Vehicle Parameter Identification through Particle Filter using Bridge ...

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ScienceDirect Procedia Engineering 188 (2017) 64 – 71

6th Asia Pacific Workshop on Structural Health Monitoring, 6th APWSHM

Vehicle Parameter Identification through Particle Filter using Bridge Responses and Estimated Profile Haoqi Wanga, Tomonori Nagayamaa,*, Di Sua a

The Univiersity of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan

Abstract The weight of vehicles driving over bridges needs to be evaluated because the vehicle load potentially causes problems such as fatigue and large vibration. Bridge Weigh-In-Motion systems to evaluate vehicle load by measuring strain response of bridges have been proposed. However, the installation of strain gauges at bridge members are often costly and time consuming, limiting practical applications. This paper investigates the identification of vehicle weight using bridge acceleration response data at different sensor locations through the numerical simulation of vehicle-bridge interaction system. The identified parameters are not limited to the vehicle weight; suspension stiffness and damping coefficients are also identified. A data assimilation technique known as particle filter based on the Bayesian theory is employed to identify the parameters. The time history of the profile is estimated using the same particle filter technique from the dynamic response of a probe vehicle equipped with sensors. The proposed method is shown to have robustness against noises for the mass parameter identification. ©2016 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of the 6th APWSHM. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 6th APWSHM Keywords: Particle Filter; Parameter Identification; Bridge Response; Profile Estimation; Vehicle-Bridge Interaction

1. Introduction Vehicle-induced vibration refers to the bridge vibration caused by dynamic forces generated by vehicles passing over a bridge. If it is not well controlled, this unpleasant bridge vibration will cause fatigue problems or even early damage to the bridge. Since this kind of vibration is directly caused by the passing vehicle, the dynamic properties of vehicles, which are reflected by a series of dynamic properties, have a great influence on vehicle-induced vibration. The knowledge of vehicle parameters, especially the mass, is of great importance in at least two ways. The first is that

* Corresponding author. Tel.: +81-03-5841-6144; fax: +81-03-5841-7454. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 6th APWSHM

doi:10.1016/j.proeng.2017.04.458

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the evaluation of real dynamic load is highly dependent on these parameters and valuable information can be provided to the bridge designers for the setting of load. The second is for the purpose of monitoring the vehicle weight on the bridge to see whether the bridge is suffering from overload vehicles that may shorten the working life of road pavement and bridge structure [1]. However, due to the variation of the vehicle-induced load in time and space and the limitation of available bridge response data, this is usually not an easy task. Therefore, how to obtain the vehicle weight as well as other vehicle parameters accurately and efficiently becomes a crucial issue for evaluating the bridge conditions [2]. This problem has drawn much attention in recent years [3-9]. However, it should be noted that the bridge profile is the main vibration source of the vehicle-bridge coupled system. Therefore, having a good knowledge about profile is of great importance. In this paper, a new method is proposed, where the bridge profile is first estimated using a sensorequipped probe car and then serves as the profile input in the vehicle parameter identification problem. Both the profile estimation and vehicle identification problems are solved using particle filter technique [10-12]. This paper consists of four parts. In section 2, the theory and background of particle filter is introduced. Some concepts are introduced in section 3, including the generation of profile, the methods to calculate vehicle and bridge response and how the vehicle-bridge interaction is considered in this paper. Section 4 shows the details about profile estimation, in which a sensor-equipped vehicle with known parameters is used. The estimated profile in section 4 is then used as the known input to identify vehicle parameters from bridge response. The conclusions are summarized in section 6. 2. Particle Filter In the particle filter technique, the system is usually expressed in state-space, where a system equation determines the evolution of the state, as shown in Eq. (1) xk 1 f k ( xk )  w(k ) (1) where xk stands for the state vector that represents the state of the system at time step k. fk represents the state transition function of the system. w(k) stands for the system error which follows a zero-mean Gaussian distribution. The observation equation shown in Eq. (2) links the system state and the observation data. (2) yk h k ( xk )  v(k ) in which yk is the measurement vector at time step k and hk is the measurement function representing the relation between state vector xk and measurement vector yk. v(k) is the corresponding observation error also following zeromean Gaussian distribution. In Bayesian theory, the objective of estimating the system state is to construct the posterior probability density function (PDF) of the state. This posterior PDF requires information of available observed data. Two stages, known as prediction and update, are necessary to construct PDF. These two procedures are shown in Eq. (3) and Eq. (4) respectively

p( xk | y1:k 1 )

³ p( x

p( xk | y1:k )

k

| xk 1 ) p( xk 1 | y1:k 1 )dxk 1

p( yk | xk ) p( xk | y1:k 1 ) p( yk | y1:k 1 )

(3) (4)

where p(xk|y1:k) is the conditional posterior PDF of state vector when measurement data up to time step k is known. p(xk|y1:k-1) is called prior PDF which means the measurement data is not available at the current step. The posterior PDF of step k-1 p(xk-1|y1:k-1) is assumed to be known. In the update equation, the observed data yk is introduced to modify the distribution of state from prior PDF to the objective posterior PDF. Particle filter technique uses a large number of particles to represent the state PDF following a monte-carlo method [10]. In this process, each particle is first passed through the system equation to obtain the prior PDF of next step, which is described in Eq. (3). The objective posterior PDF is calculated through a resampling procedure. When observed data become available, the likelihood of each particle is evaluated and normalised by Eq. (5). A new set of particles are then resampled according to the normalised likelihood. These new particles represent the posterior PDF.

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qi

p( yk | xk (i )) N

¦ p( y

k

(5)

| xk (i ))

i 1

In this paper, the particle filtering technique is used in both the profile estimation problem and the vehicle parameter identification problem. The details will be discussed in the following sections. 3. Vehicle Bridge Dynamics 3.1. Road Profile Simulation The bridge profile is the main excitation input of the vehicle passing and exciting the bridge [2]. For road profile simulations in this paper, the method based on power spectral density (PSD) proposed in ISO 8608 [13] is adopted, in which the road profile is expressed in terms of the sum of a series of harmonics as shown in Eq. (6) N

h( x )

¦

2G(ni )'n cos(2S ni x  Ii )

(6)

i

In Eq. (6), x is the distance along the road, h(x) is the road profile and Δn is the frequency spacing determined by the total length of the profile. Φi is the phase angle which follows a uniform distribution from 0 to 2π. G(ni) is the onesided PSD according to different road classes defined in ISO 8608. 3.2. Vehicle Model and Vehicle Response A half-car vehicle model, which represents the ordinary two-axle vehicle, is adopted as the model of moving vehicles on the bridge, with 4 degrees of freedom including vehicle body vertical movement, two vehicle tire movement and a vehicle body rotation, shown in Fig. 1.

Fig. 1. Half-car model.

The equation of motion of this half-car model is shown in Eq. (7), where M, C, and K are mass, damping and stiffness matrix respectively. P(t) is the excitation of vehicle system and U(t) is the response of the vehicle. The definition of these matrices can be found in Reference [14]. (7) MU (t )  CU (t )  KU (t ) P(t ) 3.3. Bridge Model and Bridge Response A simply supported beam is adopted as the beam model for analysis in this paper due to its simplicity and good accuracy in this problem. The dynamic equation of bridge is expressed in Eq. (8)

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w 2 y ( x, t ) wy ( x, t ) w 4 y ( x, t ) mb  cb  EI wt 2 wt wx 4

p ( x, t )

(8)

in which mb, cb are mass and viscous damping per length and EI is the flexural stiffness, y is the bridge displacement which relates to both time and space, and p is the moving dynamic load on the bridge. This partial differential equation is numerically solved by using modal decomposition analysis. The details of this method can be found in Reference [15]. 3.4. State-Space Expression This section describes the method of calculating system response from current step to the next step. This is necessary because all the responses are calculated in state-space in the particle filter method. This is achieved by adopting Newmark-beta method assuming average acceleration between two steps. The procedure is shown in Eq. (9) [16]. 1

uk uk uk

§ ( 't )2 · 't K ¸  Pk  CBC  KBK ¨M  C  2 4 © ¹ u u uk 1  k 1 k 't 2 u u uk 1  uk 1't  k 1 k ( 't ) 2 4

(9)

in which uk is the state displacement response of vehicle or bridge, Δt is the time difference between two steps and BC, BK are expressed in Eq. (10).

uk 1 't 2

BC

uk 1 

BK

u uk 1  uk 1't  k 1 ( 't )2 4

(10)

3.5. Vehicle Bridge Interaction When the vehicle is passing on the bridge, it will be excited from the bridge profile and starts to vibrate in vertical direction. The dynamic force on the bridge will give rise to the vibration of the bridge, which changes the initial input profile and affect the vehicle response. This coupling effect is known as the vehicle-bridge interaction [2]. In this paper, the vehicle-bridge coupling dynamic equation is solved through an iteration method, which is first proposed by Green and Cebon to consider interaction between heavy vehicles and bridges [17]. The process of this iteration is shown in the following figure.

Fig. 2. Vehicle-bridge interaction.

The initial bridge profile is first used to calculate the vehicle response, which is then used to obtain the dynamic force on the bridge through Eq. (11). The bridge displacement response under this load is added to the initial profile to represent the new input and the vehicle response in recalculated. This iteration process continues until the difference of two sequential results is smaller than the threshold value predetermined. The definition of each variable in Eq. (11)

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is in Fig. 1.

F

(m  m f  mr ) g  ktf ( x f  y f )  ktr ( xr  yr )

(11)

4. Estimation of Bridge Profile 4.1. Problem Description in state-space equation For the purpose of estimating the bridge profile, a probe car with the driving speed of 20m/s is simulated to pass over the bridge and two types of vehicle responses are measured including vehicle body acceleration and the body angular velocity. The bridge profile is estimated from the vehicle responses through a particle filter. In this section, the problem of bridge profile estimation is described in state-space equation. Because the road profile acts as the input and remains to be estimated in particle filter, the profile has to be included in the evolution process of the system state, which is expressed as

ª¬U U U

X

hf

hr º¼

T

(12)

where X is the system state, U is the vehicle displacement responses and hf, hr are independent profile input at front and rear tire. As the bridge profile is a random process, the profile inputs at two sequential steps do not have any theoretical relation, but it is reasonable to assume there is not a large difference between the profile value. In this case, the distance between two input points is set to be 0.02m, which is small enough for this assumption to be true. Therefore, it is assumed that the mean value of profile PDF at time step k+1 is equal to the estimated value of time step k, as expressed in Eq. (13).

h f ,k 1

h f ,k  [ f ,k

hr ,k 1

hr ,k  [r ,k

(13)

in which ξ is the system error for the profile state evolution to control the change of profile in each step. The ξ value is related to the concerning profile and for the target profile to be estimated in this section, this value is empirically set to be 0.005. Although it is known that rougher profile requires higher level of ξ value, this noise level is not theoretically investigated in this paper. The state response U evolves according to Eq. (14) with a state error term set as 0.1% of the RMS value of each state quantity, for the purpose of making up for the incompleteness coming from the dynamic model [18]. This value is dependent of the time step of the discretization of dynamic equation, which, in this case, is set to be 0.001s, given the vehicle speed of 20m/s.

ª¬U k ,U k ,U k º¼

T

T

f k ( ª¬U k 1,U k 1,U k 1 º¼ )  w(k )

(14)

4.2. Procedures of Estimation The bridge profile generated from section 3.1 is used as the input values which is to be estimated from vehicle response. The vehicle body acceleration and angular velocity response at d = 0.3m are calculated under this input with the consideration of vehicle bridge interaction following section 3.5. The distance of 0.3m means that the sensor is put on the dashboard. A noise is added to the response based on a Gaussian distribution whose mean value is zero and standard deviation is 10% of RMS response value to mimic the sensor noise before it is used as the measured response. The vehicle is assumed to be known here, with the parameters shown in Table 1. These parameters are half-car model parameters shown in Fig. 1. Bridge parameters are set as follows: mb = 16381kg/m, length L = 40m, EI = 1.67×1011N灄m2, and the damping ratio of each mode is assumed to be 0.05. Table 1. Vehicle Parameters (in SI unit) Name Value

M(kg) 1.2×104kg

Iy 1.2×104kg·m2

mf = mr 3.6×103kg

kf = kr 2.4×105N/m

ktf = ktr 2.4×106N/m

cf = cr 6×104N·s/m

Lf 1m

The particle filtering process are summarized for the bridge profile estimation problem as follows. Step 1. Generate N particles, each of which representing a state shown in Eq. (12).

Lr 1.4m

d 0.3m

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Step 2. Pass each particle through system equation expressed in Eq. (13) and Eq. (14) to form the prior PDF of the state (Prediction stage). Step 3. Calculate corresponding vehicle response for each particle from Eq. (2) following the method proposed in section 3.4. The observation error should be close to the noise level. Step 4. Introduce the measurement data, which is the vehicle body acceleration and angular velocity in this case, to calculate the weight of each particle through Eq. (5) Step 4. Resample a new series of particles following the weight of each particle. These new particles represent the posterior PDF (Update Stage). Step 5. Return to Step 2 to start another step of iteration until all steps are finished. 4.3. Results and Discussions Following the procedures proposed in last section, the comparison in both time and frequency domain of estimated profile and real profile is shown in Fig. 3. The estimated input is obtained from the front tire because it is closer to the sensor and is more accurate. Due to the existence of sensor noise, the estimated profile is not exactly the same as the real profile. The estimation error of time history is 25.6%, calculated by dividing the RMS value of the difference between estimated and real profile over the RMS value of the real profile.

Fig. 3. Comparison of estimated profile and real profile in time and frequency domain

5. Vehicle Identification from Estimated Profile 5.1. Problem Description in state-space The central idea of this section is to use the bridge response caused by vehicle passage to identify the vehicle parameters. Because the bridge response is induced by the passing vehicle, it is theoretically feasible to obtain vehicle information from the bridge response. The vehicle parameters are assumed to be unknown and remain to be estimated through the particle filter process. For the purpose of parameter identification, the parameters are included in the system state, shown in Eq. (15)

X

ª¬U U U

y

y

y 4º¼

T

(15)

where U and y are vehicle and are bridge response respectively and Θ is the unknown vehicle parameter vector expressed as:

4 ª¬m I y

mf

mr

kf

kr

ktf

ktr

cf

cr º¼

T

(16)

It is worth noting that when the vehicle parameters are introduced into the state, the state equation shown in Eq. (1) becomes nonlinear. It is no longer amenable to linear theories such as the conventional Kalman filter. Similar to the profile estimation problem, a series of particles are adopted to represent the system state to employ particle filter

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technique to identify unknown vehicle parameters. The details are summarized in next section. 5.2. Procedures of Identification A vehicle with pre-determined parameters is simulated to pass over a bridge and the bridge acceleration responses at mid-span, 1/4 span, and 3/4 span are calculated. Zero-mean white noise with standard deviation of 10% of RMS response is added to the response. These bridge acceleration signals are used as the observed value. The vehicle-bridge interaction is also considered using the iteration method described in section 3.4. Step 1. Generate N particles including all unknown parameters following initial distribution. Step 2. Calculate vehicle and bridge response from the initial input. Step 3. Evolve the parameters according to Eq. (17). Here the system error w(k) is set to be 0.5% of nominal values of each unknown parameter. Evolve also the response terms by Eq. (9). (17) 4k 1 4k  w(k ) Step 4. Calculate bridge acceleration response at mid-span, 1/4 span and 3/4 span for each particle. Step 5. Use the measured bridge acceleration response to calculate weight for each particle Step 6. Resample a new series of particles according to the weights and estimate the parameter values at this step by mean value of all particles. Step 7. Return to Step 2 to start another step of iteration until all steps are finished. 5.3. Results and Discussion Following the procedures stated above, the half-car parameters are identified with the percentage errors listed in Table 2. The converging process of three vehicle mass parameters are shown in Fig. 4. The parameters converge to the real value with an error listed in Table 2 within around 500 cycles of iteration, which is equal to around 0.5 second in this case. Robustness against various sources of noise is also well observed. Uniform distribution is used to represent the initial distribution of each parameter, with the lower limit and upper limit set as around 30% and 150% of the real value. Although the effect of initial distribution on the result is not investigated in this paper, good robustness is observed when choosing different initial distribution.

Fig. 4. Converging process of mass parameters. Table 2. Target values and errors of vehicle parameters identification (in SI unit) Name Value Error

m 1228.4kg 5.35

mf

mr 368.5kg 2.7% 2.1%

Iy 1228.4kg·m2 5.9%

kf

kr 24568N/m 7.1% 5.8%

ktf ktr 245680 N/m 1.2% 2.8%

cf cr 6142N·s/m 2.6% 2.5%

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6. Conclusions In this paper, a new method for vehicle parameter identification is proposed. The particle filtering technique is first used for estimating the bridge profile, which is then adopted as the known input for vehicle parameter identification problem. The vehicle-bridge interaction is considered in the whole process. The results show good accuracy and robustness against noise. Acknowledgements We gratefully acknowledge the CSC scholarship (No. 201506260191) for supporting this research. This work was partially supported by Council for Science, Technology and Innovation, “Cross-ministerial Strategic Innovation Promotion Program (SIP), Infrastructure Maintenance, Renovation, and Management” (funding agency: JST) and the Kajima Foundation’s Research Grant. References [1] Cebon, D. "Theoretical road damage due to dynamic tyre forces of heavy vehicles part 2: simulated damage caused by a tandem-axle vehicle."Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 202.2 (1988): 109-117. [2] Zhu, X. Q., and S. S. Law. "Recent developments in inverse problems of vehicle–bridge interaction dynamics." Journal of Civil Structural Health Monitoring 6.1 (2016): 107-128. [3] Moses, Fred. "Weigh-in-motion system using instrumented bridges." Journal of Transportation Engineering 105.3 (1979). [4] O'Connor, Colin, and Tommy Hung Tin Chan. "Dynamic wheel loads from bridge strains." Journal of Structural Engineering 114.8 (1988): 1703-1723. [5] Law, S. S., Tommy HT Chan, and Q. H. Zeng. "Moving force identification: a time domain method." Journal of Sound and vibration 201.1 (1997): 1-22. [6] Law, S. S., Tommy HT Chan, and Q. H. Zeng. "Moving force identification—a frequency and time domains analysis." Journal of dynamic systems, measurement, and control 121.3 (1999): 394-401. [7] Kalman, Rudolph Emil. "A new approach to linear filtering and prediction problems." Journal of basic Engineering 82.1 (1960): 35-45. [8] Hoshiya, Masaru, and Osamu Maruyama. "Identification of running load and beam system." Journal of engineering mechanics 113.6 (1987): 813-824. [9] Chen, Tsung-Chien, and Ming-Hui Lee. "Research on moving force estimation of the bridge structure using the adaptive input estimation method." Journal of Structural Engineering 8 (2008): 20-208. [10] Gordon, Neil J., David J. Salmond, and Adrian FM Smith. "Novel approach to nonlinear/non-Gaussian Bayesian state estimation." IEE Proceedings F-Radar and Signal Processing. Vol. 140. No. 2. IET, 1993. [11] Carpenter, James, Peter Clifford, and Paul Fearnhead. "Improved particle filter for nonlinear problems." IEE Proceedings-Radar, Sonar and Navigation 146.1 (1999): 2-7. [12] Arulampalam, M. Sanjeev, et al. "A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking." IEEE Transactions on signal processing 50.2 (2002): 174-188. [13] ISO-8608 1995. "Mechanical vibration-Road surface profiles-Reporting of measured data." [14] Jazar, Reza N. Vehicle dynamics: theory and application. Springer Science & Business Media, 2013. [15] Chopra, Anil K. "Dynamics of structures: theory and applications to earthquake engineering. 2007." (2007). [16] Kwon, Young W., and Hyochoong Bang. The finite element method using MATLAB. CRC press, 2000. [17] Green, M. F., and D. Cebon. "Dynamic interaction between heavy vehicles and highway bridges." Computers & structures 62.2 (1997): 253264. [18] Matsuoka, Kodai, et al. "Estimation of Bridge Deflection Response Under Passing Train Loads Based on Acceleration." Journal of Japan Society of Civil Engineers, Ser. A1 (Structural Engineering & Earthquake Engineering (SE/EE)) 69 (2013): 527-542.

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