A Novel Big-data-based Estimation Method of Side-slip Angles

for Autonomous Road Vehicles

D

´

aniel F

´

enyes, Bal

´

azs N

´

emeth and P

´

eter G

´

asp

´

ar

Institute for Computer Science and Control, Hungarian Academy of Sciences,

Kende u. 13-17, H-1111 Budapest, Hungary

Keywords:

Side-slip Estimation, Regression Analysis, Big Data, Kalman Filtering.

Abstract:

In the paper a novel side-slip estimation algorithm, which is based on big data approaches, is proposed. The

idea of the estimation is based on the availability of a large amount of information of the autonomous vehicles,

e.g. yaw-rate, accelerations and steering angles. The signiﬁcant number of signals are processed through

big data approaches to generate a simpliﬁed rule for the side-slip estimation using the onboard signals of the

vehicles. Thus, a subset selection method for time-domain signals is proposed, by which the attributes are

selected based on their relevance. Furthermore, a linear regression using the Ordinary Least Squares (OLS)

method is applied to derive a relationship between the attributes and the estimated signal. The efﬁciency of the

estimation is presented through several CarSim simulation examples, while the WEKA data-mining software

is used for the OLS method.

1 INTRODUCTION AND

MOTIVATION

The spread of autonomous driving is predicted to be a

future tendency of intelligent transportation systems.

Several research institutes have focused on the new

challenges posed by autonomous vehicles, such as en-

vironment detection and the accurate estimation of

vehicle states. One of these signals is the side-slip

angle, which has relevance in the evaluation of the

vehicle stability. In several research projects ﬁltering

methods and observers are designed to estimate the

side-slip angle, see (Stephant et al., 2004; Coyte et al.,

2014). The precise estimation using Kalman ﬁltering

requires sensor fusion with GPS measurements, but

these solutions suffer from the loss of signals in urban

locations and tunnels (Grip et al., 2009).

Therefore, several further techniques have been

published in the literature. Big data were used in the

prediction of vehicle slip through the combination of

individual measurements of the vehicle and database

information (Jeon et al., 2015). In (Sasaki and Nishi-

maki, 2000) a layered neural network was developed

to compute the side-slip angle. An artiﬁcial neural

network method for slip estimation using accelera-

tion, velocity, inertial and steering angle information

was also proposed in (Kato et al., 1994). Moreover,

in (Boada et al., 2015) an adaptive neuro-fuzzy infer-

ence system approach was applied with various signal

measurements. Another formulation of the neural net-

works, such as the general regression for the side-slip

angle estimation, was used in (Wei et al., 2016).

In this paper a novel side-slip angle estimation

method which is based on linear regression is pre-

sented. As a ﬁrst step, a subset selection method

is proposed, which is able to prioritize the attributes

based on their relation to the estimated signal. In

the method the time-domain measurements of the at-

tributes are processed through probability-based com-

putations. Secondly, the OLS method is used to ex-

press the relationship between the attributes and the

estimated signal in a linear form. In this process

the pace regression algorithms of the WEKA data-

mining software are performed (Wang and Witten,

1999). The advantage of the method is that it requires

little on-line computation, while the complex opera-

tions are solved off-line. Moreover, in the estimation

method only the onboard signals of the vehicle are

used, which are available without a loss in communi-

cation.

The structure of the paper is the following. Sec-

tion 2 provides a subset selection method, by which

priorities between the attributes can be set. The re-

sults of the selection are used through linear regres-

sion, which is presented in Section 3. The results of

the big-data-based method are illustrated through var-

ious simulations in Section 4. Finally, the contribu-

420

Fényes, D., Németh, B. and Gáspár, P.

A Novel Big-data-based Estimation Method of Side-slip Angles for Autonomous Road Vehicles.

DOI: 10.5220/0006849504200426

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 420-426

ISBN: 978-989-758-321-6

Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

tions of the paper and further challenges are summa-

rized in Section 5.

2 SUBSET SELECTION FOR THE

SIDE-SLIP ESTIMATION

The purpose of subset selection is to ﬁnd the attributes

which have relevance in the estimation of the side-slip

angle. A difﬁculty in the subset selection method may

be the high number of attributes, which can lead to

the infeasibility of the problem. Thus, in the gener-

ation of subset models, it is necessary to reduce the

complexity of the big data analysis, while informa-

tion loss from the data is negligible (Wang and Wit-

ten, 2002). Several ranking algorithms have been pro-

posed for the relationships between event-based at-

tributes (Shibata, 1981; Thompson, 1978). However,

in the side-slip estimation problem the attributes con-

tain time-domain instances. Thus, it requires the eval-

uation of the measurement in the time-domain, which

is proposed in the following.

Consider a simple linear equation (1) with only

one variable.

y = c

k,0

+ c

k,1

x

k

, (1)

where y is the output (estimated attribute), x

k

∈ X is

a selected variable from the set of the measured at-

tributes (X), c

k,0

and c

k,1

are the ﬁtting parameters.

Since, in this case, the attributes are time-domain vari-

ables, the relative derivatives can be derived from (1).

d

dt

y

x

k

= c

k,1

. (2)

Then, the statistical parameters of c

k,1

are computed,

such as σ

c

k,1

variance and ˆc

k,1

mean. It can be seen

that if the linear model (1) provides the perfect ﬁtting

(the correlation between the two variables is 1)

σ

c

k,1

= 0, (3a)

ˆc

k,1

= c

∗

k,1

, (3b)

where

∗

denotes the parameters of the real model.

Furthermore, if the linear model gives the worst ﬁtting

(the correlation between the two variables is 0)

σ

c

k,1

→ ∞, (4a)

ˆc

k,1

= 0, (4b)

If y 6= 0, x

k

6= 0 and y 6= b + x

k

, where b is a constant

bias parameter.

As the previous equations show, the relevance of

an attribute depends on the variance and the mean of

its c

k

1

parameter. If the normal distribution of c

k,1

is

assumed, then the relationship between these param-

eters is given by the gaussian distribution function:

G

k

(c

k,1

) =

1

σ

c

k,1

√

2π

e

−

1

2

c

k,1

−ˆc

k,1

σ

c

k,1

2

(5)

The relevance of an attribute can be expressed by

integrating the gaussian function and taking into ac-

count the sign of c

k,1

.

D

k

(c

k,1

) =

Z

∞

−∞

sign(c

k,1

)G(c

k,1

)dc

k,1

= (6)

=

1

σ

c

k,1

√

2π

Z

∞

−∞

sign(c

k,1

)e

−

1

2

c

k,1

−ˆc

k,1

σ

2

dc

k,1

D

k

(c

k,1

) yields a value D

k

(c

k,1

) ∈ [0,1], which indi-

cates the correctness of c

k,1

according to its sign, but

ignores the variance σ

c

k,1

. Therefore, the relevance of

an attribute is formed as:

R

k

=

D

k

(c

k,1

)

σ

c

k,1

(7)

Finally, the attribute with the highest relevance on y is

the attribute whose R is the highest.

max

x

(R ) (8)

3 ESTIMATION OF THE

SIDE-SLIP ANGLE BASED ON

THE OLS METHOD

The background of the estimation of the side-slip an-

gle is the combination of the ordinary linear regres-

sion method and the subset selection procedure. In

the following the most important features are summa-

rized and a detailed description is found in (Wang and

Witten, 1999).

In the method a dataset with n independent in-

stances is considered with k input variables and one

output variable. The instances are written in the form

of an n ×k design matrix X. Furthermore, the param-

eter vector of the true model is ζ

∗

. Using the paame-

ter vector of the true model and the matrix of the in-

stances, the output vector y is formed as

y = X ζ

∗

+ ε (9)

where ε is the noise vector, of which elements have

normal distribution N(0, σ

2

). Its variance is σ, and σ

2

is assumed to be known or estimated. The estimation

of σ is denoted by

ˆ

σ

2

. (9) describes the true model

M (ζ

∗

), which is approximated with a ﬁtted, linear

model M (ζ), which has a unique parameter vector ζ.

A Novel Big-data-based Estimation Method of Side-slip Angles for Autonomous Road Vehicles

421

The goal of the estimation problem is to ﬁnd a

model from the entire model space M = {M (ζ) : ζ ∈

R

k

}, whose predictive accuracy is the greatest on the

given dataset. The task can be solved through several

algorithms, such as the OLS method, the OLS subset

selection, shrinkage, RIC, CIC methods, see (Wang

and Witten, 1999). These methods can reduce the di-

mension of the models by discarding the redundant

variables.

It is deﬁned a distance D between the candidate

model and the true model for the evaluation of the

estimation method. D is deﬁned in the form of

D(M (ζ

∗

),M (

ˆ

ζ)) =

||y − ˆy||

2

σ

2

, (10)

where ||·|| denotes the L

2

norm and σ

2

is replaced by

its estimated value

ˆ

σ

2

. Moreover, ˆy is the prediction

output vector.

Thus, the estimation problem is to minimize the

distance between the models, which is represented by

the following expression

min

ζ

D(M (ζ

∗

),M (ζ)) (11)

In the OLS regression the prediction is expressed

in the following way

ˆy = X

ˆ

ζ (12)

where

ˆ

ζ is the parameter vector of the prediction

model. The distance (10) can be reformulated as

D(M (ζ

∗

),M (

ˆ

ζ)) =

||y −X

ˆ

ζ||

2

σ

2

, (13)

which can be expressed by the following form of the

training sets:

(y −X

ˆ

ζ)

2

σ

2

=

ω

∑

i=1

||y

i

−X

i

ˆ

ζ||

2

σ

2

(14)

where y

i

and X

i

are one of the training sets, while ω is

the number of the training sets.

The solution of the optimization task (11) can be

achieved by the partial derivation of (13) according to

ˆ

ζ. The solution is the following:

ˆ

ζ = (X

T

X)

−1

X

T

y. (15)

Thus, the predicted output ˆy in the vector form ˆy can

be expressed by using the orthogonal projection ma-

trix

P = X (X

T

X)

−1

X

T

such as

ˆy = P y. (16)

The estimation of the side-slip angle β(k) is based

on the optimization. The prediction model is formed

as

ˆ

β(k) = ζ

1

+

k

∑

i=k−j

ζ

2,i

a

x

(i) +

k

∑

i=k−j

ζ

3,i

a

y

(i)+

+

k

∑

i=k−j

ζ

4,i

˙

ψ(i) +

k

∑

i=k−j

ζ

5,i

δ

s

(i) (17)

in which past signals are applied in the estimation to

take into consideration the possible regressions. Al-

though their incorporation may improve the accuracy

of the prediction, the increase in j results in a more

complex model.

The number of the past elements can be assisted

through a spectrum analysis of β(k), with which the

dominant frequencies can be determined. It results in

the sampling time T and the entire time horizon of the

past is j ·T .

The measured elements of X

k

are

X

k

=

a

x

(k) a

y

(k)

˙

ψ(k) δ

s

(k)

T

with

• longitudinal acceleration a

x

,

• lateral acceleration a

y

,

• yaw-rate

˙

ψ,

• angle of the steering wheel δ

s

.

Furthermore, the side-slip angle y

k

= β(k) as a ref-

erence for the OLS regression analysis is also mea-

sured. The result of the optimization is the vector

ˆ

ζ = [ζ

1

ζ

2,k−j

... ζ

2,k

ζ

3,k−j

... ζ

3,k

...

ζ

4,k−j

... ζ

4,k

ζ

5,k−j

... ζ

5,k

]

T

. (18)

4 SIMULATION RESULTS OF

THE ESTIMATION METHODS

The efﬁciency of the proposed estimation method is

demonstrated through simulation examples, using the

high-ﬁdelity vehicle dynamic software CarSim.

Several simulations have been performed, which

resulted in a training set for big data analysis. In the

simulations a D-class sedan passenger car has been

used, whose sprung mass is 1320kg, during the col-

lection of the signals the noises of the sensors were

considered.

The car has been driven along the Michigan Wa-

terford Hill Race Course (Figure 1) several times at

various longitudinal velocities (Figure 2), which has

resulted in the collection of more than 2 million in-

stances.

The prediction model also contains the past val-

ues of the attributes, see the relationship (17). The

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

422

−200 −100 0 100 200 300

−200

−150

−100

−50

0

50

100

150

X (m)

Y (m)

Figure 1: Simulation scenario.

0 20 40 60 80 100 120 140 160

Time (s)

0

10

20

30

40

50

60

V (km/h)

Figure 2: Velocity during the scenario.

Figure 3: Spectrum analysis of β.

T sampling time of the signals from the past is com-

puted through the Fourier spectrum analysis, whose

results are found in Figure 3.

They show that the frequency domain of the power

spectral density of the side-slip signal is between 0−8

Hz. Thus, the sampling time of the past values, ac-

cording to the Nyquist-Shannon sampling criteria, has

been chosen at 16 Hz. Moreover, 6 past values of all

attributes have been considered in the model estima-

tion, which corresponds to 0.375 sec horizon back-

wards. The relevance of the selected attributes is eval-

uated by the algorithm presented in Section 2.

Figures 4 shows the relative derivatives of the

measured signals. It can be seen that the means of the

acceleration signals (a

x

,a

y

) are close to zero and their

variances are relatively high, which indicates low rel-

evance on the estimated variable (β). The mean of the

yaw rate (

˙

ψ) and the steering angle (δ) are above zero

and their variances are signiﬁcantly smaller, which

implies that these signals have high relevance on the

selected output.

0 20 40 60 80 100

Time (s)

-4

-2

0

2

4

6

Relative derivatives

0 20 40 60 80 100

Time (s)

-3

-2

-1

0

1

2

3

Relative derivatives

Figure 4: Relative derivatives of the selected attributes.

Table 1 summarizes the computed relevance of all

signals. In the following the predeﬁned order of the

attributes is determined by their calculated relevance.

Furthermore, the OSL subset selection method has

created a model using the predeﬁned order of the at-

tributes, whose correlation coefﬁcient is 0.911.

Figure 5 illustrates a scenario of the slip estima-

tion, in which the car is driven along the test track

at various velocities. In the example the bias and the

variances of the sensors are selected in the following

way:

• inertial sensor: bias = 0.15 m/s

2

, variance = 0.1

2

• qyroscope: bias = 0.01 rad/s, variance = 0.01

2

The error of the estimation is very low, although

the velocity of the vehicle varies signiﬁcantly, see Fig-

A Novel Big-data-based Estimation Method of Side-slip Angles for Autonomous Road Vehicles

423

Table 1: Relevance of attributes.

T R

a

x

R

a

y

R

˙

ψ

R

δ

t 0.00013 0.00449 0.14336 0.58502

t-1 0.0002 0.00266 0.01528 0.00082

t-2 0.00095 0.00074 0.00103 0.00086

t-3 0.00216 0.00003 0.00236 0.00592

t-4 0.001 0.0007 0.00022 0.01225

t-5 0.00284 0.00217 0.00156 0.00455

t-6 0.00044 0.00287 0.00165 0.00133

0 50 100 150

Time (s)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

(deg)

Reference

Predicted

Figure 5: Estimation of the side-slip angle.

ure 2. In the following, the accuracy and the capabil-

ity of the generated predictive model are examined in

various situations.

4.1 Increased Noise

In the ﬁrst situation the impact of the increasing noise

on the predictive accuracy is examined. In practice,

the inertial sensors and the gyroscope are signiﬁcantly

affected by sensor noises, while the angle of the steer-

ing wheel can be relatively well measured. In this sce-

nario the parameters of the sensors are signiﬁcantly

modiﬁed to:

• inertial sensor: bias = 0.3 m/s

2

, variance = 0.4

2

• gyroscope: bias = 0.02 rad/s, variance = 0.04

2

Figure 6 shows the result of the simulation with the

modiﬁed sensor parameters. It can be seen that the

variation of the noise has only a slight effect on the

prediction. The estimated model predicts the side-slip

angle accurately despite the increased noises.

4.2 Variation of Vehicle Mass

Secondly, the effect of the car mass variation on the

accuracy of the prediction is investigated. The nomi-

nal sprung mass of the passenger car is m = 1320kg,

which is modiﬁed in two ways: the mass is reduced to

0 50 100 150

Time (s)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

(deg)

Reference

Predicted

Figure 6: Effect of the increased noise on the prediction.

m = 1000kg and then it is increased to m = 1740kg.

The results of the changes can be seen in Figure 7.

Apart from a short section, the applied model has high

predictive accuracy, which means that the variation of

the mass has no signiﬁcant inﬂuence on the predic-

tion. Therefore, the calculated model can resist the

change in the vehicle mass.

0 50 100 150

Time (s)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

(deg)

Reference

Predicted

Figure 7: Effect of the variation of mass on the prediction

(m = 1000kg).

4.3 Variation of the Adhesion

Coefﬁcient

In the third case the variation of the adhesion coefﬁ-

cient µ is simulated. The initial value of the adhesion

coefﬁcient is µ = 1. In the simulations its value is de-

creased to µ = 0.7 and then to µ = 0.4. The results of

the simulations are shown in Figure 8. In the case of

µ = 0.7 it can be seen that the model predicts the side-

slip angle as accurately as in the normal case, see Fig-

ure 5. In the case of µ = 0.4 the model also operates

appropriately apart from two short sections between

80 −90s and 110 −120s. In these sections the vehi-

cle reaches its stability boundary, and therefore, the

effect of the steering angle on the lateral dynamics

decreases. Nevertheless, the applied model operates

accurately in all other sections.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

424

0 50 100 150

Time (s)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

(deg)

Reference

Predicted

Figure 8: Effect of the variation of the adhesion coefﬁcient

on the prediction (µ = 0.4).

4.4 Highway Analysis

Below, the predictive model is tested on a highway

route. In the example the highway between Ulm and

Stuttgart in Germany, which is one of the most hilly

highway sections in Europe, is selected. Since all of

the regression models can be easily overﬁtted, it is

important to guarantee that the calculated model op-

erates in other cases. In this case the passenger car

is driven at high velocity v

x

∼ 130 km/h. The side-

slip angle and its prediction are shown in Figure 9.

The simulation shows that the applied model is able

to predict the side-slip angle accurately. Its predic-

tive accuracy is still relatively high despite the low

side-slip values. It means that the proposed predictive

model is generally able to predict the side-slip angle.

0 50 100 150

Time (s)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

(deg)

Reference

Predicted

Figure 9: Side-slip angles on the highway.

4.5 Comparison of the Results with

Kalman Filtering

Finally, the OLS estimation is compared with the

results of a Kalman ﬁltering technique. Although

Kalman ﬁltering and the presented OLS method are

close to each other (Otter, 1978), the design technique

of the ﬁlter and the computation algorithm with the

huge number of data can lead to slightly different re-

sults. For example, in case of the Kalman ﬁlter de-

sign the selection of the weights has an important role.

Moreover, in this application of the Kalman ﬁltering

a sensor fusion algorithm is used, which incorporates

in the signals of GPS and IMU. The purpose of sen-

sor fusion is to use the precise longitudinal and lateral

velocity information of the GPS module, whose sam-

pling time is relatively high. Therefore, in the fusion

the less precise longitudinal and lateral acceleration

signals a

x,IMU

,a

y,IMU

of the IMU with low sampling

time are also incorporated (Ryu et al., 2002). The

goal of the comparison is to illustrate that the big data

based estimation can also be an acceptable method for

the estimation problem of vehicle side-slip angle.

The simulation results are illustrated in Figure 10.

It can be seen that both methods can provide accurate

results. However, the Kalman ﬁlter has higher preci-

sion at increased side-slip signals, while the proposed

big-data-based approach is more efﬁcient at low β val-

ues.

0 20 40 60 80 100 120 140

Time (s)

-2

-1

0

1

2

3

(deg)

Measured

Predicted

Kalman-Filter

Figure 10: Comparison of the Kalman ﬁlter method and the

prediction method.

5 CONCLUSIONS

In the paper a big-data-based algorithm for vehicle

side-slip estimation has been proposed. Vehicle dy-

namic simulations have illustrated that the proposed

method can be an efﬁcient alternative algorithm to

the conventional Kalman-ﬁltering techniques. First,

a subset selection method has been proposed, which

provided priorities between the different attributes,

such as longitudinal and lateral accelerations, the

front wheel steering angle and the yaw rate. Second, a

linear regression ﬁtting based on the OLS method has

been performed. As a future challenge in the research,

the proposed method will be demonstrated through

hardware-in-the-loop simulations and real test vehi-

cle measurements.

A Novel Big-data-based Estimation Method of Side-slip Angles for Autonomous Road Vehicles

425

ACKNOWLEDGEMENTS

This work was supported by the GINOP-2.3.2-15-

2016-00002 grant of the Ministry of National Econ-

omy of Hungary.

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