road profile estimation using wavelet neural network … · road profile estimation using wavelet...

Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0812-x

Road profile estimation using wavelet neural network and 7-DOF vehicle

dynamic systems†

Ali Solhmirzaei1,*, Shahram Azadi2 and Reza Kazemi2 1Technical and Engineering Department, Mapna Locomotive Company, Mapna Group, Tehran, Iran

2Department of Mechanical Engineering, K. N. Toosi University, Tehran, Iran

(Manuscript Received October 1, 2011; Revised March 18, 2012; Accepted May 2, 2012)

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Abstract

Road roughness is a broad term that incorporates everything from potholes and cracks to the random deviations that exist in a profile.

To build a roughness index, road irregularities need to be measured first. Existing methods of gauging the roughness are based either on

visual inspections or using one of a limited number of instrumented vehicles that can take physical measurements of the road irregulari-

ties. This paper more specifically focuses on the estimation of a road profile (i.e., along the "wheel track"). This paper proposes a solution

to the road profile estimation using a wavelet neural network (WNN) approach. The method incorporates a WNN which is trained using

the data obtained from a 7-DOF vehicle dynamic model in the MATLAB Simulink software to approximate road profiles via the accel-

erations picked up from the vehicle. In this paper, a novel WNN, multi-input and multi-output feed forward wavelet neural network is

constructed. In the hidden layer, wavelet basis functions are used as activate function instead of the sigmoid function of feed forward

network. The training formulas based on BP algorithm are mathematically derived and a training algorithm is presented. The study inves-

tigates the estimation capability of wavelet neural networks through comparison between some estimated and real road profiles in the

form of actual road roughness.

Keywords: Road profile; Simulation; Wavelet neural network; BP algorithm; Estimation

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1. Introduction

A road profile is one of the most effective vehicle environ-

mental conditions that influences ride, handling, fatigue, fuel

consumption, tire wear, maintenance costs, and vehicle delay

costs. Therefore, establishment of methods for road profile

measurement is completely essential. Currently, many rou-

tines are available for road profile measurement. Most of them

measure vertical deviations of the road surface along the trav-

eling wheel path. The American Society of Testing and Mate-

rials (ASTM) standard E867 [1] defines road roughness as the

deviations of a pavement surface from a true planar surface

with characteristic dimensions that affect vehicle dynamics,

ride quality, dynamic loads, and drainage.

About some of the road profile measuring methods and

tools, their accuracy is affected by inaccurate vehicle manu-

facturer's data and insufficient degrees of freedom. Further-

more, both of these approaches demand formulating the in-

verse of a dynamic model. To avoid these problems,

Ngwangwa et al. [2] developed an artificial neural network

(ANN) based technique to reconstruct the road profile. They

used displacement responses of a quarter car model as inputs

to a two-layer Narx network. They concluded that the tech-

nique is capable of reconstructing the road profile within a

margin of error of 45%. They also indicated that with other

considerations, the error may decrease to 20%.

The applications of ANN based methods are rapidly in-

creasing in various fields of science. They are able to ap-

proximate complicated systems. As for vehicle technology,

neural network has contributed many solutions to areas such

as control and dynamic simulations. The following is a brief

summary of some of the neural network contributions to the

vehicular field.

In 1993, Kageyama [3] used a three-layer feed forward neu-

ral network to transform a group including 17 state variables

of a vehicle model to four state variables of force. The outputs

of the network were properly in agreement with the values

resulting from the simulation.

In 1994, Palkovics and his team [4] examined the ability of

neural networks and also compared the feedforward and feed-

back neural network accuracy in simulation of a tire under

vertical dynamic load. Due to lack of experimental data for

training the network, they used results from simulation of a

magic formula (MF)-tire model, which was proposed by Pace-

*Corresponding author. Tel.: +98 26 36774160-254, Fax.: +98 26 36774160-257

E-mail address: [email protected] † Recommended by Editor Yeon June Kang

© KSME & Springer 2012

3030 A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036

jka and Takahashi in 1992. Their research showed feedfor-

ward methods are more accurate in estimation but not as ro-

bust as feedback methods [5]. In conclusion, they showed that

the majority of complicated models can be replaced with the

tire model created by the neural network [5].

In 1994, Wurtenberger and Iserman [5] used a feedforward

neural network for obtaining a tire and vehicle model to study

the lateral vehicle dynamics.

In 1996, Ghazizadeh and Fahim [6] utilized a two-layer

feedforward neural network to obtain a quasi-static roll model

to study vehicle roll-over behavior. They used a vehicle model

in which inputs were vehicle velocity and steering angle, and

outputs were lateral acceleration, yaw rate, and quasi-static

load transfer in the rear and front axles. They used network

outputs with the number of time delays as feedback to the

input.

In 1997, Pasterkamp and Pacjeka [7] compared feedforward

and radial basis networks to estimate the slip angle and the

friction between the tire and road considering self-aligning

torque and tire loads. They showed that although both net-

works perform reasonable estimation, feedforward networks

are preferred because they have a smaller structure and are

more robust. In addition, research pointed out that for deter-

mining the suspension system behavior, implementation of a

suitably learned feedforward neural network is more appropri-

ate than methods in which, to save on cost of measuring in-

struments, real-time complicated kinematic calculation has to

be done.

In 2009 Yousefzadeh and Azadi and Soltani [8] proposed a

solution to the road profile estimation using an artificial neural

network (ANN) approach. The method incorporates an ANN

which is trained using the data obtained from a validated vehi-

cle model in the ADAMS software to approximate road pro-

files via the accelerations picked up from the vehicle. The

study investigates the estimation capability of neural networks

through comparison between some estimated and real road

profiles in the form of actual road roughness and power spec-

tral density. They showed that all of the combinations indicate

road profile PSDs have higher correlation in comparison to the

real road profiles (in time domain).

The models of natural phenomenon and physical system

which include a nonlinear feature have been linearized via

various linearizing techniques, because of their convenience of

analysis. However, the nonlinear models have been driven by

the improvement of the processor and the development of new

mathematical theories. One of them is to utilize the neural

networks as identification technique.

The performance of identification technique depends on the

type and learning algorithm of neural networks. The most

popular neural networks are multi-layer perceptron network

(MLPN). However, the MLPN has large structures. It induces

the increase of calculation effort. Therefore, the wavelet neu-

ral network (WNN), which is a powerful tool as an estimator,

was introduced by Zhang and Benveniste recently [10]. The

WNN with a simple structure has excellent performance com-

pared with the MLPN.

But conventional back-propagation neural networks

(BPNN) most frequently used in practical applications have

low learning speed, difficulty in choosing the proper size of

network, and easy to fall into local minima. WNN combines

the time-frequency characteristic of wavelet transformation

with the self-learning of conventional neural network. The

basis of WNN is using a wavelet function as the activation

function of neurons and combining wavelet with neural net-

work directly [10]. As wavelet analysis employs mainly the

expansion and contraction of basis function to detect simulta-

neously the characteristics of global and local of the measured

signal [11], WNN inherits these characters from wavelet

analysis, and has stronger approximating, tolerance and classi-

fication capacity than a conventional neural network, which

makes it have strong advantages in dealing with nonlinear

mapping and on-line estimates [12]. This paper selects the

Mexican hat wavelet as the basis function, using the error BP

algorithm to train the network.

The data used in training and testing the WNN were ob-

tained from the simulation of the MATLAB Simulink model,

which was excited by road profiles generated via MATLAB.

2. Dynamic modeling of the vehicle

2.1 Full car vibrating model

A general vibrating model of a vehicle is called the full car

model. Such a model that is shown in Fig. 1 includes the body

bounce x, body roll φ, body pitch θ, wheels hop x1, x2, x3, x4

and independent road excitations y1, y2, y3, y4. A full car vi-

brating model has 7-DOF with the following equations of

motion.

(a)

(b)

Fig. 1. (a) Full car vibrating model of a vehicle; (b) Full car vibrating

model of a vehicle.

A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036 3031

.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

The body of the vehicle is assumed to be a rigid slab of

mass m, which is the total body mass, a longitudinal mass

moment of inertia Ix and a lateral mass moment of inertia Iy.

The moments of inertia are only the body mass moments of

inertia not the vehicle’s mass moments of inertia. The wheels

have a mass m1, m2, m3, and m4, respectively. The front and

rear tires stiffness is indicated by ktf and ktr , respectively. Be-

cause the damping of tires is much smaller than the damping

of shock absorbers, we may ignore the tires’ damping for sim-

pler calculation. The suspension of the car has stiffness kf and

damping cf in the front and stiffness kr and damping cr in the

rear. It is common to make the suspension of the left and right

wheels mirror. So, their stiffness and damping are equal. The

vehicle may also have an antiroll bar in front and in the back,

with a torsional stiffness kRf and kRr . Using a simple model,

the antiroll bar provides a torque mr proportional to the roll

angle φ. Definitions of the employed parameters are indicated

in Table 1.

3. Wavelet neural network and training algorithm

The wavelet theory was proposed in multi-resolution analy-

sis in the early 1980s for improving the defect of the Fourier

series by Mallet. The WNN, which has a wavelet function, is

one type of neural network [9]. Combining the wavelet trans-

form theory with the basic concept of neural networks, a new

mapping network called adaptive wavelet neural network

(WNN) is proposed as an alternative to feedforward neural

networks for approximating arbitrary nonlinear functions [13].

A wavelet neural network generally consists of a feedfor-

Table 1. Full vehicle model parameter.

Parameter Value

Front suspension stiffness kr = 16088 (N/m)

Rear suspension average stiffness kr = 15401 (N/m)

Front tire vertical stiffness ktf = 160880 (N/m)

Rear tire vertical stiffness ktr = 154010 (N/m)

Anti roll bar stiffness kR = kRf = kRr

= 20000 N m/rad

Front vertical damping cf = 2305 (N.s/m)

Rear vertical damping cr = 1226 (N.s/m)

Body vehicle mass m = 930 kg

Front wheel mass mf = m1 = m2 = 31.5 kg

Rear wheel mass mr = m3 = m4 = 29 kg

Pitch moment of inertia 1243 (kg.m2)

Roll moment of inertia 298 (kg.m2)

Wheel base 3.45 m

Body vertical motion coordinate x (m)

Front right wheel vertical motion coordinate x1 (m)

Front left wheel vertical motion coordinate x2 (m)

Rear right wheel vertical motion coordinate x3 (m)

Rear left wheel vertical motion coordinate x4 (m)

Body pitch motion coordinate θ (rad/s)

Body roll motion coordinate φ (rad/s)

Road excitation at the front right wheel y1 (m)

Road excitation at the front left wheel y2 (m)

Road excitation at the rear right wheel y3 (m)

Road excitation at the rear left wheel y4 (m)

Body longitudinal mass moment of inertia Ix = 298 (kg.m2)

Body lateral mass moment of inertia Iy = 1243 (kg.m2)

Distance of C from front axle a1 = 1.7 (m)

Distance of C from rear axle a2 = 1.75 (m)

Distance of C from right side b1 = 0.7 (m)

Distance of C from left side b2 = 0.705 (m)


ward neural network, with one hidden layer, whose activation

functions are drawn from an orthonormal wavelet family. The

WNN algorithms consist of two processes: the self-

construction of networks and the minimization of error. In the

first process, the network structures applied for representation

are determined by using wavelet analysis [14]. The network

gradually recruits hidden units to effectively and sufficiently

cover the time-frequency region occupied by a given target.

Simultaneously, the network parameters are updated to pre-

serve the network topology and take advantage of the later

process. Each hidden unit has a square window in the time-

frequency plane. The optimization rule is only applied to the

hidden units where the selected point falls into their windows.

Therefore, the learning cost can be reduced.

3.1 Wavelet neural network

Wavelet networks use three-layer structure (input layer,

hidden layer, output layer) and wavelet activation function.

The wavelet function is a waveform that has limited duration

and average value of zero. The WNN architecture shown in

Fig. 3 approximates any desired signal y(t) by generalizing a

linear combination of a set of mother wavelets φa,b(t). The

mother wavelet is composed of the translation factor and

the dilation factor ai , where the subscript i indicates the ith

wavelet and n indicates the nth input signal [13]:

1( ) .n i

n

ii

u bu

aaψ ϕ

−=

(8)

Note that the dilation factor a > 0.

In this work, the Mexican hat wavelet is used for the wave-

let neural network. Compared with other wavelet functions,

the Mexican hat wavelet function has several characteristics

that are advantageous in this work: (1) it has an analytical

expression and therefore can be used conveniently for decom-

posing multidimensional time series, (2) it can be differenti-

ated analytically, (3) it is a non-compactly supported but rap-

idly vanishing function (Jiang and Adeli 2004b), and (4) it is

computationally efficient. The Mexican hat wavelet function

is expressed as follows (Fig. 2) [15]:

2

2( ) (1 )2

tt t ex pϕ

= −

(9)

where

.X b

ta

−= (10)

The structure of a wavelet neural network is very similar to

that of a (1+1/2) layer neural network. That is, a feedforward

neural network, taking one or more inputs, with one hidden

layer and whose output layer consists of one or more linear

combiners or summers (see Fig. 3). The hidden layer consists

of neurons, whose activation functions are drawn from a

wavelet basis. These wavelet neurons are usually referred to as

wavelons.

There are two main approaches to creating wavelet neural

networks [13]. In the first, the wavelet and the neural network

processing are performed separately. The input signal is first

decomposed using some wavelet basis by the neurons in the

hidden layer. The wavelet coefficients are then output to one

or more summers whose input weights are modified in accor-

dance with some learning algorithm. The second type com-

bines the two theories. In this case the translation and dilation

of the wavelets along with the summer weights are modified

in accordance with some learning algorithm.

In general, when the first approach is used, only dyadic dila-

tions and translations of the mother wavelet form the wavelet

basis. This type of wavelet neural network is usually referred

to as a wavenet. We will refer to the second type as a wavelet

network.

The input in this case is a multidimensional vector and the

wavelons consist of multidimensional wavelet activation func-

tions. They will produce a non-zero output when the input

vector lies within a small area of the multidimensional input

space. The output of the wavelet neural network is one or

more linear combinations of these multidimensional wavelets.

Fig. 4 shows the form of a wavelon. The output is defined as:

. (11)

This wavelon is in effect equivalent to a multidimensional

wavelet. The architecture of a multidimensional wavelet neu-

ral network is shown in Fig. 3. The hidden layer consists of M

wavelons. The output layer consists of K summers. The output

Fig. 2. Mexican hat wavelet function [15].

Fig. 3. Structure of a wavelet neural network.


of the network is defined as

. (12)

Therefore, the input-output mapping of the network is de-

fined as:

(13)

where M is the number of windowing wavelets, Wi is the

weight coefficients, K is a number of outputs, and N is a num-

ber of input.

3.2 Identification method for nonlinear systems

In this paper, we employ the serial-parallel method for iden-

tifying model of the nonlinear system. Fig. 5 represents the

identification structure. The inputs of the WNN for the identi-

fying model consist of the current input, the past inputs, and

the past outputs of the nonlinear system. The current output of

the WNN represents as follows:

(14)

where Ny indicates the number of the past outputs and Nu

describes the past inputs. And also, dk (n) is the nonlinear sys-

tem output and u1 (n) is the identification input. In this re-

search Ny and Nu are 2.

3.3 Wavelet neural network training algorithm

A nonlinear optimization algorithm, such as gradient de-

scent, conjugate gradients or Byden-Fletcher-Goldfarb Shanno

(BFGS), could be applied to training a wavelet neural network.

However, the advantage of the wavelet neural network archi-

tecture is that it can be trained in stages using linear optimiza-

tion algorithms, which allows for faster training and improved

convergence compared with nonlinear alternatives.

One method often used to vary the weights and biases is

known as the backpropagation algorithm, in which the

weights and biases are modified so as to minimize an average

quadratic error function of the form:

2

1 1

1( ) ( )

2

N N

k k

n k

E d n y n= =

= − ∑∑ (15)

where dk (n) is the expected output of WNN. The backpropa-

gation algorithm actually adopts gradient descent to minimize

E and the corresponding iterative formulas are presented as

the following [17]:

(16)

(17)

(18)

(19)

(20)

(21)

where η refers to wik , ai and bi learning rate parameter; µ re-

fers to momentum their own factor 0 < µ < 1.

4. Profile estimation using wavelet neural network

In this section, we apply the proposed algorithm to vehicle

dynamic systems.

Choosing an appropriate architecture of WNN is dependent

Fig. 4. A wavelet neuron with a multidimensional wavelet activation

function.

Fig. 5. Identification structure using the WNN.


on the type of system being modeled. In this work, the

MATLAB software was used to model the intended WNN.

The inverse of a vehicle model was used to construct the

WNN model, where the inputs were accelerations of a vehicle

moving along a road and the outputs were the road profiles.

The network is a dynamic WNN. In addition to the vehicle

accelerations, the delayed version of the vehicle accelerations

and the road profiles were also input to the WNN block.

The number of hidden layers and their nodes is an arbitrary

parameter that cannot be determined according to a specified

rule. In other words, an optimized network size can be

achieved using trial and error and by considering the accuracy

of the results and the training convergence speed. In this work,

a network including one hidden layer with fifteen Mexican hat

wavelet function nodes and an output layer with linear transfer

function nodes was created.

The outputs of the network were four road profiles related to

each of the vertical displacements of the wheels during the ve-

hicle trip. The network input consisted of two groups. The first

group, called independent input, consisted of seven vehicle

accelerations. The second group, dependent input, was the feed-

back of the network output and independent input, both with

one and two delay elements. A dependent input was used be-

cause state variables in a dynamic system depend not only on

the current inputs of the system but also on the state variables in

previous time. In this model, several different delays and their

combinations were analyzed, and, finally, the combination of

one and two delays was found to be suitable. Neural networks

adjust the values of weights and biases through a process that is

referred to as a learning rule or training algorithm.

5. Training data collection

Network training data were gathered using the vehicle dy-

namic systems model in the MATLAB Simulink software.

Two road profiles, generated in MATLAB, excited the front

wheels. The other two road profiles for exciting the rear

wheels were mostly similar to those of the front wheels but

with some delays caused by the distance between the front and

rear axles of the vehicle.

These four road profiles with a specified vehicle velocity

were applied to the vehicle model to derive seven accelera-

tions, including three accelerations of the body (roll, bounce

and pitch) and four accelerations of the wheels. The generated

road profiles were considered as network output training data.

Input training data consisted of independent and dependent

data, which were introduced in the previous section.

6. Training and testing the network

For training and testing the network, road profiles similar to

the types C and D of ISO 8608 standard [16] were used. As

shown in Table 3, four groups of road profiles were created.

Each of the road profiles consisted of four series of data for

exciting the wheels of the vehicle in the MATLAB Simulink

software.

The road profiles were originally 1000 m long, but with

considering 3.45 m distance between the rear and front axles,

996.55 m of road profiles were used. Furthermore, the fre-

quency content of the road profiles was around 0.1-10 cy-

cles/m, as was intended from the beginning. After applying

the roads to the vehicle at different speed and deriving the

accelerations, the transient span of the accelerations should be

removed to have proper network training.

As shown in Table 4, four combinations of road data from

Table 3 with their corresponding accelerations were used to

train and test four different networks. The transient parts of the

data were removed before training and testing the network.

The criterion is a minimum value for the normalized root of

the sum of the square of errors (RSSE) for all training data

points as follows [15]:

(22)

where yp and ym represent the measured and predicted outputs,

and P

y is the mean of the measured outputs and Na is the

total number of training samples. RSSE is also an indicator of

the performance of the WNN model. In this research, the

value of RSSE < ε = 0.005 is specified. Training usually starts

Table 2. Simulation parameters and the results for WNN.

Simulation condition Model

Number of wavelet node 12

Number of past inputs 2

Number of past output of plant 2

Sampling rate 0.01

Table 3. Road profiles used for training and testing the network.

Road profile No. Road type Application

1 C Training

2 D Training

3 C Test

4 D Test

Table 4. Combination sets used for evaluating the networks.

Combination

No. Training set

Vehicle

velocity

(m/s)

Testing set

1 Road profile No.1 30 Road profile No. 3





from a random set of weights and proceeds until a specified

value of the RSSE is met or a maximum number of iterations

are reached. In most cases, the cost function displays many

local minima, and the training result depends on the initial

weight values. Figs. 6-13 indicate the networks in predicting

the target profile.

7. Conclusion

We have proposed a BP based training algorithm for the

WNN. To verify the effectiveness of the proposed algorithm,

we applied it to train the parameters of the WNN. And then

using the WNN, we executed the identification for the vehicle

system dynamic. In addition, the WNN training by the pro-

Fig. 6. Combination 1 front left wheel.

Fig. 7. Combination 1 front right wheel.

Fig. 8. Combination 2 front left wheel.






posed algorithm adapts well to the abrupt change and the high

nonlinearity of the chaotic systems, because the proposed

theorem concerns the learning rates of each parameters of the

WNN, respectively.

In this paper, the idea of road profile estimation using neural

network algorithm is presented. Due to lack of equipment,

such as a four-post laboratory and a vehicle provided with

accelerometers and accurate distance measuring system, a full

ride model in the MATLAB Simulink software was used for

simulations.

In the next stage of this research, using road data including

frequency contents with much longer wavelengths is planned

to identify the effect of profiles’ frequency contents and vehi-

cle speed on estimation accuracy.

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Ali Solhmirzaei received his BSc in

Railway Engineering (Rolling Stock)

from Iran University of Science and

Technology in 2008, and his MSc in

Mechanical Engineering from K.N.T

University of Technology, Iran, in 2011.

His research is mainly focused on vehi-

cle dynamics, railway vehicle dynamics,

finite elements and fatigue analysis of railway structures.

Shahram Azadi received his B.S. and

M.S. in Mechanical Engineering from

Sharif University of Technology, Iran,

in 1988 and 1992, respectively. He then

received his Ph.D from Amirkabir Uni-

versity of Technology, Iran, in 1999. Dr.

Azadi is currently an assistant professor

in the faculty of Mechanical Engineer-

ing at K.N.Toosi University of Technology in Tehran, Iran.


road profile estimation using wavelet neural network … · road profile estimation using wavelet...

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