Road profile estimation using wavelet neural network … · Road profile estimation using wavelet...
Transcript of 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)
3032 A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036
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.
A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036 3033
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.
3034 A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036
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
2 Road profile No.1 30 Road profile No. 4
3 Road profile No.2 30 Road profile No. 3
4 Road profile No.2 30 Road profile No. 4
A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036 3035
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.
Fig. 9. Combination 2 front right wheel.
Fig. 10. Combination 3 front right wheel.
Fig. 11. Combination 3 front right wheel.
Fig. 12. Combination 4 front right wheel.
3036 A. Solhmirzaei et al. / Journal of Mechanical Science and Technology 26 (10) (2012) 3029~3036
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.
Fig. 13. Combination 4 front right wheel.