CN103591902B - A kind of wheel diameter of urban rail vehicle detecting device based on laser sensor and method - Google Patents

Abstract

The invention discloses a kind of wheel diameter of urban rail vehicle detecting device based on laser sensor and method.This device includes CPU and multiple laser sensor, and described laser sensor is all connected with CPU；The rail shift outward of detector segments, and guard rail is set inside the rail of this detector segments, tangent inside guard rail and wheel rim；Laser sensor is arranged between region and the guard rail that rail skew is vacated, and the probe of laser sensor is along the arrangement of rail direction and is respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement.The method uses multiple laser sensors, is arranged on below wheel by it according to certain geometrical relationship, and sensor detecting wheel simultaneously obtains sensing point, obtains initial diameter by least square fitting, and then initial diameter being averaged obtains wheel diameter.On line non contact measurement of the present invention has the advantage that speed is fast, precision is high, measurement diameter range is big.

Description

A kind of wheel diameter of urban rail vehicle detecting device based on laser sensor and method

Technical field

The present invention relates to railway wheel detection field, particularly a kind of wheel diameter of urban rail vehicle detecting device based on laser sensor and method.

Background technology

City rail vehicle there will be abrasion in various degree in the process run, and wheel safe operation can be produced impact by abrasion, and it is particularly critical wherein to wear away the wheel diameter change caused.During train main track runs, coaxially and with steering framing wheel footpath differ from all limited requirement, the difference excessive wheel that is easily caused in coaxial wheels footpath is to scratch, and wheel footpath excessive being also easy to of difference is caused flange wear or train abnormal vibrations by same wheel, therefore to the measurement of wheel diameter to safe train operation important in inhibiting.

Conventional arc radius measuring method includes slide calliper rule method and arc height chord length method, and wherein slide calliper rule method is applicable to the occasion that required precision is not high, measures the scope restriction by arc length, and slide calliper rule range is by the restriction of located lateral frame；And the operation of arc height chord length method is comparatively laborious, these two kinds of methods are generally used for workpiece is done the off-line measurement of static state.Chinese patent CN201159640Y (diameter measurement device of railway wheel, application number: 200820055350.8, the applying date: 2008-02-02) disclose a kind of arc height chord length method measurement radius of wheel device, repair method belongs to hand dipping and off-line is measured automatically, when needing regularly to send workshop to overhaul after wheels travel a period of time.This static off-line measurement adopts special measuring tool or omnipotent measurer manual detection, there is the shortcomings such as testing result error is big, poor accuracy, rework rate height, inefficiency, labor intensity are big.

Diameter or rail wheels geometric parameters are grown up by contactless on-line measurement wheel gradually, Chinese patent CN1899904A (detector for train wheel pair size online test method and device, application number: 200510035961.7 applyings date: 2005-07-20), the laser displacement sensor of certain distance is installed in the both sides of every one steel rail, sensor is measured obliquely from the bottom side of rail, thus recording wheel tread data, and the speed calculation moved based on train obtains diameter through two laser sensor chord lengths.The shortcoming of the method is, it is necessary to utilize train speed information simultaneously, it is impossible to the measurement of complete independently diameter, and utilizes single laser sensor record tread information, can cannot be accurately positioned diameter position due to the change of tread.Chinese patent CN101219672A (the wheel diameter non-contact type dynamic measurement method based on laser, application number: 200810056339.8 applyings date: 2008-01-16) adopt two laser displacement sensor direct irradiation wheel tread rolling surfaces, wheel diameter is measured by installing the geometry site of sensor, the shortcoming of the method is not solved by alignment issues for detection line, and it is similar to method of cutting sth. askew equally, it is impossible to wheel diameter is accurately described.To sum up, current noncontact wheel diameter measurement technology yet suffer from certainty of measurement not high, measure the shortcomings such as response speed is slow, engineering construction is difficult.

Summary of the invention

It is an object of the invention to provide a kind of high-precision wheel diameter of urban rail vehicle detecting device based on laser sensor and method, adopt non-contact measurement, detection speed is fast, it is big to measure scope.

The technical solution realizing the object of the invention is: a kind of wheel diameter of urban rail vehicle detecting device based on laser sensor, including CPU and multiple laser sensor, described laser sensor is all connected with CPU；The rail shift outward of detector segments, and guard rail is set inside the rail of this detector segments, tangent inside guard rail and wheel rim；Laser sensor is arranged between region and the guard rail that rail skew is vacated, and the probe of laser sensor is along the arrangement of rail direction and is respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement.

A kind of wheel diameter of urban rail vehicle detection method based on laser sensor, comprises the following steps:

1st step, is installed on each laser sensor rail and offsets the region vacated, makes the probe of each laser sensor along the arrangement of rail direction and be respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement, and laser sensor is designated as P_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then the coordinate of laser sensor is (x_i,y_i), each laser sensor probe is θ relative to the mounted angle of X-axis_i；

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

Compared with prior art, the present invention has the great advantage that (1) is based on laser detection system, by the algorithm of least square fitting, it is achieved to the online non-cpntact measurement of train wheel, certainty of measurement is high；(2) automatically obtained any many point coordinates of wheel by laser sensor, by corresponding data Processing Algorithm, it is thus achieved that institute's measuring car wheel diameter instantly, simple, convenient fast；(3) there is the advantage that detection speed is fast, measurement scope is big.

Accompanying drawing explanation

Fig. 1 is the postrun abrasion schematic diagram of wheel tread.

Fig. 2 is the present invention structural representation based on the wheel diameter of urban rail vehicle detecting device of laser sensor.

Fig. 3 is the schematic diagram of rail switching place in wheel diameter of urban rail vehicle detecting device of the present invention.

Fig. 4 is the distance Q size broken face schematic diagram with guard rail of rail of the present invention skew.

Fig. 5 is the wheel diameter detection schematic diagram that in embodiment 1, laser sensor circular arc normal is installed.

Fig. 6 is the relation of the measured value of each laser sensor t in time (ms) in embodiment 1.

Fig. 7 is a certain moment detection sequence point (X in embodiment 1_i,Y_i) and matching after circle.

Fig. 8 is whole diameters that in embodiment 1, all effective measured data values matchings are obtained.

Fig. 9 is 20 diameter acquired results schematic diagrams of repeated measure in embodiment 1.

Figure 10 is the vertically arranged wheel diameter detection schematic diagram of laser sensor circular arc in embodiment 2.

Figure 11 is the relation of the measured value of each laser sensor t in time (ms) in embodiment 2.

Figure 12 is a certain moment detection sequence point (X in embodiment 2_i,Y_i) and matching after circle.

Figure 13 is whole diameters that in embodiment 2, all effective measured data values matchings are obtained.

Figure 14 is 20 diameter acquired results schematic diagrams of repeated measure in embodiment 2.

Figure 15 is the wheel diameter detection schematic diagram that in embodiment 3, laser sensor straight incline is installed.

Figure 16 is the relation of the measured value of each laser sensor t in time (ms) in embodiment 3.

Figure 17 is a certain moment detection sequence point (X in embodiment 3_i,Y_i) and matching after circle.

Figure 18 is whole diameters that in embodiment 3, all effective measured data values matchings are obtained.

Figure 19 is 20 diameter acquired results schematic diagrams of repeated measure in embodiment 3.

Figure 20 is the vertically arranged wheel diameter detection schematic diagram of laser sensor straight line in embodiment 4.

Figure 21 is the relation of the measured value of each laser sensor t in time (ms) in embodiment 4.

Figure 22 is a certain moment detection sequence point (X in embodiment 4_i,Y_i) and matching after circle.

Figure 23 is whole diameters that in embodiment 4, all effective measured data values matchings are obtained.

Figure 24 is 20 diameter acquired results schematic diagrams of repeated measure in embodiment 4.

Figure 25 is the wheel diameter detection schematic diagram of laser sensor designated mounting in embodiment 5.

Figure 26 is the relation of the measured value of each laser sensor t in time (ms) in embodiment 5.

Figure 27 is a certain moment detection sequence point (X in embodiment 5_i,Y_i) and matching after circle.

Figure 28 is whole diameters that in embodiment 5, all effective measured data values matchings are obtained.

Figure 29 is 20 diameter acquired results schematic diagrams of repeated measure in embodiment 5.

Detailed description of the invention

Below in conjunction with drawings and the specific embodiments, the present invention is described in further detail.

Fig. 1 indicates the tread profile after certain wheel runs and tread profile when just putting into operation, can be seen that distance wheel rim side place 70mm is for abrasion concentration place, this place is measurement diameter position conventional in engineering, and wheel diameter often controls between 770～840mm, therefore laser sensor sensing point is chosen for the wheel circumference at this place.

The present invention is based on the wheel diameter of urban rail vehicle detecting device of laser sensor, and including CPU and multiple laser sensor, described laser sensor is all connected with CPU；The rail shift outward of detector segments, and guard rail is set inside the rail of this detector segments, tangent inside guard rail and wheel rim；Laser sensor is arranged between region and the guard rail that rail skew is vacated, and the probe of laser sensor is along the arrangement of rail direction and is respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement.

As shown in Figure 2, in detector segments by outer for rail 6 inclined, vacate certain area, laser sensor probe 3 is arranged on below the measurement point of wheel 1, guard rail 5 is set inside wheel rim to prevent taking turns in S or axial float from causing derailing, laser sensor probe 3 is fixed by clamp of sensor 4, it is possible to adjust position and the inclination angle of laser sensor probe 3, the corresponding test point that the laser beam 2 that each laser sensor probe 3 sends can be detected simultaneously by wheel.

As it is shown on figure 3, switching place of rail shift outward is arc, train is conducive to enter and exit detecting area.Fig. 4 describes the concrete size Q of rail shift outward, and for wheel tread and 60 rails, Q controls between 50～65mm so that track centerline is without departing from the outer rim of wheel.Guard rail exceeds the size P of wheel rim, controls between 30～50mm.The distance of the wheel circumference distance wheel rim side carrying out diameter measurement is 70mm.

Due to wheel to be measured and track Long Term Contact, smooth surface roughness is low, therefore relates to utilize laser scanning testing head that the metal curved surface that direct reflection is very strong is carried out profile measurement, and this measurand is a difficult point in current topography measurement field.Zhang Liang etc. analyze the existing several laser feeler measurement capability to metal surface, show that cone light polarization holography probe and oblique fire formula triangle probe are relatively suitable for measuring metal curved surface (Zhang Liang, Fei Zhigen, Guo Junjie. laser scanning testing head is to metal curved surface measurement Research, lathe and hydraulic pressure, the 39th volume the 9th phase: in May, 2011).Therefore the laser sensor that the present invention relates to, it is preferable that cone light polarization holography probe and oblique fire formula triangle probe, the quantity of laser sensor be 3～10 and the probe of all laser sensors be fixed on below wheel by clamp of sensor.

Use the above-mentioned method carrying out wheel diameter detection based on the wheel diameter of urban rail vehicle detecting device of laser sensor, comprise the following steps:

1st step, is installed on each laser sensor rail and offsets the region vacated, makes the probe of each laser sensor along the arrangement of rail direction and be respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement, and laser sensor is designated as P_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then the coordinate of laser sensor is (x_i,y_i), each laser sensor probe is θ relative to the mounted angle of X-axis_i；

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；Adopting method of least square to be fitted circle, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\Σ}({X_i}^2+{Y_i}^2)+{\mathrm{a\ΣX}}_i+{\mathrm{b\ΣY}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C=n\ΣX_i^2-\ΣX_i\ΣX_i\\ D=n\ΣX_i{Y}_i-\ΣX_i\ΣY_i\\ E=n\ΣX_i^3+n\ΣX_i{Y_i}^2-\Σ(X_i^2+{Y_i}^2)\ΣX_i\\ G=n\Σ{Y_i}^2-\ΣY_i\ΣY_i\\ H=n\ΣX_i^2{Y}_i+n\ΣY_i^3-\Σ({X_i}^2+{Y_i}^2)\ΣY_i\end{array}\right\},i=1,\mathrm{2...}n$

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

Below in conjunction with specific embodiment, introduce sensor respectively and adopt that circular arc normal, circular arc are vertical, straight incline, straight line are vertical and the wheel diameter of urban rail vehicle detecting device of designated mounting mode and method, the present invention is described in further detail.

Embodiment 1

The present embodiment is wheel diameter of urban rail vehicle detecting device and the method for sensor circular arc normal installation.

As it is shown in figure 5, the probe of n laser sensor along the arrangement of rail direction and to be distributed on chord length be L, radius be on the circular arc of R, each laser sensor probe is measured along the normal direction of circular arc, the center of circle of detection beam alignment installation circular arc.In Fig. 5, c represents the center of circle of wheel rolling circle, and w represents sensor P₁、P₂、P₃、P₅The center of circle of place circular arc, c₁Representative sensor P₁Project the point on wheel rolling circle, c₂Representative sensor P₂Project the point on wheel rolling circle, c₃Representative sensor P₃Project the point on wheel rolling circle.

The external factor affecting laser triangulation sensor accuracy mainly includes measured surface inclination, lustrous surface, roughness, color and scanning speed etc..Sensor adopts arc method wire type to install so that laser sensor probe can be directed at tested surface simultaneously, effectively inhibits tested surface to tilt the error brought；Be conducive to the preferred process of installation parameter is analyzed simultaneously, reduce analysis difficulty.

The installation parameter of laser sensor meets the following conditions: the number of laser sensor is n and 3≤n≤10, chord length installed by laser sensor is L and n × 30mm≤L≤1800mm, and point to the vertical dimension of track of installing along first laser sensor in rail direction is | y₁| and | y₁| >=100mm, laser sensor install arc radius be R and

The wheel diameter of urban rail vehicle detecting device installed based on above sensor circular arc normal carries out the method that detects, comprises the following steps:

1st step, n laser sensor is installed on rail and offsets the region vacated, the probe of laser sensor is along the arrangement of rail direction and to be distributed on chord length be L, radius be on the circular arc of R, each laser sensor probe is measured along the normal direction of circular arc, detection beam alignment installs the center of circle of circular arc, and laser sensor is designated as P respectively_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then each laser sensor probe is relative to the mounted angle θ of X-axis_iDetermined by following formula:

${\mathrm{\θ}}_i=\frac{\mathrm{\π}}{2}+(i-1-\frac{n-1}{2})\×\mathrm{\θ},i=1,2...n$

Wherein θ is chord length L and the radius R arc angle determined,

Coordinate (the x of laser sensor_i,y_i) determined by following formula:

$\left\{\begin{array}{c}{x}_i=\frac{L}{2}-R\×{\mathrm{cos\θ}}_i\\ y_i=y_1-{\mathrm{Rsin\θ}}_i+\sqrt{R^2-\frac{L^2}{4}}\end{array}\right.,i=1,2...n$

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；Being fitted circle, adopt method of least square, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\Σ}({X_i}^2+{Y_i}^2)+{\mathrm{a\ΣX}}_i+{\mathrm{b\ΣY}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C=n\ΣX_i^2-\ΣX_i\ΣX_i\\ D=n\ΣX_i{Y}_i-\ΣX_i\ΣY_i\\ E=n\ΣX_i^3+n\ΣX_i{Y_i}^2-\Σ(X_i^2+{Y_i}^2)\ΣX_i\\ G=n\Σ{Y_i}^2-\ΣY_i\ΣY_i\\ H=n\ΣX_i^2{Y}_i+n\ΣY_i^3-\Σ({X_i}^2+{Y_i}^2)\ΣY_i\end{array}\right\},i=1,\mathrm{2...}n$

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

According to engineering reality and the analysis to measurement error, 4 parameters are preferably as follows (unit: mm):

$\left\{\begin{array}{c}{y}_1=-100\\ n=6\\ L=800\\ R=2500\end{array}\right.$

Thus obtaining the coordinate (x of each sensor_i,y_i) (unit: mm) and mounted angle θ_i(unit: °):

θ_i=[80.793184.475988.158691.841495.524199.2069]

$\left\{\begin{array}{c}{x}_i=\[\begin{array}{cccccc}0& 159.3373& 319.6685& 480.3315& 640.6627& 800\end{array}\]\\ y_i=\[\begin{array}{cccccc}-100& -120.5968& -130.9165& -130.9165& -120.5968& -100\end{array}\]\end{array}\right.$

If the sampling period of laser sensor is 1kHz, measure random error 0.1mm, by the tested vechicle wheel measurement data that computer simulation generation diameter is 800 as shown in Figure 6, measurement data export diameter according to following steps:

(1.1) all laser sensor output point sequence S are collected_i, and select data when 6 sensors effectively detect.Certain moment wheel is through out-of-date virtual value:

S_i=[309.2010188.2974137.8491138.1852189.8197312.4783]

(1.2) for the output valve S of sensor_iAnd point coordinates (x is installed_i,y_i), inclination angle theta_i, push away to obtain point coordinates (X on camber line_i,Y_i)；Fig. 7 depicts S in (1.1)_iMoment wheel is through the sequence of points (X in central point moment_i,Y_i) and matching after circle:

$\left\{\begin{array}{c}{X}_i=\[\begin{array}{cccccc}49.5169& 177.4748& 324.1039& 475.8919& 622.4061& 750.0586\end{array}\]\\ Y_i=\[\begin{array}{cccccc}205.4936& 66.9409& 7.0473& 7.1771& 68.1715& 208.1128\end{array}\]\end{array}\right.$

(1.3) by sequence of points (X_i,Y_i) to obtain, according to least square fitting circle, the wheel diameter that this moment surveys be 800.44mm.Fig. 8 is all effective corresponding wheel diameter values measured in the moment, and under present embodiment, the wheel that diameter is 800 being measured effective measure dot number is 137 points, and in effective range, the data in all moment calculate that to obtain diameter be D be 799.6mm～800.4mm.

(1.4) data in Fig. 8 are averaged, obtain the output diameter D of one-shot measurement_final=799.93mm.Analogue measurement 20 times, obtains the measurement result shown in Fig. 9, and from this measurement result, this embodiment can realize the high-acruracy survey of wheel diameter, and measurement error is < 0.1mm when being left out alignment error.

Embodiment 2

The present embodiment is the vertically arranged wheel diameter of urban rail vehicle detecting device of sensor circular arc and method.

As shown in Figure 10, the probe of n laser sensor is along the arrangement of rail direction and to be distributed on chord length be L, radius be on the circular arc of R, and the detection beam orthogonal rail of each laser sensor is upwards.In Figure 10, c represents the center of circle of wheel rolling circle, and w represents sensor P₁、P₂、P₃The center of circle of place circular arc, c₁Representative sensor P₁Project the point on wheel rolling circle, c₂Representative sensor P₂Project the point on wheel rolling circle, c₃Representative sensor P₃Project the point on wheel rolling circle.

The installation parameter of laser sensor meets the following conditions: the number of laser sensor is n and 3≤n≤10, chord length installed by laser sensor is L and n × 30mm≤L≤1800mm, and point to the vertical dimension of track of installing along first laser sensor in rail direction is | y₁| and | y₁| >=100mm, laser sensor install arc radius be R and

The method carrying out based on the vertically arranged wheel diameter of urban rail vehicle detecting device of above sensor circular arc detecting, comprises the following steps:

1st step, n laser sensor is installed on rail and offsets the region vacated, the probe of laser sensor is along the arrangement of rail direction and to be distributed on chord length be L, radius be on the circular arc of R, and upwards, laser sensor is designated as P to the detection beam orthogonal rail of each laser sensor respectively_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then each laser sensor probe is relative to the mounted angle θ of X-axis_iIt is 90 °, the coordinate (x of laser sensor_i,y_i) determined by following formula:

$\left\{\begin{array}{c}{x}_i=\frac{L}{2}-R\&times;{\mathrm{cos\&beta;}}_i\\ y_i=y_1-{\mathrm{Rsin\&beta;}}_i+\sqrt{R^2-\frac{L^2}{4}}\end{array}\right.,i=1,2...n$

Wherein,

In formula, β is chord length L and the radius R arc angle determined,

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；Adopting method of least square to be fitted circle, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\&Sigma;}({X_i}^2+{Y_i}^2)+{\mathrm{a\&Sigma;X}}_i+{\mathrm{b\&Sigma;Y}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C=n\&Sigma;X_i^2-\&Sigma;X_i\&Sigma;X_i\\ D=n\&Sigma;X_i{Y}_i-\&Sigma;X_i\&Sigma;Y_i\\ E=n\&Sigma;X_i^3+n\&Sigma;X_i{Y_i}^2-\&Sigma;(X_i^2+{Y_i}^2)\&Sigma;X_i\\ G=n\&Sigma;{Y_i}^2-\&Sigma;Y_i\&Sigma;Y_i\\ H=n\&Sigma;X_i^2{Y}_i+n\&Sigma;Y_i^3-\&Sigma;({X_i}^2+{Y_i}^2)\&Sigma;Y_i\end{array}\right\},i=1,\mathrm{2...}n$

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

According to engineering reality and the analysis to measurement error, 4 parameters are preferably as follows:

$\left\{\begin{array}{c}{y}_1=-100\\ n=6\\ L=600\\ R=3000\end{array}\right.$

Thus obtaining the coordinate (x of each sensor_i,y_i) (unit: mm):

$\left\{\begin{array}{c}{x}_i=\&lsqb;\begin{array}{cccccc}0& 119.93& 239.97& 360.03& 480.07& 600\end{array}\&rsqb;\\ y_i=\&lsqb;\begin{array}{cccccc}-100& -105.76& -108.65& -108.65& -105.76& -100\end{array}\&rsqb;\end{array}\right.$

If the sampling period of laser sensor is 1kHz, measure random error 0.1mm, by the tested vechicle wheel measurement data that computer simulation generation diameter is 800 as shown in Figure 11, measurement data export diameter according to following steps:

(2.1) all laser sensor output point sequence S are collected_i, and select data when 6 sensors effectively detect.Certain moment wheel is through out-of-date virtual value:

S_i=[234.0412151.6957118.6378118.5963153.0630236.7570]

(2.2) for the output valve S of sensor_iAnd point coordinates (x is installed_i,y_i), inclination angle theta_i, push away to obtain point coordinates (X on camber line_i,Y_i)；Figure 12 depicts S in (2.1)_iSequence of points (the X determined_i,Y_i) and this moment matching after circle:

$\left\{\begin{array}{c}{X}_i=\&lsqb;\begin{array}{cccccc}0& 119.8072& 239.9036& 360.0964& 480.1928& 600.0000\end{array}\&rsqb;\\ Y_i=\&lsqb;\begin{array}{cccccc}134.2381& 42.2560& 4.2916& 4.7120& 43.1836& 136.5666\end{array}\&rsqb;\end{array}\right.$

(2.3) by sequence of points (X_i,Y_i) according to least square fitting circle, to obtain the wheel diameter in this moment be 798.782mm.Figure 13 is all effective corresponding wheel diameter values measured in the moment, and in effective range, the data in all moment calculate that to obtain diameter be D be 798.5mm～801.5mm.

(2.4) data in Figure 13 are averaged, obtain the output diameter D of one-shot measurement_final=799.89mm.And analogue measurement 20 times, obtain the measurement result shown in Figure 14, from this measurement result, this embodiment can realize the high-acruracy survey of wheel diameter, and measurement error is left out < 0.25mm when alignment error.

Embodiment 3

The present embodiment is that linear sensor tilts the wheel diameter of urban rail vehicle detecting device and method installed.

As shown in figure 15, the probe of laser sensor is along the arrangement of rail direction and is distributed on the horizontal line that length is E, arranges each laser sensor probe mounted angle θ relative to X-axis according to the coordinate of laser sensor_i, enable each laser sensor probe to measure along different directions and detection light beam arrive wheel simultaneously.In Figure 15, c represents the center of circle of wheel rolling circle, c₂Representative sensor P₂Project the point on wheel rolling circle, c₃Representative sensor P₃Project the point on wheel rolling circle.

The installation parameter of laser sensor meets the following conditions: the number of laser sensor is n and 3≤n≤10, horizontal line length installed by laser sensor is E and n × 30mm≤E≤1800mm, and point to the vertical dimension of track of installing along first laser sensor in rail direction is | y₁| and | y₁|≥100mm。

Tilt, based on above sensor vertical, the method that the wheel diameter of urban rail vehicle detecting device installed carries out detecting, comprise the following steps:

1st step, each laser sensor is installed on rail and offsets the region vacated, the probe of laser sensor is along the arrangement of rail direction and is distributed on the horizontal line that length is E, each laser sensor probe is measured along different directions and detection light beam can arrive wheel simultaneously, and each laser sensor is designated as P respectively_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And it is perpendicular to rail upwards for Y-axis, the then coordinate (x of laser sensor_i,y_i) determined by following formula:

$\left\{\begin{array}{c}{x}_i=(i-1)\&times;E/(n-1)\\ y_i=y_1\end{array}\right.,i=1,2...n$

Coordinate according to laser sensor arranges each laser sensor probe mounted angle θ relative to X-axis_iSo that all detection light beams can arrive wheel simultaneously；

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；Adopting method of least square to be fitted circle, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\&Sigma;}({X_i}^2+{Y_i}^2)+{\mathrm{a\&Sigma;X}}_i+{\mathrm{b\&Sigma;Y}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C=n\&Sigma;X_i^2-\&Sigma;X_i\&Sigma;X_i\\ D=n\&Sigma;X_i{Y}_i-\&Sigma;X_i\&Sigma;Y_i\\ E=n\&Sigma;X_i^3+n\&Sigma;X_i{Y_i}^2-\&Sigma;(X_i^2+{Y_i}^2)\&Sigma;X_i\\ G=n\&Sigma;{Y_i}^2-\&Sigma;Y_i\&Sigma;Y_i\\ H=n\&Sigma;X_i^2{Y}_i+n\&Sigma;Y_i^3-\&Sigma;({X_i}^2+{Y_i}^2)\&Sigma;Y_i\end{array}\right\},i=1,\mathrm{2...}n$

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

According to engineering reality and the analysis to measurement error, 4 parameters are preferably as follows:

$\left\{\begin{array}{c}{y}_1=-100\\ n=6\\ E=1000\end{array}\right.$

Thus obtaining the coordinate (x of each sensor_i,y_i) (unit: mm) and mounted angle θ_i(unit: °):

θ_i=[45609090120135]

$\left\{\begin{array}{c}{x}_i=\&lsqb;\begin{array}{cccccc}0& 200& 400& 600& 800& 1000\end{array}\&rsqb;\\ y_i=\&lsqb;\begin{array}{cccccc}-100& -100& -100& -100& -100& -100\end{array}\&rsqb;\end{array}\right.$

If the sampling period of laser sensor is 1kHz, measure random error 0.1mm, by the tested vechicle wheel measurement data that computer simulation generation diameter is 800 as shown in figure 16, measurement data export diameter according to following steps:

(3.1) all laser sensor output point sequence S are collected_i, and select data when 6 sensors effectively detect.A certain moment wheel is through out-of-date virtual value:

S_i=[383.1241244.0411153.031199.8297140.6533243.1484]

(3.2) for the output valve S of sensor_iAnd point coordinates (x is installed_i,y_i), inclination angle theta_i, push away to obtain point coordinates (X on camber line_i,Y_i)；Figure 17 depicts S_iSequence of points (the X determined_i,Y_i) and this moment matching after circle:

$\left\{\begin{array}{c}{X}_i=\&lsqb;\begin{array}{cccccc}170.8398& 269.9187& 400.0000& 600.0000& 677.3753& 727.9330\end{array}\&rsqb;\\ Y_i=\&lsqb;\begin{array}{cccccc}170.9372& 111.2730& 52.7153& -0.1772& 22.1058& 71.9640\end{array}\&rsqb;\end{array}\right.$

(3.3) by sequence of points (X_i,Y_i) after according to least square fitting circle, to obtain the diameter in this moment be 798.853mm.Figure 18 is all effective corresponding wheel diameter values measured in the moment, and in effective range, the data in all moment calculate that to obtain diameter be D be 797mm～803mm.

(3.4) data in Figure 18 are averaged, obtain the output diameter D of one-shot measurement_final=800.17mm.Analogue measurement 20 times, obtains the measurement result shown in accompanying drawing 19, and from this measurement result, this embodiment can realize the high-acruracy survey of wheel diameter, and measurement error is left out < 0.35mm when alignment error.

Embodiment 4

The present embodiment is the vertically arranged wheel diameter of urban rail vehicle detecting device of linear sensor and method.

As shown in figure 20, the probe of n laser sensor is along the arrangement of rail direction and is distributed on the horizontal line that length is F, and the detection beam orthogonal rail of each laser sensor is upwards.In Figure 20, c represents the center of circle of wheel rolling circle, c₁Representative sensor P₁Project the point on wheel rolling circle, c₂Representative sensor P₂Project the point on wheel rolling circle, c₃Representative sensor P₃Project the point on wheel rolling circle.

The installation parameter of laser sensor meets the following conditions: the number of laser sensor is n and 3≤n≤10, horizontal line length installed by laser sensor is F and n × 30mm≤F≤D, D is wheel diameter, and point to the vertical dimension of track of installing along first laser sensor in rail direction is | y₁| and | y₁|≥100mm。

The method carrying out based on the vertically arranged wheel diameter of urban rail vehicle detecting device of above linear sensor detecting, comprises the following steps:

1st step, each laser sensor is installed on rail and offsets the region vacated, the probe of laser sensor is along the arrangement of rail direction and is distributed on the horizontal line that length is F, and upwards, each laser sensor is designated as P to the detection beam orthogonal rail of each laser sensor respectively_i, it being followed successively by 1,2, Kn along rail direction i, n is the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then each laser sensor probe is relative to the mounted angle θ of X-axis_iIt is 90 °, the coordinate (x of laser sensor_i,y_i) determined by following formula:

$\left\{\begin{array}{c}{x}_i=(i-1)\&times;F/(n-1)\\ y_i=y_1\end{array}\right.,i=1,2...n$

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2, Kn；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ₁) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；Adopting method of least square to be fitted circle, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\&Sigma;}({X_i}^2+{Y_i}^2)+{\mathrm{a\&Sigma;X}}_i+{\mathrm{b\&Sigma;Y}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C=n\&Sigma;X_i^2-\&Sigma;X_i\&Sigma;X_i\\ D=n\&Sigma;X_i{Y}_i-\&Sigma;X_i\&Sigma;Y_i\\ E=n\&Sigma;X_i^3+n\&Sigma;X_i{Y_i}^2-\&Sigma;(X_i^2+{Y_i}^2)\&Sigma;X_i\\ G=n\&Sigma;{Y_i}^2-\&Sigma;Y_i\&Sigma;Y_i\\ H=n\&Sigma;X_i^2{Y}_i+n\&Sigma;Y_i^3-\&Sigma;({X_i}^2+{Y_i}^2)\&Sigma;Y_i\end{array}\right\},i=1,\mathrm{2...}n$

The multiple valid data groups collected are fitted obtaining a series of wheel diameter, are averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final by the 6th step_final。

According to engineering reality and the analysis to measurement error, 3 parameters are preferably as follows:

$\left\{\begin{array}{c}{y}_1=-100\\ n=6\\ F=600\end{array}\right.$

Thus obtaining the coordinate (x of each sensor_i,y_i) (unit: mm):

$\left\{\begin{array}{c}{x}_i=\&lsqb;\begin{array}{cccccc}0& 120& 240& 360& 480& 600\end{array}\&rsqb;\\ y_i=\&lsqb;\begin{array}{cccccc}-100& -100& -100& -100& -100& -100\end{array}\&rsqb;\end{array}\right.$

If the sampling period of laser sensor is 1kHz, measure random error 0.1mm, by the tested vechicle wheel measurement data that computer simulation generation diameter is 800 as shown in figure 21, measurement data export diameter according to following steps:

(4.1) all laser sensor output point sequence S are collected_i, and select data when 6 sensors effectively detect.A certain moment wheel is through out-of-date virtual value:

S_i=[233.9894142.0952104.0036104.3825143.4801236.6142]

(4.2) for the output valve S of sensor_iAnd point coordinates (x is installed_i,y_i), inclination angle theta_i, push away to obtain point coordinates (X on camber line_i,Y_i)；Figure 22 depicts S in (4.1)_iSequence of points (the X determined_i,Y_i) and this moment matching after circle:

$\left\{\begin{array}{c}{X}_i=\&lsqb;\begin{array}{cccccc}0& 120& 240& 360& 480& 600\end{array}\&rsqb;\\ Y_i=\&lsqb;\begin{array}{cccccc}133.9894& 42.0952& 4.0036& 4.3825& 43.4801& 136.6142\end{array}\&rsqb;\end{array}\right.$

(4.3) by sequence of points (X_i,Y_i) to obtain, according to least square fitting circle, the wheel diameter that this moment surveys be 799.354mm.Figure 23 is all effective corresponding wheel diameter values measured in the moment, and in effective range, the data in all moment calculate that to obtain diameter be D be 798.5mm～801.5mm.

(4.4) data in Figure 23 are averaged, obtain the output diameter D of one-shot measurement_final=800.13mm.Analogue measurement 20 times, obtains the measurement result shown in Figure 24, and from this measurement result, this embodiment can realize the high-acruracy survey of wheel diameter, and measurement error is < 0.3mm when being left out alignment error.

Embodiment 5

The present embodiment is wheel diameter of urban rail vehicle detecting device and the method for sensor designated mounting.

As shown in figure 25, wherein c represents the center of circle of wheel rolling circle, c₁Representative sensor P₁Project the point on wheel rolling circle, c₂Representative sensor P₂Project the point on wheel rolling circle, c₃Representative sensor P₃Project the point on wheel rolling circle, c₄Representative sensor P₄Project the point on wheel rolling circle, c₅Representative sensor P₅Project the point on wheel rolling circle.Set the coordinate (x of each sensor_i,y_i) and mounted angle θ_i:

θ_i=[45 ° 60 ° 90 ° 120 ° 135 °]

$\left\{\begin{array}{c}{x}_i=\&lsqb;\begin{array}{ccccc}0& 300& 500& 700& 1000\end{array}\&rsqb;\\ y_i=\&lsqb;\begin{array}{ccccc}-100& -100& -100& -100& -100\end{array}\&rsqb;\end{array}\right.$

If the sampling period of laser sensor is 1kHz, measure random error 0.1mm, by the tested vechicle wheel measurement data that computer simulation generation diameter is 800 as shown in figure 26, measurement data export diameter according to following steps:

(5.1) all laser sensor output point sequence S are collected_i, and select data when 5 sensors effectively detect.A certain moment wheel is through out-of-date virtual value:

S_i=[306.6595140.094499.8658140.5186307.6415]

(5.2) for the output valve S of sensor_iAnd point coordinates (x is installed_i,y_i), inclination angle theta_i, push away to obtain point coordinates (X on camber line_i,Y_i)；Figure 27 depicts S in (5.1)_iSequence of points (the X determined_i,Y_i) and matching after circle:

$\left\{\begin{array}{c}{X}_i=\&lsqb;\begin{array}{ccccc}216.8410& 370.0472& 500.0000& 629.7407& 782.4646\end{array}\&rsqb;\\ Y_i=\&lsqb;\begin{array}{ccccc}116.8410& 21.3253& -0.1342& 21.6926& 117.5354\&rsqb;\end{array}\&rsqb;\end{array}\right.$

(5.3) by sequence of points (X_i,Y_i) to obtain, according to least square fitting circle, the wheel diameter that this moment surveys be 801.243mm.Figure 28 is all effective corresponding wheel diameter values measured in the moment, and under present embodiment, the wheel that diameter is 800 being measured effective measure dot number is 137 points, and in effective range, the data calculating in all moment obtains diameter D is 797mm～803mm.

(5.4) data in Figure 28 are averaged, obtain the output diameter D of one-shot measurement_final=800.15mm.Analogue measurement 20 times, obtains the measurement result shown in Figure 29, and from this measurement result, this embodiment can realize the high-acruracy survey of wheel diameter, and measurement error is < 0.4mm when being left out alignment error.

In sum, the present invention is based on laser detection system, by the algorithm of least square fitting, it is achieved to the online non-cpntact measurement of train wheel, certainty of measurement is high；Automatically obtained any many point coordinates of wheel by laser sensor, by corresponding data Processing Algorithm, it is thus achieved that institute's measuring car wheel diameter instantly, simple, convenient fast；And there is the advantage that detection speed is fast, measurement scope is big.

Claims (2)

1. the wheel diameter of urban rail vehicle detection method based on laser sensor, it is characterised in that including CPU and multiple laser sensor, described laser sensor is all connected with CPU；The rail shift outward of detector segments, and guard rail is set inside the rail of this detector segments, tangent inside guard rail and wheel rim；Laser sensor is arranged between region and the guard rail that rail skew is vacated, and the probe of laser sensor is along the arrangement of rail direction and is respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement, specifically comprise the following steps that

1st step, is installed on each laser sensor rail and offsets the region vacated, makes the probe of each laser sensor along the arrangement of rail direction and be respectively positioned on below wheel, and all laser sensors are coplanar with the wheel circumference carrying out diameter measurement, and laser sensor is designated as P_i, it is followed successively by 1 along rail direction i, 2 ... n, n are the number of laser sensor；

2nd step, sets up two-dimensional coordinate system in the plane at the wheel circumference carrying out diameter measurement: be X-axis along rail direction, through first laser sensor P₁And being perpendicular to rail upwards for Y-axis, then the coordinate of laser sensor is (x_i,y_i), each laser sensor probe is θ relative to the mounted angle of X-axis_i；

3rd step, gathers the output valve of all laser sensors, and selects the valid data group { S simultaneously having n sensor output value_i, S_iFor i-th sensor P_iOutput valve, i=1,2 ... n；

4th step, according to sensor P_iOutput valve S_i, coordinate figure (x_i,y_i), mounted angle θ_iDetermine respective sensor P on wheel_iMeasurement point coordinates (X_i,Y_i):

(X_i,Y_i)=(x_i,y_i)+(S_i×cosθ_i,S_i×sinθ_i) i=1,2 ... n

5th step, measures point coordinates (X according on wheel n_i,Y_i) it is fitted circle, obtain the wheel diameter D of this measurement position；

6th step, repeats to be fitted obtaining a series of wheel diameter by the multiple valid data groups collected, is averaged by a series of wheel diameters obtained, obtain the wheel diameter D that this measurement position is final_final。

2. the wheel diameter of urban rail vehicle detection method based on laser sensor according to claim 1, it is characterised in that measure point coordinates (X according on wheel n described in the 5th step_i,Y_i) it is fitted circle, adopt method of least square, formula is as follows:

$D=\sqrt{a^2+b^2+4\frac{\mathrm{\&Sigma;}(X_i^2+Y_i^2)+{\mathrm{a\&Sigma;X}}_i+{\mathrm{b\&Sigma;Y}}_i}{n}},i=1,2...n$

Wherein, a is the center of circle abscissa x after matching₀-2 times i.e. a=-2x₀, b is the center of circle vertical coordinate y after matching₀-2 times i.e. b=-2y₀, and

$a=\frac{HD-EG}{CG-D^2}$

$b=\frac{HC-ED}{D^2-GC}$

Wherein C, D, E, G, H are intermediate parameters, as follows respectively:

$\left.\begin{array}{c}C={\mathrm{n\&Sigma;X}}_i^2-{\mathrm{\&Sigma;X}}_i{\mathrm{\&Sigma;X}}_i\\ D={\mathrm{n\&Sigma;X}}_i{Y}_i-{\mathrm{\&Sigma;X}}_i{\mathrm{\&Sigma;Y}}_i\\ E={\mathrm{n\&Sigma;X}}_i^3+{\mathrm{n\&Sigma;X}}_i{Y}_i^2-\mathrm{\&Sigma;}(X_i^2+Y_i^2){\mathrm{\&Sigma;X}}_i\\ G={\mathrm{n\&Sigma;Y}}_i^2-{\mathrm{\&Sigma;Y}}_i{\mathrm{\&Sigma;Y}}_i\\ H={\mathrm{n\&Sigma;X}}_i^2{Y}_i+{\mathrm{n\&Sigma;Y}}_i^3-\mathrm{\&Sigma;}(X_i^2+Y_i^2){\mathrm{\&Sigma;Y}}_i\end{array}\right\},i=1,2...n.$