of the positive arrow and the point n + 1 which is the end point of the chord at that time, the movement amount of both points is 0 in order to ensure the continuity of the trajectory.

In order to establish this formula, it is known that the following two formulas (Formula 7 and Formula 8) may be satisfied in setting the design arrow d _i .

However, considerable skill is required to set the design arrow d _i that satisfies the above two conditions after satisfying linear conditions such as the relaxation curve length and movement restrictions at each point. Therefore, the accumulation method is a difficult calculation method.

Another method for calculating the amount of movement is the averaging method. Average method, to have a width in the design versine d _i, with geometrical calculation principles described above, a partial and visual solved going process. In the averaging method, the point i is moved so as to smooth the unevenness (adjacent positive arrow difference = curvature difference) appearing in the trapezoidal positive arrow diagram. This method is named “Average Method” because the unevenness of the measurement arrow is averaged.

The calculation principle of the average method is the same as that of the cumulative method. That is, versine value _{m i} of the point i, when moved the point _{i + g i, m 'i} = m i + g i , and the point i-1 and the point i + 1 before and after versine values m' _i-1 , M ′ _{i + 1} vary according to Equation 3 and Equation 4. At this time, the points i−1 and i + 1 do not move. In order to visually recognize such a change process of the arrow and proceed with the calculation, a curve correction computer has been developed (for example, see Patent Document 1).

  This trial calculation is repeated, and the relaxation curve length and shape (which can be visually confirmed), the movement limitation at each point, and the continuous conditions of measurement points 1 and n (the two points are structures that cannot be physically moved) Determine the amount of movement while checking.

  Since the average method performs an operation with a certain range for the design arrow value, individual differences occur in the calculation result of the movement amount. On the other hand, the average method has an advantage that it is not necessary to be aware of the conditions of Equations 7 and 8 that are essential in the accumulation method. For this reason, the intersection method used at track maintenance sites is exclusively an average method.

  As described above, the cumulative method and the average method in the cross method have advantages and disadvantages, respectively. Furthermore, there is a common problem with the cross method, and even if the inevitable measurement error that occurs during the Masaya measurement is small, a superimposed phenomenon that accumulates several times in the process of calculation occurs. Theoretically it is considered inappropriate. The rationale is as follows.

The movement amount gn of the _n- th measuring point by the intersection method is given by the following equation according to Equation 5, regardless of whether the design arrow d _i is determined by the accumulation method or the average method.

Here, the measurement error included in m _i and m _e, mean = 0, assuming a white noise of standard deviation = sigma, errors influence component g _en included in the calculated amount of movement g _n is It is given by

With the dispersion of the addition theorem, variance Var of the error influence component _{g en [g en]} is given by the following equation.

Effect, with respect to the variance of the measurement error m _e (= σ ^2), rapidly increases with increasing measuring points n. Therefore, the movement amount g _n which is calculated by the crossing method has been said to not be a theoretically that practical use of track maintenance.

  In addition, the chord length of 10 m and the moving amount of about several tens of mm that are normally used for “the point i−1 and the point i + 1 do not move when the point i is moved”, which is the premise of the geometrical principle. Then, although there is no intuitive discomfort due to gauge rigidity, etc., it is theoretically not clear at all.

  Although the crossing method is a calculation method with more construction results than in the past, there has been little theoretical analysis on its results, and its use is inappropriate due to the statistically derived measurement error superposition problem. As a result, it has reached a great deal of technical contradiction between practical results and theoretical evaluation.

  There is a restoration method other than the intersection method. The restoration method applies digital signal processing technology and, like the intersection method, calculates the actual shape of the trajectory using Masaya data obtained by trajectory error inspection. In 1987, Yoshimura of the Railway Technical Research Institute Published by Dr. Akiyoshi (see Non-Patent Document 1).

  In the restoration method, 10 m string Masaya data is wavelength-decomposed by Fourier transform, multiplied by the inverse of the detection magnification of 10 m string corresponding to each wavelength, and then recombined to obtain the actual shape of the orbit.

  The restoration method has the following characteristics.

  Of the wavelength-resolved trajectory error, the actual shape calculation of the trajectory focused only on the wavelength band to be managed and repaired can be performed uniquely without individual differences.

  The trajectory before and after the section to be calculated by the FIR (Finite Impulse Response) filter, which combines the detection characteristics of the 10m string Masaya method and the inverse characteristics of the selected wavelength band, developed to avoid the effects of the phase difference that occurs at this time. Is ensured.

  Superposition of Masaya measurement error, which is the biggest problem in the crossing method, hardly occurs.

  An example of the FIR filter is shown in FIG. The restoration wavelength by this FIR filter is up to 100 [m]. In FIG. 31, the horizontal axis represents the distance [m] from the data center, and the vertical axis represents the amplification magnification.

  In the restoration method, due to the characteristics of the FIR filter, additional measurement of about 350 to 500 [m] is performed according to the digital data at intervals of 0.25 to 1 [m] and the longest wavelength of the orbital deviation restored before and after the target section. Is required. Since such a large amount of measurement data is required, the use of track inspection vehicle data is a prerequisite for the use of the reconstruction method. The restoration method using track inspection data has been introduced as a method for calculating the amount of travel since about 2000 by railway operators that have introduced a dedicated system.

  Even before the reconstruction method was put to practical use, the crossing method was theoretically unsuitable, but the improvement of the crossing method curve correction with experience and experience in the field was made in each direction. By using the travel calculated by the law for track maintenance work, long wavelength orbital deviation is effectively suppressed and effective.

  For example, a correction amount calculation system that calculates a correction amount using an inverse matrix from a design front arrow and a measurement front arrow based on an intersection method is known (see Patent Document 2). However, in this correction amount calculation system, it is necessary to calculate the inverse matrix by a conventional inverse matrix calculation method (sweep method, Gauss-jordan method, etc.), and as the number of measurement points increases, the calculation of the inverse matrix becomes remarkable. It becomes difficult.

  In this correction amount calculation system, in order to reduce the error, after calculating the correction amount, the filter removes wavelength components of a predetermined wavelength or more. Although such error reduction is effective in practice, it has been carried out without theoretically elucidating how to avoid the effect of superimposing measurement errors, which has been considered theoretically impossible in the intersection method.

  Conventionally, the application of the crossing method other than curve correction in railways has not been studied at all.

Japanese Utility Model Publication No. 55-126306 JP 2014-109088 A

Akiyoshi Yoshimura "Examination of theoretical basis for restoration of trajectory of original trajectory and its application" Report No. 1336, 1987

  An object of the present invention is to solve the above-described problem, and an object of the present invention is to make it possible to process measurement data such as measurement Masaya by a simple calculation by applying a crossing method.

A standing wave component extraction system for measurement data according to the present invention is a system for extracting standing wave components from measurement data, an input unit for receiving measurement data, and a processing unit for processing measurement data input from the input unit. , And an output unit that outputs a standing wave component extracted by the processing unit, and the measurement data is n data measured at a predetermined spatial interval or time interval, and is represented by an n-dimensional vector x, The processing unit calculates a product of a matrix calculation step for calculating an n-dimensional eigenspace matrix S _i (i is an integer of 1 to n), the eigenspace matrix S _i and the vector x, and the measurement data And the extraction step of extracting the standing wave component S _i x having the i-th long standing wave wavelength from the first to second, and in the matrix calculation step, the (i, k) component S _{i, (j, k) of the} eigenspace matrix S _i is executed. Is calculated by the following equation: S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1)), where the i-th standing wave wavelength is The standing wave wavelength Λ _i of the long standing wave component is Λ _i = 2p (n + 1) / i. The measurement data standing wave decomposition method of the present invention is a method of decomposing the n pieces of measurement data into a sum of n pieces of standing wave components using the measurement data standing wave component extraction system, and represents the measurement data. The vector x is a sum of the standing wave component S _i x having the i- _th longest standing wave wavelength from i = 1 to i = n.

The movement amount calculation system of the present invention is a system for calculating a movement amount from a design arrow and a measurement arrow at a plurality of measurement points at a predetermined interval on a railway track, and receives an input of the design arrow and the measurement arrow. It includes an input unit, a processing unit for processing the design versine and measured versine inputted from the input unit, and an output unit for outputting the moving amount of the processing unit is calculated, the design versine, the plurality A positive arrow value determined at a station, represented by an n-dimensional vector d, and the measured positive arrow is a positive arrow value measured at the plurality of stations, represented by an n-dimensional vector m, The movement amount is a correction value for correcting the measurement positive arrow to a design positive arrow, and is represented by an n-dimensional vector g, and the processing unit calculates an n-dimensional intersection method matrix T ^−1. calculate the amount of movement and law matrix calculating ^step, the ^{g = T -1 (d-m} ) Perform a moving amount calculating step of, at the crossing method matrix calculating step, the crossing method matrix ^{T -1} (i, j) component ^T -1 (i, j) is the 1 ≦ i ≦ j ≦ n Then, T ⁻¹ (i, j) = 2 / (n + 1) × i (n−j + 1) is calculated. When 1 ≦ j <i ≦ n, T ⁻¹ (i, j) = 2 / (n + 1) ) × j (n−i + 1).

The moving amount standing wave component extraction system of the present invention is a system for extracting a moving amount standing wave component from a design arrow and a measurement arrow at a plurality of measurement points at a predetermined interval on a railway track. An input unit that receives an input of a positive arrow, a processing unit that processes a design positive arrow and a measurement positive arrow that are input from the input unit, and an output unit that outputs a stationary wave component of a movement amount extracted by the processing unit the design versine, the a versine value defined in multiple stations is represented by n-dimensional vector d, the measuring versine includes a component versine value measured by the plurality measurement points The movement amount is a correction value for correcting the measurement positive arrow into a design positive arrow, and is expressed by an n-dimensional vector g. _(the i 1 to n an integer) eigenspace matrix S _i and its solid Calculates a matrix and eigenvalues calculating step calculates a value lambda _i, the eigenspace matrix _{S i} and its eigenvalues lambda _i and versine difference (d-m) the product of the _{λ i S i (d-m} ), an extraction step of extracting a standing wave component λ _i S _i m having the i-th long standing wave wavelength is performed, and in the matrix and eigenvalue calculation step, the (j, k) component S _{i, (j of the} eigenspace matrix S _i is performed. _{, K)} is calculated by the following equation: S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1)), the eigenvalue λ _i Is calculated by the following equation: λ _i = 1 / (1-cos (iπ / (n + 1))), and the standing wave wavelength Λ _i of the standing wave component having the i-th long standing wave wavelength is Λ _i = 2p (n + 1) ) / I. The moving amount standing wave decomposition method of the present invention is a method for decomposing the n moving amounts into a sum of n standing wave components using the moving amount standing wave component extraction system, wherein the moving amount is: It is the sum of the standing wave component λ _i S _i (dm) having the i- th longest standing wave wavelength from i = 1 to i = n. The movement amount error reduction method of the present invention is a method for reducing an error included in the movement amount by using the movement amount standing wave decomposition method, and from the movement amount decomposed into a sum of standing wave components, The standing wave component (1 ≦ r <n) having a long standing wave wavelength from the first to the r th is removed. In this movement amount error reduction method, the r is preferably selected from integers of 1 to 5.

  According to the method and system of the present invention, measurement data such as measurement arrows can be processed by simple calculation using a matrix, and standing wave component extraction, movement amount calculation, and the like can be performed.

The figure explaining the definition of a term and a variable. (A) is a figure explaining the partial orthogonal coordinates in a curve, (b) is the elements on larger scale of (a). (A) is a figure explaining the partial orthogonal coordinate in the both ends in a curve, (b) is the elements on larger scale of (a). (A) (b) is a figure explaining the principle of the intersection method as a coordinate transformation problem. (A) is an intersection method matrix having an even number of dimensions, and (b) is an intersection method matrix having an odd number of dimensions. (A) is a figure explaining positive arrow measurement, (b) is a figure explaining a standing wave eigenvector. The figure which compared the calculation precision with respect to an orbit deviation component equivalent. The figure which compared the calculation precision with respect to a curve basic component equivalency. The figure which compared the calculation precision with respect to the 10-m chord deviation in the curve of curve radius 500m. (A) is a figure which shows the example of matrix spectrum decomposition | disassembly, (b) is a figure which shows the example of the discrete Fourier transform of the same data. The figure which illustrates the linear separation in a curve in case the zone | band of a band pass filter is longer than a relaxation curve length. The figure which illustrates the linear separation in a curve when the zone | band of a band pass filter is shorter than the relaxation curve length. . The figure which illustrates linear separation in a curve at the time of using matrix spectrum decomposition. (A) is a diagram illustrating a DC input value and a value obtained by decomposing and recombining the input value using matrix spectral decomposition, and (b) is a diagram illustrating a standing wave obtained by decomposing the DC. The side view which shows the string arrangement | positioning of a multiple tie tamper (MTT). MTT arrow transformation matrix. The graph which shows the change of the condition number (kappa) with respect to the dimension number n. Graph showing the statistical prediction error wave g _e maximum value. (A) is a figure which shows the input measurement error which has a white noise spectrum, (b) is a figure which shows each standing wave component of the error wave. (A) is a figure which shows another input measurement error which has a white noise spectrum, (b) is a figure which shows each standing wave component of the error wave. The graph which shows the residual rate of the error influence by rank removal. A graph showing an increase in the maximum error wave and a decrease in the residual rate. It is an example of the necessity examination in use judgment of a crossing method operation value, (a) is the amount of positive arrows, (b) is a figure showing the amount of movement. It is an example which can be used in the use determination of a cross method calculation value, (a) is a positive arrow amount, (b) is a figure which shows a movement amount. The block block diagram of the standing wave component extraction system of the measurement data which concerns on one Embodiment of this invention. The figure which shows the measurement of a trajectory error by 10 m string Masaya method. The figure which shows the relationship between the real shape of a plane curve, and Masaya. The figure which illustrates the false deviation in a track | orbit real shape and a Masaya figure. The characteristic view of the detection magnification in the 10 m string Masaya method. The figure which shows the geometrical calculation principle of an intersection method. The figure which shows an example of a FIR filter (until restoration wavelength 100m).

  The inventor of the present invention has made a theoretical redefinition of the crossing method to arrive at the present invention. Before describing the configuration of the present invention, the theoretical redefinition of the intersection method will be described.

(Definition of terms and variables, etc.)
Prior to the theoretical redefinition, refer to Figure 1 to define terms, including those already mentioned. The symbol indicates a scalar, vector, or matrix depending on the context.

  “Track” indicates the rail portion of the track structure.

  "Point" is used as a term indicating a place on the orbit.

  The “string” indicates a string having two different points on the track (the rail tops or the positions 14 mm below the rail top) at the start and end. This means the shortest distance between the two points.

  “Arrow” indicates the distance from the perpendicular to the intersection on the orbit at an arbitrary location on the string. The case where the arbitrary point is the middle point of the string is particularly called “Saya”.

  “Measurement point” is used as a term indicating a point on a trajectory where Masaya measurement is performed.

  n: Shows the total number of Masaya measurement points. Also called the number of stations. Corresponds to the number of dimensions of a matrix or vector.

  i, j, k: As a general rule, symbols used to indicate station numbers are used. Unless otherwise specified, 1 ≦ i, j, k ≦ n.

  L: String length [m]. L = 10 [m] unless otherwise specified.

  p: Measurement interval [m]. In the Masaya method, p = 1/2 × L [m].

Measurement positive arrow m _i : The positive arrow value observed at the measurement point i. Indicated as m as an n-dimensional vector (n-dimensional column vector).

Design positive arrow d _i : A positive arrow value determined from the design alignment or the like at the measurement point i. Denote d as an n-dimensional vector.

Movement amount g _i: in stations i, representing a correction value needed to correct the measured versine m _i design versine d _i. Indicated as g as an n-dimensional vector.

Actual shape y _i : Defines one y (i) axis perpendicular to the string and passing through the measurement point i, and one arbitrary x (i) axis parallel to the string (not necessarily passing through the measurement point i). The y (i) axis coordinate value of the point i in this orthogonal coordinate is represented. y is expressed as an n-dimensional vector.

  Λ: indicates wavelength [m].

  λ: Indicates the eigenvalue of the matrix.

  Other symbols, sequences, vectors, and matrices are defined each time.

(Coordinate definition and recurrence formula in railway curve)
When handling various curve shapes used in railways as a function, it is complicated to express the function in one orthogonal coordinate system. Therefore, a distance axis is set along the curve, and the first or second derivative at the point i is set. (Or difference) is represented by partial orthogonal coordinates with the x (i) axis at the same point and the perpendicular to the y (i) axis. Since the positive arrow method is a second-order difference amount of a real shape, a coordinate axis parallel to the positive arrow measurement string passing through the measurement point i can be used as the x (i) axis.

2 (a) and 2 (b), if the y (i) axis coordinate of each actual shape measurement point in the i-th partial orthogonal coordinate is y _i−1 , y _i , y _{i + 1} , from equation 13), a recurrence formula representing the relation between _{m i} and the unknown actual shape _{y i} (equation 14) is satisfied.

  In addition, as shown in FIGS. 3A and 3B, both the outer orbital line shape and the connection direction are expressed by the two orthogonal coordinates at the two ends of the target section to be corrected, at the measurement points 1 and n. It is necessary to ensure smooth continuity without changing. In this case, since the primary difference at both measurement points can be considered as the x (1) axis and the x (n) axis of the partial orthogonal coordinates, the following Expression 15 and Expression 17 are established.

  By using Equations 14, 16, and 17, it is possible to define the relationship between the positive arrow measurement value (known) and the actual shape (unknown) at all measurement points.

(Masaya transformation matrix T)
Next, the intersection method definition as a vector linear transformation will be described. The matrix T is defined as follows using Equations 14, 16, and 17. Hereinafter, the matrix T is referred to as “Masaya transformation matrix”.

  The square arrow transformation matrix T is a square matrix, the diagonal component is 1, the component adjacent to the diagonal component is 1/2, and the other components are 0.

  If the positive arrow transformation matrix T is used, the relationship between the known measured positive arrow vector m and the vector y representing the unknown actual shape can be written as Equation 19 as a linear transformation.

Given the design positive arrow vector _d , the relationship between the design positive arrow vector d and the vector yd representing the unknown actual shape will be described. Here, the "observation design versine d _i in certain portions orthogonal coordinate system x (i) -y (i) ", instead of thinking as the principle of geometric cross method "point i moves" as shown in FIG. 4 (a) (b), "the x (i) axis with the strings parallel to the chord at a distance of _{m i} (FIG. 4 (a)), the parallel strings and strings _{d i} "What does it look like when an unknown shape is observed from the partial orthogonal coordinate system _xd (i) -y (i) translated to the _xd (i) axis at a distance" (Fig. 4 (b))? The idea changes to a coordinate transformation problem. Thus, the magnitude relationship of the measured chord length and the moving amount, or trajectories today rigidity such as from "adjacent points moves, does not move" emerged from sensory discussion such, the relationship between the vector d and the vector y _d, chord length L [ m] can be generalized to the entire Masaya method.

Again, the mutual relationship between the vector d and the vector y _d, like the equation 19 given by the following equation 20.

Here, the vector y means a set of y (i) coordinates of the actual shape as seen from the x (i) -y (i) coordinate system. The vector _{y d, y d} means a set of _{x d (i) -y (i} ) of the actual shape viewed from the coordinate system y (i) coordinate.

  Equation 21 is derived from Equation 20 and Equation 19.

Here, if a new vector g is defined by Equation 22, g is the coordinate transformation performed based on the design positive arrow d and the measurement positive arrow m, and between the unknown real shapes observed in each partial orthogonal coordinate. This is the difference in the y (i) axial direction, and this is the movement amount vector to be obtained.

(Intersection matrix T ^-1 )
Since the square arrow transformation matrix T is a square matrix and its determinant is not 0 (det (T) ≠ 0), an inverse matrix T ⁻¹ exists. An example of the inverse matrix T ⁻¹ is shown in FIGS. The inverse matrix T ⁻¹ shown in FIG. 5A is an even matrix, and the dimension number n is an even number (for example, n = 6). The inverse matrix T ⁻¹ shown in FIG. 5B is an odd matrix, and the dimension number n is an odd number (for example, n = 7). The inventor of the present invention has found that the n-dimensional inverse matrix T ⁻¹ has the following three characteristics. These features have not been known so far (see, for example, Patent Document 2).

As a first feature, the inverse matrix T ⁻¹ is a non-negative real symmetric matrix having two symmetric axes.

  As a second feature, all components become natural numbers when multiplied by (n + 1) / 2.

  As a third feature, all components can be easily calculated with only the dimension number n.

That is, the inverse matrix T ⁻¹ can determine the components without using the inverse matrix calculation method (sweep method, Gauss-jordan method, etc.). When the hatched components in FIGS. 5A and 5B are calculated, the other components are determined by the symmetry of the inverse matrix T ⁻¹ (the second feature described above). If the component of the inverse matrix T ⁻¹ is expressed by an equation, the following equation is obtained. The proof of the regularity of the inverse matrix component expressed by Equation 23 will be described later.

Hereinafter, the inverse matrix T ⁻¹ is referred to as “intersection matrix”. If the intersection matrix T- ¹ is used, the movement amount vector g can be obtained by the following equation.

According to Equation 24, the crossing method is redefined as a linear transformation between the known measurement arrow vector m and the design arrow vector d and the unknown movement vector g. Further, from Equation 18 and Equation 23, the positive arrow transformation matrix T and the crossing method matrix T ⁻¹ are both biaxial real symmetric matrices, and various mathematical advantages of the symmetric matrix can be utilized.

(Analysis of the nature of the intersection method as an eigenvalue problem)
As shown in the following equation, a vector x whose length is changed without changing its direction and a scalar magnification λ thereof are called an eigenvector and an eigenvalue of the matrix A, respectively, for the linear transformation of the matrix A, that is, the coordinate transformation action. It is known that there are n dimensions.

  Ax = λx

Considering the result of Equation 24, which is the first-order transformation by the intersecting matrix T− ¹ , in practice, both the measured arrow and the amount of movement obtained by calculation are always measured strings (= partial orthogonal coordinates x (I) A direction orthogonal to the axis). It can be said that this is an eigenstate whose direction is not changed by the action of the matrix T ⁻¹ (the one whose direction is changed by 180 degrees is interpreted as a minus scalar multiplication). Therefore, it is possible to analyze the essence of the crossing method by considering the operation of the crossing method as an eigenvalue problem of the matrix T- ¹ , obtaining eigenvectors / eigenvalues, and considering them. Since the Masaya transformation matrix T and the intersection method matrix T ⁻¹ are real symmetric matrices, all eigenvalues and eigenvectors can be obtained as real numbers.

Derivation of eigenvalues and eigenvectors will be described. When the i-th eigenvalue of the matrix T is λ _Ti , the eigenvector is x _Ti , the i-th eigenvalue of the inverse matrix T ⁻¹ is λ _i , and the eigenvector is x _i , the following relationship is established. Therefore, if one eigenvalue / eigenvector of the matrix T and the inverse matrix T- ¹ is derived, the other eigenvalue / eigenvector is easily derived. In the derivation of the inverse matrix T ⁻¹ eigenvalue / eigenvector, it is more efficient to solve the eigen equation of T, which is a sparse matrix with many components.

  The eigen equation is derived as follows. Here, E indicates a unit matrix.

In order for the vector x in Equation 26 to have a solution other than the 0 vector, there must be no inverse matrix in (T−λ _n E), and the eigen equation is as follows:

  By solving this eigen equation, eigenvalues and eigenvectors are derived.

The eigenvalue λ _i of the intersection method matrix T ⁻¹ is derived as follows.

The eigenvalue λ _Ti of the positive arrow transformation matrix T is derived as follows:

The component u _ij of the normal eigenvector u _i is derived as follows:

  Each of the eigenvalue and eigenvector components has a trigonometric function with only the dimension number n as a variable, and can be accurately obtained (Equation 28 to Equation 30).

In particular, the normalized component of the i-th eigenvector u _i (Equation 30) matches the equation of the i-th natural vibration mode (hereinafter “i-th standing wave”) of the both-end fixed string of the chord length n + 1.

  Derivation of eigenspace matrix and spectral decomposition will be described.

If the normal eigenvector u _i and the transposed vector ^t u _i are used, the eigenspace matrix S _{i of} Expression 31 is derived from S _i = u _i ^t u _i .

S _i is a special matrix called projection element matrix, linear transformation by the matrix, u _i component (in this case, the i standing wave component) Any vector having the property of extracting.

By using the above eigenvalues λ _i , λ _Ti and eigenspace matrix S _i , T ⁻¹ and T can be expressed as the linear form of the eigenvalue and eigenspace matrix S _i , Equation 32, Equation 33. This linear format is hereinafter referred to as “matrix spectral decomposition”.

The matrix spectral decomposition of the intersection method matrix T ⁻¹ is expressed by the following equation.

  The matrix spectral decomposition of the Masaya transformation matrix T is expressed as follows.

(Engineering meaning and application of cross-method theory analysis)
Next, the values derived from the various theoretical analyzes described above, the engineering meaning and merits of the equations, and their applications are described.

(Eigenvalue and Masaya detection magnification)
Assuming that the longest wavelength extracted by matrix S ₁ is Λ ₁ [m], the first term obtained by decomposing the matrix spectrum by Equations 32 and 33 with the number of measurement points n and the interval p [m] observed, As shown in FIGS. 6A and 6B, Λ ₁ [m] is the wavelength of the first standing wave having a chord length (n + 1) p [m], and thus the following equation is derived.

Similarly, the i-th longest standing wave wavelength Λ _i [m] (1 ≦ i ≦ n) decomposed by the matrix S _i is obtained by the following equation.

On the other hand, the detection magnification h _Λi of the Lm chord Masaya in the sine wave of wavelength Λ _i [m] is given by Equation 2. By substituting the chord length L = 2p and Equation 35 into Equation 2, L and Λ _i are eliminated, and the following equation is obtained.

  The right side of Equation 36 is equal to the right side of Equation 29. Therefore, the relationship between the eigenvalue and the detection magnification is obtained as follows.

That is, the i-th eigenvalue lambda _i of intersection method matrix ^{T -1} is the chord length at the wavelength _{Λ i [m] L (=} 2p) [m] reciprocal of the detected magnification h _.LAMBDA.i the versine (hereinafter, "recovery ratio ”).

The i-th eigenvalue λ _Ti of the positive arrow transformation matrix T matches the detection magnification h _Λi of the chord length L (= 2p) [m] positive arrow at the wavelength Λ _i [m].

(Operation of intersection method matrix T- ¹ and Masaya transformation matrix T, its engineering meaning)
Substituting Equation 37 into the spectral decomposition of Equation 32 yields:

According to Equation 38, the operation of the intersection method matrix T ⁻¹ is the extraction of the standing wave component of Λ _i [m] contained in the arbitrary vector by the eigenspace matrix S _i and the reconstruction factor of the Lm chord Masaya at the same wavelength. It can be interpreted as the sum of amplification by (1 / h _Λi ). This is similar to the concept of the restoration method described above, and is different from the restoration method in that it does not depend on the Fourier transform.

  On the other hand, the following formula is obtained by substituting the formula 37 into the spectral decomposition of the formula 33.

According to this equation, the action of the Lm string positive arrow transformation matrix T is to extract the standing wave component of Λ _i [m] contained in an arbitrary vector from the eigenspace matrix S _i and to detect the Lm string positive arrow detection magnification (h _It can be interpreted as the sum of amplification by _Λi ). Therefore, the past than, the detection ratio h _.LAMBDA.i of versine of a single sine wave in which has been described in FIGS. 29 and Equation 2, of the two actions of amplification and degradation with the versine transformation matrix T, the eigenvalue It can be said that the amplification effect by λ _Ti is expressed.

Here, the standing wave decomposition action by the eigenspace matrix S _i shared by T ⁻¹ and T can be explained as follows.

By using the S _i, any n-dimensional vector a can be decomposed itself into n standing wave (matrix spectral resolution). Since the target vector a to be decomposed into standing waves is an arbitrary n-dimensional vector, it is not limited to the measurement positive arrow m, the design positive arrow d, the movement amount g, the actual shape y, etc. For example, a predetermined time interval or space It may be n physical quantities measured (sampled) at intervals. Therefore, matrix spectral decomposition can be used in a wide range of fields other than curve correction.

  The matrix spectral decomposition (formula 41) discovered by the inventors of the present application is similar to a conventionally known discrete Fourier transform (DFT) in that it has a function of decomposing into a sine wave. , It is a different transformation.

  Compare DFT and matrix spectral decomposition for transforming n discrete values measured at interval p. In DFT, n discrete values are decomposed into n-1 sine waves. In matrix spectrum decomposition, n discrete values are decomposed into n sine waves (standing wave oscillation mode).

In DFT, the longest wavelength is Λ ₁ = p (n−1) [m]. In matrix spectral decomposition, the longest wavelength is Λ ₁ = 2p (n + 1) [m].

In DFT, the shortest wavelength is Λ _n−1 = 2p [m]. In matrix spectral decomposition, the shortest wavelength is Λ _n = 2p (n + 1) / n [m].

  A DFT generally has a direct current component (a constant value). In matrix spectral decomposition, the DC component is always zero.

  In DFT, the sine wave of each wavelength generally has a phase difference. In matrix spectral decomposition, the phase difference of the sine wave of each wavelength is always zero.

  As described above, the matrix spectrum decomposition is a special sine wave decomposition method having various features such as all being decomposed into standing waves (that is, calculated only with real numbers) and no DC component.

  The inventor of the present invention theoretically redefined the crossing method and discovered matrix spectral decomposition, so that the Masaya method and the crossing method that have been used for a long time in railway track maintenance have all measured values. 2 × (measurement extension p (n−1) [m] + own string length 2p [m]) = 2p (n + 1) [m] is fundamentally decomposed into n standing waves. It became clear that The essential difference of the crossing method with respect to the restoration method using DFT lies in the difference in wavelength decomposition function by matrix spectral decomposition.

(Engineering merit of primary conversion using T- ¹ )
The trajectory engineering merit and meaning of the linear transformation using the intersection method matrix T ⁻¹ and the eigenspace matrix S _i will be described. The amount of calculation, which is an index of how much memory of the computer is consumed until the component of the inverse matrix T ⁻¹ is determined, is that the component is determined in one calculation if the dimension number n of the matrix T is determined ( Formula 23), the order O (n ³ ) is significantly improved from O (n ² ) to O (n ² ) or more.

In general, since a large number of divisions are required for the calculation of the inverse matrix, it is not recommended to solve the multiple simultaneous linear equations directly from the mathematical expression 24, which is an inverse transformation, due to the influence of rounding errors. By contrast component of T ^-1 In Equation 23, the component of S _i with a standing wave extraction action spectrally decompose T ^-1 is the equation 31, since both can be calculated with a small number of division in which only the number of dimensions n and variable , Not only saves the computer's memory consumption, but also minimizes the effects of rounding errors during computation, improving calculation accuracy.

This effect was verified using two types of input values (data number n = 31) simulating the positive arrow value observed in orbit. As shown in FIGS. 7 and 8, both provide a known input values, the matrix spectral decomposition again after synthesized value with S _i (in the legend of the graph is denoted as "degradation and resynthesis value") The difference was used to quantify the improvement in calculation accuracy. In FIG. 7 and FIG. 8, input / output inversion display is used to improve the discrimination. In the first condition, using the S _i obtained by equation 31. In the second condition, the conventional inverse matrix calculation method (Gauss-Jordan method) and eigenvector calculation method (QR method) were used.

When the input values and trajectory deviation component corresponds, the difference between the input value and the value again synthesized after matrix spectral decomposition, the use of S _i obtained by equation 31, a maximum 1.6 × 10 ^-14, and the conventional When used with the calculation method, the maximum value was 1.0 × 10 ⁻³ (see FIG. 7).

When the input value is equivalent to the curve basic component of the curve radius R = 250 [m] and the relaxation curve length TCL50 [m], the difference between the value synthesized again after the matrix spectrum decomposition and the input value is obtained by Equation 31. _{When i} is used, the maximum is 9.2 × 10 ⁻¹⁴ , and when using _i , the maximum is 6.3 × 10 ⁻³ (see FIG. 8).

Although both computations have an accuracy that is not problematic in practice, the difference reaches 10 ⁻¹⁰ or more. Therefore, when the matrix spectral decomposition is used, a clear computation accuracy improvement effect is confirmed by determining the components with a small number of divisions.

According to Equation 32, the movement amount g output using the intersection method matrix T ⁻¹ is a superposition of n standing waves by matrix spectrum decomposition. A standing wave has zero amplitude at the end of the string. Therefore, in the intersection method, regardless of the number of measurement points n and the input positive arrow (difference) value, the chord starts at point 1 (point 0) and ends at the end of string (point n + 1) at point n. The amount becomes zero. This means that the amount of movement calculated by the crossing method is always guaranteed to be continuous with the preceding and following trajectories regardless of the sampling interval, and is a very useful characteristic in track maintenance. It is.

Matrix spectral decomposition using the intersection matrix T- ¹ and wavelength decomposition using FFT (Fast-Fourier-Transfer), which is a kind of DFT, were compared using the same data. As shown in FIG. 9, data imitating a 10 m chordal deviation in an R = 500 [m] curve with n = 32 was used. Using the difference between the known input value and the numerical value after decomposing and recombining, we compared the calculation accuracy of both. There was no clear difference between the two in the accuracy of calculation.

  The difference between the sine waves used for the decomposition of both is shown in FIGS. 10 (a) and 10 (b). Both sinusoidal decomposition bases are quite different. In addition, the amount of movement at the beginning of the string at point 1 (point 0) and the end of the string at point n (point n + 1) is 0 in the matrix spectral decomposition (FIG. 10A), but 0 in FFT. (FIG. 10B) is not guaranteed.

  In the actual situation, this difference becomes apparent as the difference in sampling extension. In other words, when FFT is used, it is necessary to perform additional sampling with a corresponding extension for ensuring the continuity of the trajectory outside the interval to be evaluated and the filter calculation processing for smoothing as shown in FIG. Become. On the other hand, in the matrix spectrum decomposition, since all the sine waves used for the decomposition are stationary waves, additional sampling for eliminating the phase difference is unnecessary.

Furthermore, in the matrix spectral decomposition, the premise of partial orthogonal coordinates using the primary difference at both ends of the target section shown in FIG. 2A, that is, the amount of movement at station 1 and station n does not occur ( By establishing ₍ g ₁ = g _n = 0), it is possible to determine not only the continuity with the preceding and following trajectories but also the movement amount that keeps the continuity without changing the direction. For this reason, it has been handed down for a long time that the design arrow d _i that simultaneously establishes Equation 7 and Equation 8 is established. Until was elucidated by matrix spectral decomposition, it could not be said that it was elucidated.

Focusing on the coefficients in the first and nth rows of the intersection matrix T- ^1, the two necessary conditions (Equation 7 and Equation 8) can be obtained from the symmetry of the matrix and the fact that the matrix component can be a natural number. It can be proved succinctly and theoretically. Details of the proof will be described later.

Specific Design of d _i satisfying the above requirements (Equation 7 and Equation 8), conventionally, a linear separation method (a method of separating the trajectory deviation from the measured versine amount) is using the band-pass filter. In the linear separation method using such a bandpass filter, when the band is longer than the relaxation curve length, as shown in FIG. 11, the design curve of the relaxation curve is apparently changed to a shape that extends. End up. That is, the trajectory error can be separated, but the relaxation curve length (TCL) cannot be maintained.

  In the linear separation method using a band-pass filter, the relaxation curve length (TCL) is taken into consideration, and when the band is shortened on the contrary, as shown in FIG. Crazy separation is incomplete.

  Linear separation in railway curves using filters often tends to fall into such a trade-off relationship.

  On the other hand, as shown in FIG. 13, by using a linear separation method (matrix spectrum decomposition) based on the classic crossing method (Formula 7 and Formula 8), there exists a relaxation curve length (TCL) and exists. It is possible to separate the curve basic alignment while ensuring a smooth connection that is not affected by trajectory error and does not change the direction of the front and rear trajectories.

  This effect is particularly manifested as a difference in ride comfort in the left-right direction after maintenance is applied in the relaxation curve portion. Kant is strictly managed especially in the relaxation curve, so when the design Masaya is easily obtained with a filter, etc., Kant and Masaya, there is a shift in the mutual change, and the level of the same section as one wave Equivalent to generating madness. The occurrence of level deviation, particularly in a narrow conventional line between gauges, has a significant effect on vehicle lateral vibration. With matrix spectral decomposition, Masaya and Kant's changes can be matched. Matching the changes between Masaya and Kant is an important technical requirement in terms of riding comfort management on curves in which various forces such as excess centrifugal force act on passengers.

(Other merits of operations using the intersection matrix T- ¹ )
In track maintenance work, there is almost no case where the movement amount plan g ideally calculated from the design arrow d can be used as it is, and there are some restrictions such as absolute value of movement and movement direction due to various movement restriction conditions at the site. Added. Knowing what kind of finish will result when construction is performed with the corrected travel amount g 'with this restriction added (hereinafter, the "finished Masaya" symbol d' will be used), and depending on the situation, the constraint may be relaxed Measures to do this (for example, shaving the platform head or adjusting the position of the overhead line) are required.

When the corrected movement amount g ′ obtained by optimizing the constraint given to the movement amount plan (ideal) g is obtained, the finished Masaya d ′ (unknown) is g ′ = T ⁻¹ (d′−m). Thus, it is possible to examine the necessity of various additional measures for the purpose of improving the quality of track maintenance work, such as shaving stones, based on the size and shape of d ′ calculated and calculated by the following equation.

  The calculation and concept described above can be more effective when applied to the maintenance of high and low. When track maintenance work is performed for ups and downs, except for special cases, the rail is lifted to the desired height (hereinafter referred to as “working up”), and ballast (gravel) is formed in the gap generated under the sleeper. (Hereinafter referred to as “squeeze”) to smooth the wheel running surface of the train.

  The leveling work by MTT (multiple tie tamper) is the same, and the trajectory deviation is corrected only by lifting and tightening the rail and sleeper. Therefore, in the MTT relative reference leveling work that does not have information on the actual trajectory shape of the entire work section in advance, in order to make the MTT reliably sense the amount of the rise, the tip of the MTT string is 10 m string high and low trajectory error throughout the entire section. The work is continuously performed with the standard deviation set at a position higher than the current level by about 2 to 3 times the standard deviation. As a result, not only the correction of the trajectory deviation up to about Λ ≦ 30 [m] that can be sensed by the MTT chord length, but also the problem that the entire work section is shifted upward occurs.

Therefore, the crosswise matrix T- ¹ is used to efficiently grasp the actual shape in the height direction in the work section, and this is combined with optimization calculation that gives (-) direction movement = 0 and the allowable amount of rise at each point. By doing so, it is possible to suppress an unnecessary amount of rising generated in the MTT relative reference leveling operation.

  Specifically, by performing a reverse calculation using Equation 42, g ′ that does not raise the higher part in the section more than the current level, g ′ that only raises the part that is lower than the average line, and design the reverse phase of the current drop. When g ′ or the like for performing the above is adopted, it is possible to grasp what kind of high and low finished Masaya d ′ is obtained, and to examine the quality of the plan.

(Conversion to long chord length)
Next, string length conversion will be described. When an arbitrary 10-meter string arrow value is given, there are many situations in which it is necessary to convert this to another string-length arrow or arrow. In such a situation, the calculation using the intersection method matrix T ⁻¹ is useful.

The first scene in which string length conversion is required is conversion of a 10 m string positive arrow value to a long wavelength string positive arrow such as a 20 m or 40 m string. Conventionally, geometrically derived double chord length formulas are known. This is derived using the intersection method matrix T- ¹ .

Any n-dimensional vector conversion source 10m chord and m _10, the actual shape y is, since the reverse sign of the movement amount g, given by the following equation.

20m chord transformation matrix T _20, since the actual shape y is given by the component intervals 5 [m], is defined as follows.

If the measurement positive arrow when the actual shape y is measured by the 20 m string positive arrow method is m ₂₀ , m ₂₀ can be obtained by the following equation.

_Here, it defined as C _{20 = ^T 20 T} ^-1, be calculated, the following equation is obtained.

To the (n-1) th row from the second row of C _20, geometrically guided to match the coefficient used in conventional multiple chord length formal matrix elements (1,2,1) appears. That is, the double chord formula can also be derived by using the intersection method matrix T- ¹ .

(DC standing wave decomposition)
Next, DC decomposition will be described. In the discrete Fourier transform, direct current (constant value) cannot be decomposed into a sine wave. On the other hand, in the matrix spectrum decomposition, an eigenspace matrix S _i can be used to decompose an arbitrary n-dimensional vector into n standing waves (Equation 41), so that direct current (n constant values) Can be decomposed into n standing waves.

FIGS. 14A and 14B show DC standing wave decomposition in the case where the number of data is n = 31. The upper data in FIG. 14A is a DC input value to be converted. When the data is decomposed using the eigenspace matrix S _i , n = 31 standing waves as shown in FIG. 14B are obtained. When the standing wave is recombined, the lower data of FIG. 14A is obtained. The difference from the input value (that is, the correct answer) of the re-synthesized data is 7.96 × 10 ⁻¹² at maximum, and is substantially zero. In FIG. 14A, the recombined values are inverted in sign so as to be easy to see.

(Conversion to Y correction used for MTT)
By using the characteristic of T ⁻¹ that can perform standing wave decomposition even for a DC input value, the actual shape y including the bending state of the entire curve can be obtained by Equation 43 using the measurement positive arrow m ₁₀ . As shown in FIG. 15, in the measurement of the positive arrow by MTT, the string is not necessarily 10 [m], and the measurement position is not necessarily the center of the string. For example, in FIG. 15, a = 8.75 [m], b = 3.75 [m], and data sampling interval P = 1.25 [m]. If the MTT arrow transformation matrix (FIG. 16) created from the MTT string arrangement is used in Expression 43, the MTT Y-corrected vector _{hy that} is given by an arbitrary 10 m string Masaya can be calculated by Expression 47.

Here, y _adj is expressed by Equation 48. F _spl is a three-dimensional spline matrix.

Note that the actual shape y of 5 [m] spacing obtained from Equation 43 is based on the spline function (Akira Sakurai, Kazumi Yoshimura, Fumio Takayama, “Spline function by computer-data analysis, taking into account the rigidity of track structures such as rails and sleepers) / CG / differential equation "P1~60 using the Tokyo reference electric University Press, Inc. 11 May 1988), 5 [m] between to match the component interval of _{T mtt} performs the conversion to the real shape _{y adj} obtained by interpolating the using ing.

  This conversion is very effective for improving the accuracy of the MTT lining work in the case where the planned Masaya allows variation within a certain range due to movement restrictions and fixed points. The Y correction value for the MTT string can be obtained according to the 10m string design Masaya of the fixed point, making it possible to automate and standardize the lining work near the fixed point that relied on the conventional manual operation (intuition). Become.

(Summarization of crossing method and summary of engineering merit)
As described above, the intersection method is theoretically redefined by the linear transformation of the vector, Equation 24, so that firstly, data other than the target section is unnecessary (minimum sampling), and secondly, Theoretically, it was clarified that a simple and highly accurate standing wave decomposition function with two advantages of ensuring orbital continuity is inherent. The crossing method using the matrix spectral decomposition discovered by the inventor of the present invention is the separation of the curve basic linearity at the minimum sampling, the selection of the wavelength to be repaired at the minimum sampling, and the measurement value at any chord length / string arrangement It can be said that it is a movement calculation method with many engineering merits that can be applied to various applications such as conversion of, without individual differences.

(Measurement error behavior)
The explanation so far is a basic function as the calculation of the crossing method, and does not consider the error of the measurement arrow vector m. However, the actual measurement arrow vector m always includes a measurement error. The influence of this measurement error was the biggest problem in the theory of the crossing method.

Vector of the true value of m _r contained in the measurement vector _m, the vector of the measurement error component if m _e, a m = m r ₊ m _e, the behavior of the error due to cross method matrix T ^-1 is formula From the 24 variations, it is expressed as follows.

Here, g _r = T ⁻¹ (d−m _r ) is a true value component of the movement amount g, and g _e = −T ⁻¹ (m _e ) is an error component of the movement amount g. However, the measurement versine true value m _r, since it is impossible to be known, the behavior of the error, will be done using the following formula 50. Here, it is assumed that the measurement error component _me has a white noise spectrum according to a Gaussian distribution with an average = 0 [mm] and a standard deviation σ [mm].

Below, a g _e referred to as "error-wave".

Since the intersection matrix T ⁻¹ is a symmetric matrix, the influence of its measurement error is evaluated by the condition number κ (Tomoki Kotani “Numerical calculation as software” p91-102 February 2006 http: / see /na-inet.jp/nasoft/chap08.pdf). The condition number κ is the ratio of the maximum eigenvalue λ ₁ to the minimum eigenvalue λ _n (the following equation).

From Equation 37, the eigenvalue λ _i of the intersection method matrix T ⁻¹ means the wavelength Λ _i = 2p (n + 1) / i standing wave restoration magnification (1 / h _Λi ) (number of dimensions n, measurement interval p [m ], Eigenvalues (maximum eigenvalue λ ₁ and minimum eigenvalue λ _n ) for calculating the condition number κ are as follows.

λ ₁ : restoration factor of first standing wave Λ ₁ = 2p (n + 1) / 1

λ _n : _n -th standing wave Λ _n = 2p (n + 1) / n reconstruction magnification

For this reason, as the number of dimensions n increases, the condition number κ defined by Equation 51 rapidly increases. FIG. 17 shows the calculation result of the condition number κ. As the number of dimensions n increases, the condition number κ increases exponentially. Such a matrix indicating a sudden increase in the number of conditions is a matrix that is called an unfavorable condition and greatly affects the measurement error. It can be said that the intersection method matrix T ⁻¹ is a typical type of matrix.

Next, the estimated statistical maximum of the error wave g _e. This is what the repeated numerical experiments by the primary conversion to g _e from m _e using equation 50, was carried out under the following conditions.

Input vector m _e : White noise vector with average = 0 [mm] and standard deviation σ = 0.3 [mm]. This value is based on the inspection accuracy of the current track inspection vehicle.

  Number of dimensions n: 10 to 255. This value corresponds to 45 [m] to 1270 [m] of normal curve maintenance extension.

  The number of trials is 15000 times / each dimension.

For the target experimental data, // g _e // (maximum norm) appears 15000 times, and the average and standard deviation are obtained for each dimension.

An error wave g _e maximum value estimated from the mean and the standard deviation of // g _e // maximum norm obtained from numerical experiments under the above conditions, the occurrence probability (hereinafter, referred to as "hazard ratio%") Is shown in FIG.

Input vector _{m e} is, even assuming that gage error σ = 0.3 [mm] extent, the n = 255, and after conversion by intersection method matrix ^{T -1,} with a maximum value of about 700 [mm] error wave g _e occurs in the risk rate of 1%. I would like the error wave to be g _e ≈0 [mm], but this event reproduced in this numerical experiment is a measurement error superposition problem that has been said from the past.

It will be described that such error superposition can be performed rationally (reduced) by using a singular value decomposition (hereinafter referred to as “SVD”) method of the intersection method matrix T− ¹ . First, using SVD, it will be explained that there is a certain regularity in the influence of error waves.

Since the intersection method matrix T ^-1 is a real symmetric matrix, the singular values and singular vectors coincide with the eigenvalues and eigenvectors, and can be used for calculation as they are (Yamamoto Yusaku, “Singular Value Analysis and Its Applications” Kobe University Graduate School of Engineering Home Page, Nagoya University Lecture Materials, May 2009).

The intersection method matrix T ⁻¹ has a function of decomposing an arbitrary input vector into standing waves (Equation 32). Therefore, the error wave g _e calculated for white noise spectrum m _e be as follows, it is decomposed into a standing wave.

As shown in FIGS. 19 (a), 19 (b) and 20 (a), 20 (b), the white noise vector _me (FIG. 19 (a) and FIG. 20 (a)) is n by the intersection matrix T- ¹ . Are output as a superposition of standing waves (FIG. 19B and FIG. 20B).

The amplitude of each standing wave constituting the error wave g _e emerges with a deviation of each wavelength that depends on the magnitude of the eigenvalues lambda _i (Equation 52). The degree of bias is calculated by the SVD approximation method using the eigenvalues of the cross matrix T- ¹ (Ryuichi Kanno, Takefumi Sasayo, Satoshi Morimoto, “Image processing using singular value decomposition and wavelets”, Research Institute of Mathematical Analysis 1529 (See Volume p28 2007.26-41) and can be quantified as follows.

That is, for the norm // g _e // all errors wave vector calculated using the equations 32 including a standing wave component (-g e) _(sign inversion equation 52), from the first standing wave until the r standing wave removed error wave vector (this, that removal of up to rank _r) was calculated using ^{_{T r -1 = λ r + 1}} S r + 1 + ... + λ n S n in the case _where, norm of -g _er ^{= T r} -1 _{m e} The ratio of // _ger // is calculated by the following equation using eigenvalues.

  This ratio is an index called relative error. Since the norm is an index indicating the size of the vector, the relative error is expressed as (maximum value of error wave from which the first to r-th standing wave components are removed) / (maximum value of error wave having all standing wave components). Represent. When rank removal is performed, it can be used as an index indicating how much the influence of the original error wave remains. Therefore, hereinafter, the relative error is expressed as “residual rate”.

  FIG. 21 shows the calculation of each residual ratio (Equation 53) obtained by removing the first standing wave to the fifth standing wave for six dimensions from n = 555 to n = 11, using Equation 53.

  According to the calculation result, the residual ratio becomes the same value according to the rank number of the standing wave to be removed regardless of the dimension number n.

In addition, the amplitude ratio (λ _i / λ ₁ ) of each standing wave with respect to the first standing wave is constant regardless of the number of dimensions n. Specifically, the numerical values shown in the fifth column of Table 1 are obtained.

The intersection method matrix T ⁻¹ is a typical ill-conditioned matrix in which a slight error is superimposed. However, as can be seen from Table 1, in the crossing method matrix T- ¹ , error superposition does not occur as divergence, but the error effect is always about 95% on the five standing waves on the long wavelength side (up to rank 5), especially The inventor of the present invention has made it clear that there is a property that occurs with a strong bias that 72% concentrates on the first standing wave (fundamental vibration).

  Next, the meaning of trajectory management for the removal of long-wave standing waves performed to reduce the above-described error influence will be described. Since the wavelength of each standing wave is given by Equation 35, for example, removing the long-wavelength-side 5 standing wave component means removing the component with i = 1 to 5 in Equation 35. However, since the wavelength of the standing wave varies depending on the dimension number n (Equation 35), the calculation result is shown in Table 2.

As an example, when n = 100, p = 5 [m], and removal rank 5 (components with i = 1 to 5 are removed), the longest wavelength is i = 6, so Λ ₆ = 2 × 5 × 101 / 6≈168.3 [m]. The wavelength component (i ≦ 5) exceeding the longest wavelength Λ ₆ is removed with both true value and error.

By the way, it is known that the natural frequency of current railway vehicles is generally in the vicinity of 1 to 2 [Hz]. For this reason, for example, when the traveling speed is 300 [km / h], the wavelength Λ [m] of the track deviation causing resonance in the train is 41.7 to 83.3 [m]. If the upper limit of the wavelength subject to trajectory management (hereinafter referred to as “Λ _dm ”) is Λ _dm = 100 [m], even if the 5 standing wave components on the long wavelength side are removed, the trajectory management target Do not remove wavelength components

  In this way, the action of removing the 5 standing wave components on the long wavelength side has a meaning in track management in that not only the effect of measurement error is reduced, but also the wavelength component that has less influence on the train vibration is cut. . Moreover, even if these wavelength components are removed, the continuity with the orbit before and after the calculated amount of movement can still be enjoyed from engineering because it is a standing wave. .

  In the above-described measurement error reduction, the relative evaluation of how much of the error effect that can be removed is described (Tables 1 and 2). In an actual scene, it is also necessary to know the absolute value of “about how many mm of the calculated maximum movement amount x [mm], the effect of the error wave remains”. Hereinafter, such an absolute error will be described.

By extending the measurement extension, that is, by increasing the number of measurement points n, the removal rank r corresponding to Λ _dm increases (Table 2), so that the residual rate (%) of error influence can be reduced. The relationship of each parameter is given by the following inequality from the modification of Equation 35.

As an example, let Λ _dm = 100 [m] be the management target wavelength (the upper limit of the wavelength that is the target of trajectory management), the measurement extension is the number of measurement points n = 255 (measurement interval p = 5 [m], extension 1270 [m] ]), The rank r to be removed is calculated from Equation 54 as r ≦ 25.6, that is, r = 1 to r = 25. In this case, the residual ratio (%) is about 0.44% from Equation 53, and can be sufficiently reduced.

On the other hand, the maximum value of the norm // _{g e} // error wave vector _{g e} computed by the crossing method matrix ^{T -1} (Equation 52), as the 15,000 numerical experiments (Fig. 18), the number of dimensions n 22 increases as shown in FIG.

For this reason, even if the measurement extension (= number of dimensions n) is extended, the rank r (formula 54) corresponding to the upper limit Λ _dm of the wavelength to be tracked is increased, and the residual rate (= relative error) is reduced. // _ger // calculated from Equation 53 is a substantially constant value. Hereinafter, // _ger // is expressed as “ _ge absolute value in Λ _dm ”.

Gage accuracy σ = 0.3 [mm], and 1% risk rate, indicating _{g e} absolute value of lambda _dm calculation results of (// _{g er} //) in Table 3.

If the Λ _{dm =} 100 [m], if gage accuracy same, can be read must expect a 1% risk of the appearance of g _e absolute value of 3 [mm] Regardless measurement number n . The value of the top row indicates the // g _e // maximum value for each measurement number n.

Next, the inspection accuracy is changed, and // g _e // the maximum value is obtained from 15000 numerical experiments in the same manner as described above, and Table 4 is obtained by Expression 53 that is multiplied by the residual rate determined from Λ _dm .

It is possible to statistically estimate the influence of error due to the calculation of the intersection method based on the inspection accuracy of the data used and the upper limit Λ _{dm of} the wavelength to be managed. This numerical value can also be used as a value indicating the S / N separation boundary in the output result when the movement amount is calculated by the intersection method.

To summarize the above, the behavior of the error wave g _e generated during crossing method used is as follows.

  Regardless of the number of measurement points n, the influence of the measurement error is always 95% of the total of the five standing waves on the long wavelength side, and particularly 72% of the first standing wave.

The absolute value of the emerging error wave g _e (mm) can be statistically estimated by long wavelength side management wavelength is on the track management request lambda _{dm [m]} and gage accuracy of the measurement system sigma (mm).

To emerge as a superposition of the error wave g _e also standing wave, even if the removal of unnecessary wavelengths, continuity with the longitudinal trajectory of the movement amount output is guaranteed.

By using the characteristics of the behavior of such errors wave g _e, the amount of movement calculated by the intersection method is whether can be used without the influence of the error, the output movement amount size (alpha of the current track deviation can be determined using a poor degree) the error wave g _e absolute value at the boundary of the S / N separation shown in Table 4. Specific examination examples will be shown with reference to FIGS. 23 (a) and 23 (b) and FIGS. 24 (a) and 24 (b).

In case that the shown in FIG. 23 (a) (b), the movement amount output becomes a less error wave g _e in Λ _{dm =} 100 [m], division of the error wave and event value is difficult, construction The plan is “need to be examined”.

On the other hand, in the case shown in FIG. 24 (a) (b), the movement amount output is much higher than the error wave g _e in Λ _{dm =} 100 [m], at least the error wave be adopted plan Since the improvement up to the _ge amplitude line can be expected, it is determined as “usable”.

  However, whether or not to actually perform maintenance is determined by whether or not improvement over a certain level can be expected from the current orbital state, rather than whether or not it is affected by error waves. Therefore, the values obtained by multiplying the numerical values in Table 4 by the “expected value for improvement” are shown in the graphs of FIG. 23B and FIG. “Judgment line”). Here, this value is doubled. This means that the maintenance can be expected to improve to half of the current level even if the influence of error waves is considered.

  If the power spectrum of the current orbit (Power Spectrum Density, hereinafter referred to as PSD (R)) is known in advance, the power spectrum of the judgment line calculated from the numerical values in Table 4 and the safety factor (hereinafter referred to as (This is expressed as PSD (e)), and a direct comparison with this is performed, whereby the usable wavelength region of PSD (R)> PSD (e) can be determined.

(Summary of theoretical redefinition of crossing method)
As described above, the inventor of the present invention uses the crossing method as an eigenvalue problem (Formula 27) in the linear transformation (Formula 24) between two vectors of a positive arrow difference vector (dm) and a movement amount vector g. ) And re-evaluated. As a result, it became clear that the intersection method has the following functions and characteristics.

(1) The real symmetric matrix T ^-1 representing the crossing method and the real symmetric matrix T representing the positive arrow method include a simple and high-accuracy standing wave decomposition function, positive arrow restoration magnification, and amplification by an eigenvalue that matches the detection magnification. The function is inherent (Equation 38, Equation 39).

(2) the error wave g _e appearing by measurement errors, despite increased and accompanied to the magnitude of the number of measurements n is rapidly increased (FIG. 18), associated with the eigenvalues of the matrix T ^-1 as a superposition of n standing wave Appear with a bias (Formula 52).

(3) appearing errors wave g _{e (standing} wave), regardless of the measurement extension, always 95% of the total error affecting the 5 standing wave on the long wavelength side, particularly in the first standing wave concentrates 72% (Table 1 ).

  (4) The magnitude (absolute value) of the error wave that appears can be statistically estimated using two parameters: the accuracy of measurement during Masaya measurement and the orbital wavelength to be managed (Table 4).

  (5) Whether or not the crossing method can be used can be theoretically determined by comparing the track state of the section to be repaired with the error wave absolute value estimated in (4).

  By utilizing the above functions and characteristics, the following effects can be obtained by using Masaya measurement data only for the repair target section.

  A simple separation between trajectory error and linear separation.

  Wavelength selection for trajectory deviation can be performed.

  There is no individual difference in the movement amount calculation result

  Can flexibly handle fixed points and movement restrictions.

  Conversion to a Y correction value for MTT (Multiple Tamper) correction can be performed.

  As described above, the inventor of the present invention theoretically redefined the intersection method and obtained new knowledge. And the inventor came to this invention based on the new knowledge.

(First embodiment)
A measurement data standing wave component extraction system and a measurement data standing wave component extraction method according to the first embodiment of the present invention will be described with reference to FIG. The standing wave component extraction system 1 extracts standing wave components from measurement data, and includes an input unit 2, a processing unit 3, and an output unit 4 as shown in FIG. The input unit 2 receives measurement data. The processing unit 3 processes the measurement data input from the input unit 2. The output unit 4 outputs the standing wave component extracted by the processing unit 3.

  The standing wave component extraction system 1 includes a computer as hardware and software executed by the computer, and executes the standing wave component extraction method of the present invention. In the present embodiment, a notebook personal computer (notebook PC) is used as the computer. The input unit 2 is a computer input device, for example, a keyboard and a mouse. The processing unit 3 is a computer processing device, and includes a CPU that executes a computer program and a memory that stores the computer program and data. In the present embodiment, a notebook PC display and printer are used as the output unit 4. The computer constituting the standing wave component extraction system 1 is not limited to a notebook PC, and may be, for example, a computer mounted on a multiple type tamper (MTT).

  The measurement data input to the input unit 2 is data measured (sampled) at a predetermined space interval or time interval (measurement interval). The quantity to be measured only needs to be quantitatively measurable and is not limited to the positive arrow value on the railroad track, and may be a physical quantity such as voltage or current in an electric circuit, for example. If the number of measurement data is n, the measurement data is represented by an n-dimensional vector. Let the n-dimensional vector be x.

The processing unit 3 executes a matrix calculation step and an extraction step. In the matrix calculation step, an n-dimensional eigenspace matrix S _i (i is an integer of 1 to n) is calculated. The (i, k) component S _{i, (j, k)} of the eigenspace matrix S _i is calculated by the following equation. This mathematical formula is the mathematical formula 31 already described.

S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1))

In the extraction step, the processing unit 3 calculates the product of the eigenspace matrix S _i and the vector x and extracts the i-th long standing wave component S _i x from the measurement data. The wavelength Λ _i of the _i -th long standing wave component is Λ _i = 2p (n + 1) / i. This mathematical formula is the mathematical formula 35 already described.

  According to the measurement data standing wave component extraction system and the measurement data standing wave component extraction method according to the first embodiment, n measurement data from the first long standing wave component to the n th long standing wave component are measured. A standing wave component can be extracted. Since the extracted standing wave component is a standing wave, it has an advantageous feature compared to the discrete Fourier transform in which there is no phase difference. The measurement data to be extracted is not limited to data for curve correction in railways.

(Second Embodiment)
A method for standing wave decomposition of measurement data according to the second embodiment of the present invention will be described. In this standing wave decomposition method, using the output of the standing wave component extraction system 1 of the first embodiment, n pieces of measurement data are decomposed into a sum of n standing wave components. In this standing wave decomposition method, the measurement data x is the sum of i = 1 to i = n of the i-th long standing wave component S _i x. This sum is Equation 41 already described. The i-th long standing wave component S _i x is the output of the standing wave component extraction system 1, that is, obtained by the standing wave component extraction method of the measurement data of the present invention.

  According to the standing wave decomposition method of measurement data according to the second embodiment, it is possible to perform spectrum decomposition of measurement data without using discrete Fourier transform. The spectrally decomposed component has an advantageous characteristic compared to the discrete Fourier transform in which the direct current component (constant value component) is zero.

(Third embodiment)
A movement amount calculation system and a movement amount calculation method according to the third embodiment of the present invention will be described with reference to FIG. This moving amount calculation system calculates a moving amount from a design arrow and a measurement arrow at a plurality of measurement points at predetermined intervals on a railroad track. The movement amount calculation system has the same hardware as in the first embodiment, and the software is different. The same code | symbol is attached | subjected to the location equivalent to 1st Embodiment. In the following description, detailed description of portions equivalent to those of the first embodiment is omitted. As shown in FIG. 25, the movement amount calculation system includes an input unit 2, a processing unit 3, and an output unit 4. The input unit 2 receives input of a design positive arrow and a measurement positive arrow. The processing unit 3 processes the design positive arrow and the measurement positive arrow input from the input unit 2. The output unit 4 outputs the movement amount calculated by the processing unit 3.

  The movement amount calculation system has a computer as hardware and software executed by the computer, and executes the movement amount calculation method of the present invention.

  The design positive arrow is a positive arrow value determined at each measurement point, and is represented by an n-dimensional vector d. The measurement positive arrow is a positive arrow value measured at each measurement point, and is represented by an n-dimensional vector m. The movement amount is a correction value for correcting the measurement positive arrow to the design positive arrow, and is represented by an n-dimensional vector g.

The processing unit 3 executes an intersection method matrix calculation step and a movement amount calculation step. The processing unit 3 calculates an n-dimensional intersection method matrix T ⁻¹ in the intersection method matrix calculation step. In this intersection method matrix calculation step, the (i, j) component T ⁻¹ (i, j) of the intersection method matrix T ⁻¹ is T ⁻¹ (i, j) = when 1 ≦ i ≦ j ≦ n. 2 / (n + 1) × i (n−j + 1), and when 1 ≦ j <i ≦ n, calculated as T ⁻¹ (i, j) = 2 / (n + 1) × j (n−i + 1). The This mathematical formula is the mathematical formula 23 already described.

In the movement amount calculating step, the processing unit 3 calculates the movement amount by g = T ⁻¹ (dm). This mathematical formula is the mathematical formula 24 already described.

  Here, the proof of the regularity of the inverse matrix component expressed by Equation 23 will be described.

Definition 1: _Let T _{n be} the following difference matrix of n × n. Also, | T _n | represents a determinant of T _n .

Lemma 2: The following holds for | T _n |.

Proof of Lemma 2: the value of the determinant of T n _{+ 1,} if attention in the leftmost column of T n _{+ 1,} can be calculated as follows by sweeping-out method.

  Obviously, the following holds.

  The general term of the recurrence formula is obtained as follows. (End of proof)

Theorem 3: _Let the inverse matrix of T _n be written as T _n ⁻¹ = (a _ij ). At this time, the following holds.

  (1) When 1 ≦ i ≦ j ≦ n

  (2) When 1 ≦ j <i ≦ n

  Proof of Theorem 3: Here, only the case of (1) above is obtained. In the case of (2), it can be proved exactly as in the case of (1).

For the inverse matrix of T _n, it is necessary to calculate the cofactor. Assume 1 ≦ i <j ≦ n. At this time, considering the i, j cofactor Δij of T _n , this has the same form as the determinant of the following matrix. This matrix is an (n−1) × (n−1) matrix obtained by removing the i-th row and the j-th row.

The upper solid line portion is the removed i-th row, and the right solid line is the j-th column. The left side from the i-th line and the left side from the j-th line remain unchanged. Therefore, T _i−j is a difference matrix of (n−i) × (n−i). Similarly, T _n−j is a difference matrix of (n−j) × (n−j). _{Xj-i, which} is the middle part, is packed one up, but not left. That is, X _j−i is a triangular matrix of (j−i) × (j−i).

That is, X _j-i has a shape in which T _j-i is shifted upward by one line. This is a natural result of removing the i-th row. Note that the determinant of the triangular matrix is a multiplication of diagonal components.

Assuming that the inverse matrix T _n ⁻¹ = (a _ij ), the component a _ij is obtained as follows. (End of proof)

Conclusion 4: T _n ⁻¹ is a biaxial symmetric matrix. That is, the following holds.

a _ij = a _ji

a _{i, n−j + 1} = a _{j, n−i + 1}

T _n ⁻¹ is a point symmetric matrix having two symmetry axes, and has the following regularity by multiplying the entire component by (n + 1) / 2.

(1) When 1 ≦ i ≦ j ≦ n, a _ij = i (n−j + 1)

(2) When 1 ≦ j <i ≦ n, a _ij = j (n−i + 1) (end of system proof)

  According to the movement amount calculation system and the movement amount calculation method according to the third embodiment, since the inverse matrix is calculated without using the conventional inverse matrix calculation method such as the sweep-out method, the calculation is possible even if the number of measurement points increases. Is easy.

(Fourth embodiment)
A moving amount standing wave component extraction system and a moving amount standing wave component extraction method according to a fourth embodiment of the present invention will be described with reference to FIG. This standing wave component extraction system extracts a standing wave component of a moving amount from a design arrow and a measurement arrow at a plurality of measurement points at predetermined intervals on a railway track. The standing wave component extraction system has the same hardware as that of the first embodiment, and the software is different. The same code | symbol is attached | subjected to the location equivalent to 1st Embodiment. In the following description, detailed description of portions equivalent to those of the first embodiment is omitted. As shown in FIG. 25, the moving amount standing wave component extraction system includes an input unit 2, a processing unit 3, and an output unit 4. The input unit 2 receives input of a design positive arrow and a measurement positive arrow. The processing unit 3 processes the design positive arrow and the measurement positive arrow input from the input unit 2. The output unit 4 outputs the standing wave component of the movement amount extracted by the processing unit 3.

  The movement amount standing wave component extraction system has a computer as hardware and software executed by the computer, and executes the movement amount standing wave component extraction method of the present invention.

  The design positive arrow is a positive arrow value determined at each measurement point, and is represented by an n-dimensional vector d. The measurement positive arrow is a positive arrow value measured at each measurement point, and is represented by an n-dimensional vector m. The movement amount is a correction value for correcting the measurement positive arrow to the design positive arrow, and is represented by an n-dimensional vector g.

The processing unit 3 executes a matrix and eigenvalue calculation step and an extraction step. In the matrix and eigenvalue calculation step, the processing unit 3 calculates an n-dimensional eigenspace matrix S _i (i is an integer of 1 to n) and its eigenvalue λ _i . In this matrix and eigenvalue calculation step, the (j, k) component S _{i, (j, k)} of the eigenspace matrix S _i is calculated by the following equation. This mathematical formula is the mathematical formula 31 already described.

S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1))

In the matrix and eigenvalue calculation step, the eigenvalue λ _i is calculated by the following equation. This mathematical formula is the mathematical formula 29 already described.

λ _i = 1 / (1-cos (iπ / (n + 1)))

In the extraction step, the processing unit 3 calculates the product λ _i S _i (dm) of the eigenspace matrix S _i , its eigenvalue λ _i, and the positive difference (dm), and the i-th long standing wave The component λ _i S _i m is extracted. This mathematical formula is the i-th term of the mathematical formula obtained by substituting the mathematical formula 32 into the mathematical formula 24 already described.

The wavelength Λ _i of the _i -th long standing wave component is Λ _i = 2p (n + 1) / i. This mathematical formula is the mathematical formula 35 already described.

  According to the moving amount standing wave component extraction system and the moving amount standing wave component extraction method according to the fourth embodiment, the moving amount standing wave component from the first long standing wave component to the n th long standing wave component is extracted. Can do. Since the extracted standing wave component is a standing wave, it has an advantageous feature compared to the discrete Fourier transform in which there is no phase difference.

(Fifth embodiment)
A moving amount standing wave decomposition method according to the fifth embodiment of the present invention will be described. In this moving amount standing wave decomposition method, n movement amounts are decomposed into a sum of n standing wave components using the output of the moving amount standing wave component extraction system of the fourth embodiment. In the moving wave standing wave decomposition method, the moving amount g is the sum of i = 1 to i = n of the i-th longest standing wave component λ _i S _i (dm). This sum is obtained by substituting Equation 32 into Equation 24 already described. The i-th long standing wave component λ _i S _i (dm) is the output of the moving amount standing wave component extraction system, that is, obtained by the moving amount standing wave component extraction method of the present invention.

  According to the moving amount standing wave decomposition method according to the fifth embodiment, the moving amount can be spectrally decomposed without using the discrete Fourier transform. The spectrally decomposed component has an advantageous characteristic compared to the discrete Fourier transform in which the direct current component (constant value component) is zero.

(Sixth embodiment)
A movement amount error reduction method according to the sixth embodiment of the present invention will be described. In this movement amount error reduction method, the movement amount standing wave decomposition method of the fifth embodiment is used to reduce errors included in the movement amount. In this movement amount error reduction method, the longest standing wave component (1 ≦ r <n) from the first to the rth is removed from the moving amount decomposed into the sum of the standing wave components.

  The fact that the error included in the movement amount is reduced by this movement amount error reduction method is shown in the mathematical formula 53 and FIG.

  In the error reduction method of the movement amount, the standing wave component (1 ≦ r <n) that is long from the first to the r-th is removed, and r is selected from an integer of 1 to 5, for example.

  As shown in Table 1, when r = 1, that is, when only the longest standing wave component (first standing wave) is removed, the error removal rate is as high as 72.2%. Increasing r further increases the throne removal rate. When r = 2, that is, when a standing wave component long from the first to the second is removed, the error removal rate is 86.2%. When r = 3, that is, when long standing wave components from the first to the third are removed, the error removal rate is 91.4%. When r = 4, that is, when long standing wave components from the first to the fourth are removed, the error removal rate is 93.9%. When r = 6, that is, when long standing wave components from the first to the sixth are removed, the error removal rate is 95.4%.

  According to the movement amount error reduction method according to the sixth embodiment, since the movement amount error can be reduced without using a filter, the change in the positive arrow shape of the relaxation curve by the filter does not occur.

  The present invention is not limited to the configuration of the above embodiment, and various modifications can be made without departing from the scope of the invention. For example, in the third to sixth embodiments, the railway may be a conventional line or a high-speed railway.

1 Standing wave component extraction system (travel amount calculation system, travel amount standing wave component extraction system)
2 Input unit 3 Processing unit 4 Output unit d Design Masaya g Movement amount m Measurement Masaya n Number of dimensions (number of measurement data)
_Si eigenspace matrix λ eigenvalue of the matrix Λ wavelength x measurement data

Claims (7)

A measurement data standing wave component extraction system for extracting standing wave components from measurement data,
An input unit for receiving measurement data;
A processing unit for processing measurement data input from the input unit;
An output unit for outputting the standing wave component extracted by the processing unit;
The measurement data is n pieces of data measured at a predetermined space interval or time interval, and is represented by an n-dimensional vector x.
The processing unit calculates a n-dimensional eigenspace matrix S _i (i is an integer of 1 to n);
An extraction step of calculating a product of the eigenspace matrix S _i and the vector x and extracting a standing wave component S _i x having the i-th long standing wave wavelength from the measurement data;
In the matrix calculation step, the (i, k) component S _{i, (j, k)} of the eigenspace matrix S _i is calculated by the following equation:
S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1))
The standing wave component extraction system for measurement data, wherein the standing wave wavelength Λ _i of the standing wave component having the i-th long standing wave wavelength is Λ _i = 2p (n + 1) / i. A measurement data standing wave decomposition method for decomposing the n measurement data into a sum of n standing wave components using the measurement data standing wave component extraction system according to claim 1,
The vector x representing the measurement data is the sum of the standing wave component S _i x having the i- _th longest standing wave wavelength from i = 1 to i = n, and the standing wave decomposition method of the measurement data, A movement amount calculation system for calculating a movement amount from a design arrow and a measurement arrow at a plurality of measurement points at a predetermined interval on a railway track,
An input unit that receives inputs of design Masaya and measurement Masaya;
A processing unit for processing the design positive arrow and the measurement positive arrow input from the input unit;
An output unit that outputs the movement amount calculated by the processing unit;
The design positive arrow is a positive arrow value determined at the plurality of measurement points, and is represented by an n-dimensional vector d.
The measurement positive arrow is a positive arrow value measured at the plurality of measurement points, and is represented by an n-dimensional vector m.
The amount of movement is a correction value for correcting the measurement positive arrow to a design positive arrow, and is represented by an n-dimensional vector g.
The processing unit includes an intersection method matrix calculation step of calculating an n-dimensional intersection method matrix T ⁻¹ ,
a movement amount calculating step of calculating a movement amount by g = T ⁻¹ (dm),
In the intersection method matrix calculation step, the (i, j) component T ⁻¹ (i, j) of the intersection method matrix T ⁻¹ is
When 1 ≦ i ≦ j ≦ n, T ⁻¹ (i, j) = 2 / (n + 1) × i (n−j + 1) is calculated,
When 1 ≦ j <i ≦ n, the movement amount calculation system is calculated by T ⁻¹ (i, j) = 2 / (n + 1) × j (n−i + 1). A moving amount standing wave component extraction system that extracts a standing wave component of a moving amount from a design positive arrow and a measuring positive arrow at a plurality of measurement points at predetermined intervals on a railway track,
An input unit that receives inputs of design Masaya and measurement Masaya;
A processing unit for processing the design positive arrow and the measurement positive arrow input from the input unit;
An output unit that outputs a standing wave component of the movement amount extracted by the processing unit;
The design positive arrow is a positive arrow value determined at the plurality of measurement points, and is represented by an n-dimensional vector d.
The measurement positive arrow is a positive arrow value measured at the plurality of measurement points, and is represented by an n-dimensional vector m.
The amount of movement is a correction value for correcting the measurement positive arrow to a design positive arrow, and is represented by an n-dimensional vector g.
The processing unit includes an n-dimensional eigenspace matrix S _i (i is an integer not less than 1 and not more than n), a matrix for calculating its eigenvalue λ _i , and an eigenvalue calculating step;
The product λ _i S _i (dm) of the eigenspace matrix S _i , its eigenvalue λ _i and the positive arrow difference (dm) is calculated, and the standing wave component λ _i S _i having the i-th longest standing wave wavelength is calculated. an extraction step for extracting m,
In the matrix and eigenvalue calculating step, the (j, k) component S _{i, (j, k)} of the eigenspace matrix S _i is calculated by the following equation:
S _{i, (j, k)} = (2 / (n + 1)) × sin (iπj / (n + 1)) × sin (iπk / (n + 1))
The eigenvalue λ _i is calculated by the following equation:
λ _i = 1 / (1-cos (iπ / (n + 1)))
The moving wave standing wave component extraction system, wherein the standing wave wavelength Λ _i of the standing wave component having the i-th long standing wave wavelength is Λ _i = 2p (n + 1) / i. A moving amount standing wave decomposition method for decomposing the n moving amounts into a sum of n standing wave components using the moving amount standing wave component extraction system according to claim 4,
The moving amount standing wave decomposition method according to claim 1, wherein the moving amount is a sum from i = 1 to i = n of the standing wave component λ _i S _i (dm) having the i-th longest standing wave wavelength . A moving amount error reduction method for reducing an error included in the moving amount using the moving wave standing wave decomposition method according to claim 5,
A moving amount error reduction method comprising: removing the standing wave component (1 ≦ r <n) having a long standing wave wavelength from the first to the r th from the moving amount decomposed into a sum of standing wave components.   The method of claim 6, wherein r is selected from an integer of 1 to 5.

Images