Method and apparatus to determine structural parameters of a railway track

Abstract

Method for determining structural parameters of a railway track having a sensor array, includes measuring vertical and/or lateral irregularities of the rail with the sensor array along the rail, thereby providing signals corresponding to geometrical irregularities at different distance from the wheel-load. A model is provided describing the rail's deflection shape, which is dependent on structural parameters of the rail and on the loads on the rail. The model is stored in a processor where geometrical irregularities are compared under different load influence for generating a measured deflection shape. At least one theoretical rail deflection shape is generated using the model by varying structural parameters and the load in the model. At least one of theoretical deflection shapes is compared with the measured deflection shape for each point of the rail, and the structural parameters of that theoretical deflection shape which best matches the measured deflection shape are determined.

Claims

1. A method of monitoring a railway track to ensure safe travel, the method comprising: determining structural parameters of a rail using a measurement system comprising: a railway car configured as a measurement vehicle, the measurement vehicle including a sensor array mounted on one side of a wheel and configured to measure at least vertical and/or lateral irregularities of the rail under influence of different loads at a plurality of points along the rail, and at different distances from the wheel of the measurement vehicle, the sensor array further configured to provide signals corresponding to said irregularities, the sensor array being positioned in the railway car adjacent to a contact point between the wheel and the rail; and a processor configured to process the signals from the sensor array; with the measurement vehicle moving on the rail, measuring at least the vertical and/or lateral irregularities of said rail with the sensor array along the rail, thereby providing the signals corresponding to the irregularities at the different distances from the wheel, and under the influence of the different loads; providing a model describing a deflection shape of the rail, wherein the deflection shape is dependent on the structural parameters of the rail and on loads on the rail, said model being stored in the processor; and comparing said irregularities under the influence of the different loads in order to separate deflection due to wheel loads from non-loaded geometrical irregularities, thus generating a measured deflection shape; generating at least one theoretical rail deflection shape using the model by varying the structural parameters and the load in the model; and comparing the at least one theoretical rail deflection shape with said measured deflection shape for each point along the rail; determining the structural parameters of the at least one theoretical rail deflection shape which best matches said measured deflection shape.

2. The method according to claim 1, wherein the loads on the rail are the wheel loads and temperature induced longitudinal loads in the rail.

3. The method according to claim 2 wherein the structural parameters are selected from track modulus, stress free temperature of the rail, bending moment of the rail, track damping, and critical ground speed.

4. The method according to claim 1 wherein the structural parameters are selected from track modulus, stress free temperature of the rail, bending moment of the rail, track damping, and critical ground speed.

5. The method according to claim 4, wherein the stress free temperature of the rail is obtained by measuring the rail temperature with a thermometer and comparing with a stress free state.

6. The method according to claim 1, wherein the sensor array for measuring said irregularities comprises any of lasers, laser and camera, laser-doppler for velocity, and radar.

7. The method according to claim 1, wherein the model describing the deflection shape of the rail is selected from a linear or nonlinear beam model with foundation or a linear or nonlinear FEM-model.

8. The method according to claim 1, wherein the step of comparing the least one of said theoretical deflection shapes includes use of any of Kalman filters, adaptive filters or sequential Monte Carlo methods.

9. The method according to claim 1, wherein in order to compare deflection shapes before and after the wheel or a bogie for critical speed determination, an additional sensor array is mounted on an other side of the wheel or the bogie in a longitudinal direction, and measuring said irregularities, thus making it possible to detect non-symmetric deflection shape when comparing the deflection shapes before and after the wheel.

10. An apparatus for determining structural parameters of a rail track having a measurement system comprising: a processor; a sensor array positioned in a railway car adjacent to a contact point between a wheel of the railway car and a rail, the sensor array configured to measure at least vertical and/or lateral irregularities of the rail under influence of different loads at a plurality of points along the rail, and at different distances from the wheel of the railway car, and to provide signals corresponding to said rail irregularities to the processor, wherein the processor is configured to process the signals from the sensor array, the processor being provided with a model describing a deflection shape of the rail, wherein the deflection shape is dependent on the structural parameters of the rail and on loads on the rail, said model being stored in the processor, the processor being further configured to perform the steps of: comparing said irregularities under influence of different loads in order to separate deflection due to wheel loads from non-loaded geometrical irregularities, thus generating a measured deflection shape; generating at least one theoretical rail deflection shape using the model by varying the structural parameters and the load in the model; comparing the at least one theoretical rail deflection shape with said measured deflection shape for each point along the rail; and determining the structural parameters of the at least one theoretical rail deflection shape which best matches said measured deflection shape.

11. The apparatus according to claim 10, wherein the sensor array comprises sensors on both sides of the wheel in a longitudinal direction.

12. The apparatus according to claim 11, further comprising: a thermometer for measuring the rail temperature.

13. The apparatus according to claim 10, further comprising: a thermometer for measuring the rail temperature.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows an example of system including a sensor array of laser/cameras fitted under a railway car body;

(2) FIG. 2 is an illustration of geometrical irregularities (vertical alignment) (dashed dotted line), deflection due to wheel load Q (solid line) and combination of the two (dashed line);

(3) FIG. 3 is an illustration of measured geometrical irregularities (vertical alignment) from four different sensors at different distance from the loaded wheel;

(4) FIG. 4 is an illustration of theoretical deflection shape and measured deflection shape at one position;

(5) FIG. 5 shows the longitudinal force resultant in vertical direction from geometrical irregularities, s(x) (dashed-dotted line), and deflection, w(x) due to a load in position 0 (solid line) at a temperature 30 C. above SFT; and

(6) FIG. 6 is a flow chart of the method according to the invention.

DESCRIPTION OF THE INVENTION

(7) The invention relates to a method for determining structural parameters of a railway track comprising a sensor array configured to measure the deflection shape of the rail, which is under influence of one or several forces, together with geometrical irregularities. Preferably the sensors are mounted under the car body of a railway vehicle which makes continuous measurements along the rail possible. The deflection shape can also be created from a model where different parameters influence the deflection shape.

(8) The invention thus is in a first aspect a method for determining structural parameters of a rail track using a measurement system comprising a sensor array configured to measure at least the vertical and/or lateral irregularities of a rail under influence of different loads at a plurality of points along the rail, and at different distances from the load/wheel. The sensor array also provides signals corresponding to said rail irregularities, the sensor array being positioned in a railway car adjacent to the contact point between a wheel and a rail; and a processor configured to process the signals from the sensor array. The method further comprises measuring at least the vertical and/or lateral irregularities of said rail with the sensor array along the rail, thereby providing signals corresponding to the geometrical irregularities at different distance from the wheel-load, i.e. under different load influence. A model is provided describing the deflection shape of a rail, wherein the deflection shape is dependent on structural parameters of the rail and on the loads on the rail, said model being stored in the processor. In the processor said geometrical irregularities are compared under different load influence in order to separate deflection due to the wheel load(s) from non-loaded geometrical irregularities, thus generating a measured deflection shape. At least one theoretical rail deflection shape is generated using the model by varying the structural parameters and the load in the model. At least one of said theoretical deflection shapes is compared with said measured deflection shape for each point of the rail, and the structural parameters of that theoretical deflection shape which best matches said measured deflection shape are determined.

(9) The loads are wheel loads and temperature induced longitudinal loads in the rail.

(10) The structural parameters that are determined are selected from track modulus, stress free temperature of the rails, bending moment of the rails, track damping, critical speed of the ground.

(11) The model describing the deflection shape of a rail, is preferably selected from linear or nonlinear beam model with foundation or linear or nonlinear FEM-model.

(12) In another aspect the invention provides an apparatus for determining structural parameters of a rail track having a measurement system comprising a sensor array configured to measure at least the vertical and/or lateral irregularities of a rail under influence of different loads at a plurality of points along the rail, and at different distances from the load/wheel, and to provide signals corresponding to said rail irregularities. The sensor array is positioned in a railway car adjacent to the contact point between a wheel and a rail; and a processor configured to process the signals from the sensor array. The processor being configured to perform the method according to any of claims 1-8, by being setup to run a program. In a preferred embodiment the sensor array comprises sensors on both sides of the wheels in the longitudinal direction.

(13) A static model usable in the invention can include the following parameters: Wheel-rail force Rail bending stiffness with extra influence of fastener clip force (proportional to the 4.sup.th derivative of the deflection shape) Temperature induced force of the rail, when the rail temperature differs from the SFT (proportional to the 2.sup.nd derivative of the deflection shape) Track modulus (proportional to the deflection shape)

(14) A dynamic model usable in the invention can in addition include: Vehicle speed Inertial terms Track and fastener damping (proportional to the 1.sup.st derivative of deflection by time) Critical speed of the track

(15) By comparing the deflection shape measured by the sensors with the deflection shape calculated from the model, varying the parameters of the model to best possible fit, the structural parameters of the track can be determined.

(16) Other parameters can also be included in the model e.g. a more detailed model of track modulus to separate the pad stiffness of the fastener from ballast and substructure modulus.

(17) As different structural parameters are proportional to different orders of the deflection shape derivatives, the correlation between measured deflection shape and correlating model deflection shape will not be under-determined and thus solvable.

(18) The sensors could be of different kinds, point lasers, line-lasers with camera and optical triangulation, grid laser with camera and optical triangulation, laser-Doppler measuring velocity (which can be integrated to displacement), distance-radar etc.

(19) Sensors can be placed in boxes/fixtures where the relative motion of the fixtures compared to the rail are measured with accelerometers and gyros.

(20) One example of a measurement setup generally designated 10 is shown in FIG. 1 (direction of movement is the x direction). The setup comprises a sensor array 12 comprising a set-up of lasers and cameras arranged in front of a pair of wheels 14. The four vertical arrows 16 illustrate line lasers lighting a cross-section of the rail 18 and the broken lines 20 illustrate the camera view of the laser line. The sensor array 12 is coupled to a processor 22 configured to process signals from the sensors according to a program for determining structural parameters of the rail.

(21) In the figure, only one rail 18 is shown, but of course both rails can be monitored at the same time with two arrays of sensors.

(22) This described measurement setup is used to determine structural parameters originating from a static model (as described previously). As the deflection shape is symmetric around the wheel (or bogie) only one side of the wheel/bogie has to be monitored.

(23) A non-contact temperature sensor such as an infrared thermometer is used to measure the rail temperature. The sensor or thermometer can then be directed towards the rail web or foot. The temperature sensor (T.sub.rail) is used to relate determined stress state (P.sub.long) of the rail to SFT (T.sub.sft), with denotations from Eq. 2.

(24) The main influencing forces creating a deflection shape, origin from the wheels. One minor influencing force taken into consideration by this invention is the vertical (z) and lateral (y) resultant coming from longitudinal (x) forces when the rail temperature differs from the SFT. This resultant force is proportional to the second derivative of the geometrical deviation with respect to x, the longitudinal elongation of the rail.

(25) One static model that can be used to describe the deflection shape is the Euler-Bernoulli beam on a Winkler foundation with the addition of longitudinal force from a rail temperature different from the SFT shown in Eq. 1.

(26) $\begin{array}{cc}\mathrm{EI}\frac{d^{4}w\left(x\right)}{{\mathrm{dx}}^4}+P_{\mathrm{long}}\frac{d^{2}w\left(x\right)}{{\mathrm{dx}}^2}+w\left(x\right)=Q\left(x\right)\left(x_0\right)-P_{\mathrm{long}}\frac{d^{2}s\left(x\right)}{{\mathrm{dx}}^2}& \left(1\right)\\ P_{\mathrm{long}}=\mathrm{EA}\left(T_{\mathrm{rail}}-T_{\mathrm{sft}}\right)& \left(2\right)\end{array}$ w(x) denotes deflection shape (either vertical or lateral). s(x) denotes vertical or lateral geometrical deviations. s(x) of equation 1, is considered as non-loaded geometry. denotes increase of rail bending stiffness due to fasteners connecting the rail to the sleepers. E is the rail-steel elastic modulus. I is the area moment of inertia of the rail cross section (different in vertical and lateral direction). P.sub.long is the longitudinal force (temperature dependent stress state) of the rail apparent when the rail temperature, T.sub.rail, differs from the stress free temperature T.sub.sft (shown in Eq. 2). denotes the track modulus. Q denotes wheel-rail force. (x.sub.0) denotes the dirac-function making sure that Q is only acting at the wheel position x.sub.0. is the coefficient of heat expansion of rail steel. A is the cross section area of the rail.

(27) E, I, A and are considered as material constants although I and A will change slightly if the rail is worn.

(28) More advanced models can also be used using e.g. more advanced beam theory, finite elements, discrete supports of the rail with masses, springs and dampers at sleeper positions, vehicle speed, critical speed of the soil and an appropriate vehicle model to represent the wheel force to mention a few possible expansions.

(29) There exist closed form solutions for equation 1 if the unknown structural parameters , , and P.sub.long are regarded as constants yielding a linear differential equation. Also Laplace/Fourier techniques can be used to solve the equation and compare/correlate with measured deflection.

(30) With varying unknown terms, Eq. 1 becomes non-linear and non-linear numerical solutions are needed.

(31) The displacement sensor array will measure a combination of the non-loaded geometrical irregularities (vertical and/or lateral alignment) and the structural deflection shape as is illustrated with circles in FIG. 2 for a measurement array with sensors at distances [0.5 1.5 2.5 3.5] meters from the wheel. The dashed-dotted line represents the unloaded vertical alignment, which are the geometrical irregularities of the rail before the train is coming. The dashed line represents the loaded vertical alignment when the wheel load is at position x=6 m. The solid line is the deflection shape due to the load at position x=6 m and naturally the difference between the unloaded and loaded vertical alignment.

(32) An illustration of readings from all four sensors is shown in FIG. 3. One specific position in the track, corresponding to the position of sensor 1 at x=9.5 m in FIG. 2, is illustrated for all four sensors with square (sensor 1), diamond (sensor 2), star (sensor 3) and circle (sensor 4). It is clear that the load influence from the wheel will increase as sensors are placed closer to the wheel. The readings from this position, forming a deflection shape measurement, is illustrated in FIG. 4 where also a theoretical deflection shape from the model in Eq. 1 is shown as a solid line. As a first try, all four sensor readings are adjusted so that sensor 1 has a zero-reading.

(33) As the measurement vehicle moves, each position along the track will have one measurement sample per sensor as illustrated in FIGS. 3-4. Each position along the track will thus have measurements with different influence from the load. This is the basis for separating geometrical irregularities s(x) from deflection w(x).

(34) If the rail temperature differs from the SFT, longitudinal forces will be present in the rail. Both geometrical irregularities and deflection shape due to a wheel load will cause the longitudinal rail force to have vertical and lateral resultants. The resultants will be proportional to the second derivative of irregularities/deflection shape as shown in Eq. 1. FIG. 5 shows an example of resultant vertical forces with the basis from FIG. 2 and a rail temperature 30 C. above current SFT.

(35) Naturally these extra forces will cause minor extra deflection, which will make it possible to determine the structural parameter of the rail stress state and SFT.

(36) Another measurement setup could be used to determine critical speed of the track. When a train approaches critical speed, the deflection shape under a wheel or bogie will not be symmetric. In order to compare deflection shape before and after the wheel or bogie for critical speed determination, a similar sensor array is mounted on the other side of the wheel/bogie, thus making it possible to detect non-symmetric deflection shape when comparing deflection shape before and after the wheel. The actual speed of the measurement vehicle is measured at the same time to relate the non-symmetric deflection shape with critical speed. Structural parameters are in this case determined with the help of a dynamic model (as described above).

(37) One possible dynamic model is described by the differential equation (3)

(38) $\begin{array}{cc}\mathrm{EI}\frac{{}^{4}w\left(x,t\right)}{x^4}+m\frac{{}^{2}w\left(x,t\right)}{t^2}+c\frac{w\left(x,t\right)}{t}+\mathrm{kw}\left(x,t\right)=Q\left(x-\mathrm{vt}\right)& \left(3\right)\end{array}$
w(x,t) is now dependent of both position (x) and time (t). Variables, not already mention previously, are: m: equivalent mass of the track per meter. c: track damping v: speed of vehicle

(39) Solving for critical speed with prior art will give the relation:

(40) $\begin{array}{cc}{v}_{\mathrm{cr}}^2=\frac{2}{m}\sqrt{\mathrm{kEI}}& \left(4\right)\end{array}$

(41) A couple of passages at different speeds are often required in order to see increasing asymmetry in the deflection shape and from that determine unknown terms for Eq. 3 and 4.

(42) Calibration

(43) The geometrical irregularities measured with the sensor array will differ from time to time for various reasons. The wheel-rail contact position will vary and cause small variation of height between the car body where sensors are mounted and the rail. The suspension system of wheel and bogie of the car will also cause variations in height. To some extent, an inertial measurement platform close to the sensor array with accelerometers and gyros can compensate for this. An accurate calibration is important where it is ascertained that the whole sensor array has the same height reference. This can e.g. be done with a steel rural with defined straightness combined with an accurate inclinometer.

(44) Remaining variations can be included as an error state in the numerical solution.

(45) Numerical Solution/Implementation of the Invention

(46) There exists a range of methods to make parameter adjustments of a model in order to fit the model to measurements and in that way estimate the parameters. One such method is the Kalman filter which exists in several different variants depending on model complexity and linearity/non-linearity. Examples of other methods are the family of adaptive filters and sequential Monte Carlo methods.

(47) A Kalman filter works on discrete data. The standard denotation of a Kalman-filter state is x. In the following n denotes discretization of the position along the track and x denotes the Kalman filter state.

(48) A Kalman filter that determines the structural parameters of Eq. 1-2 is described in Eq. 5-14. The basic equation (Eq. 5) describes the position-update (x(n+1)) solving for next position along the track and the measurement update (y(n)). In this implementation, the position update is linear with the transfer matrix F, and process noise v(n). The measurement update is non-linear for some states and linear for others. This is described with the non-linear function h( ) which is dependent on the filter states x(n) and the measurement noise e(n). Underline denotes a vector and a line above the symbol denotes a matrix.
x(n+1)=Fx(n)+v(n) y(n)=h(x(n),e(n)) (5)

(49) The covariance of v(n) is denoted Q.sub.cov(n) and the covariance of e(n) is denoted R.sub.cov(n). The state vector x(n) contains the following states:
x(n).sup.T=W(n) S(n) T.sub.SFT(n) (n) (n) Q(n)(6)

(50) The vectors W(n) and S(n) contain necessary derivatives at samples spanning the whole distance of the sensor array according to:
W(n).sup.T=w(n) w(n) w.sup.(4)(n) w(n1) . . . w(nm) w(nm) w.sup.(4)(nm)(7)
S(n).sup.T=[s(n) s(n) s(n1) s(n1) . . . s(nm) s(nm)](8)

(51) Searched parameter vectors T.sub.sft, , , Q also span the interval between n and n-m., as showed for modulus in Eq. 9.
(n).sup.T=[(n) (n1) (n2) . . . (nm+1) (nm)](9)

(52) In one example of the invention n is the position at 3.5 m to the right of the wheel, at the first sensor position, and n-m is the position at 0.5 m to the right of the wheel at the fourth (last) sensor. These four sensor positions are indicated in FIGS. 1 and 2. If the sampling frequency f.sub.s is 10 samples per meter, n=0.1 m, the number of states x, will be 279 (W-93 states, S-62 states, T.sub.sft-31 states, -31 states, -31 states, Q-31 states).

(53) The position update matrix F will consist of a number of sub-matrixes shown in Eq. 10.

(54) $\begin{array}{cc}\stackrel{\_}{F}=\left[\begin{array}{cccccc}\stackrel{\_}{F_w}& 0& 0& 0& 0& 0\\ 0& \stackrel{\_}{F_S}& 0& 0& 0& 0\\ 0& 0& \stackrel{\_}{F_T}& 0& 0& 0\\ 0& 0& 0& \stackrel{\_}{F_{}}& 0& 0\\ 0& 0& 0& 0& \stackrel{\_}{F_{}}& 0\\ 0& 0& 0& 0& 0& \stackrel{\_}{F_Q}\end{array}\right]& \left(10\right)\end{array}$

(55) F.sub.W and F.sub.Y will be built on the symmetrical numerical double derivative:
w(n)=f.sub.s.sup.2(w(n+1)2w(n)+w(n1) (11)

(56) Taking the position update (from x(n) to x(n+1)) and expanding the derivatives leads to:

(57) $\begin{array}{cc}\left(\begin{array}{c}{w}^{\left(4\right)}\left(n+1\right)=w^{\left(4\right)}\left(n+1\right)+v_{\left(w4\right)}\left(n\right)\\ w^{}\left(n+1\right)=\frac{w^{}\left(n+2\right)}{2}+\frac{w^{}\left(n\right)}{2}-\frac{w^{\left(4\right)}\left(n+1\right)n^2}{2}+v\left(w_2\right)\left(n\right)\\ w\left(n+1\right)=\begin{array}{c}\frac{w\left(n+2\right)}{2}+\frac{w\left(n\right)}{2}-\frac{w^{}\left(n+2\right)n^2}{4}-\\ \frac{w^{}\left(n\right)n^2}{4}+\frac{w^{\left(4\right)}\left(n+1\right)n^4}{4}{v}_{\left(w0\right)}\left(n\right)\end{array}\end{array}\right)\mathrm{and}& \left(12\right)\\ \left(\begin{array}{c}{s}^{}\left(n+1\right)=s^{}\left(n\right)+v_{s2}\left(n\right)\\ s\left(n+1\right)=\frac{s\left(n-1\right)}{2}+\frac{s\left(n+1\right)}{2}-\frac{s^{}\left(n\right)n^2}{2}+v_{s0}\left(n\right)\end{array}\right)& \left(13\right)\end{array}$
which fill basis for F.sub.w and F.sub.s.

(58) The remaining sub-matrixes of F, can be unity matrixes as an adequate position-update is the last estimate.

(59) The measurement update includes both linear and non-linear updates. The linear measurement update is naturally measurements from the sensors as in Equation 14. With n=0.1 m, 10 samples shift corresponds to one meter, and as seen from the equation, sensors are placed at one meters distance from each other in this example.

(60) $\begin{array}{cc}\left(\begin{array}{c}{y}_1\left(n\right)=w\left(n\right)+s\left(n\right)+e_1\left(n\right)\\ y_2\left(n\right)=w\left(n-10\right)+s\left(n-10\right)+e_2\left(n\right)\\ y_3\left(n\right)=w\left(n-20\right)+s\left(n-20\right)+e_3\left(n\right)\\ y_4\left(n\right)=w\left(n-30\right)+s\left(n-30\right)+e_4\left(n\right)\end{array}\right)& \left(14\right)\end{array}$

(61) As can be seen from the equation and also from FIGS. 2-4, sensor measurements are a combination of deflection (w(n)) and geometrical irregularities (s(n)) together with a measurement error (e(n)).

(62) Measurements of the wheel force can easily be incorporated as a measurement update if sensors for measuring wheel force are available.

(63) Finally, the non-linear measurement updates tie the model from Eq. 1 with estimated values of deflection and geometrical irregularities. This can be executed for each position spanned by the sensor array, or in order to reduce calculation complexity, only at the positions of the sensors as outlined in Eq. 15.

(64) $\begin{array}{cc}\left(\begin{array}{c}\begin{array}{c}{y}_5\left(n\right)=\left(n\right){\mathrm{EIw}}^{\left(4\right)}\left(n\right)+\mathrm{EA}\left(T_{\mathrm{sft}}\left(n\right)-T\left(n\right)\right)w^{}\left(n\right)+\\ \left(n\right)w\left(n\right)+\mathrm{EA}\left(T_{\mathrm{sft}}\left(n\right)-T\left(n\right)\right)s^{}\left(n\right)+e_5\left(n\right)\end{array}\\ \begin{array}{c}\begin{array}{c}{y}_6\left(n\right)=\left(n-10\right){\mathrm{EIw}}^{\left(4\right)}\left(n-10\right)+\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-10\right)-T\left(n-10\right)\right)w^{}\left(n-10\right)+\\ \left(n-10\right)w\left(n-10\right)+\end{array}\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-10\right)-T\left(n-10\right)\right)s^{}\left(n-10\right)+e_6\left(n\right)\end{array}\\ \begin{array}{c}\begin{array}{c}{y}_7\left(n\right)=\left(n-20\right){\mathrm{EIw}}^{\left(4\right)}\left(n-20\right)+\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-20\right)-T\left(n-20\right)\right)w^{}\left(n-20\right)+\\ \left(n-20\right)w\left(n-20\right)+\end{array}\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-20\right)-T\left(n-20\right)\right)s^{}\left(n-20\right)+e_7\left(n\right)\end{array}\\ \begin{array}{c}\begin{array}{c}{y}_8\left(n\right)=\left(n-30\right){\mathrm{EIw}}^{\left(4\right)}\left(n-30\right)+\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-30\right)-T\left(n-30\right)\right)w^{}\left(n-30\right)+\\ \left(n-30\right)w\left(n-30\right)+\end{array}\\ \mathrm{EA}\left(T_{\mathrm{sft}}\left(n-30\right)-T\left(n-30\right)\right)s^{}\left(n-30\right)+e_8\left(n\right)\end{array}\end{array}\right)& \left(15\right)\end{array}$

(65) The measurement updates y.sub.5(n)-y.sub.8(n) are formed to equal to zero.

(66) To have the wheel-force influencing the measurement equation directly, a closed form solution of Eq. 1 could be implemented as a measurement update. Another alternative is to extend the equations with states to include also the wheel position.

(67) The non-linear measurement update is implemented with for example an extended Kalman filter or an unscented Kalman filter.

(68) From an initial estimate, the Kalman filter improves the estimate as position-updates and measurement updates progress. Normally some 10-20 meters are needed to separate deflection from geometrical irregularities and to give adequate determination of structural parameters.