LOCALIZATION FOR A CAR-LIKE MOBILE ROBOT USING

NONLINEAR DYNAMIC MODEL

Niramon Ruangpayoongsak, Hubert Roth and Robert Mayr

Institute of Automatic Control Engineering, University of Siegen, Hoelderinstr. 3, D-57068 Siegen, Germany

Keywords: Localization, Mobile robotics, Kalman filter, Dynamic model.

Abstract: The problem of localization is well known in mobile robotics. A solution is to use a model-based technique

such as a kalman filter with multi sensor data fusion. For a car-like mobile robot, the nonlinear dynamic

model is suitable for robot movement representation. This work presents the discrete extended kalman filter

including a nonlinear dynamic model for the mobile robot localization. As inputs for the kalman filtering,

gyroscope and compass sensors provide the relative and absolute yaw angles. The experiments are

performed on several path types and the averages of the final position errors and the final heading errors are

proposed.

1 INTRODUCTION

The localization problem is a well known important

problem in mobile robotics. No matter how the

environment is or how the functionalities of the

robot are, whether the robot navigates autonomously

or semi-autonomously, it is useful to know where

the robot is. The mobile robot localization can be

classified into two main categories that are relative

(local) localization and absolute (global) localization

as described in (Goel et.al., 1999).

For large scale outdoor environment, the

positioning resolution is effective to have absolute

localization regarding disturbances resulting from a

missing smoothness of the ground surface. The

absolute positioning sensors such as global

positioning system (GPS), differential GPS, beacon

system, natural landmarks are exploited with the

relative positioning sensors e.g. gyroscope to obtain

accurate robot position. On the other hand, in indoor

environment with a smooth and flat surface, the

disturbance is small. However, the satellite signals in

buildings with many walls often fail regarding to

weakness and blindness. Therefore, a good

performance of at least relative localization is

necessary.

Sensors used for relative positioning are e.g.

odometer, compass and gyroscope. These sensors

have different strength and weakness (Nehmzow,

2003). Odometer is a common device used to

provide travelled distances. The relative position

provided by odometers has sometimes an

accumulated error from wheel slippage. The

gyroscope and compass sensors provide the yaw

angle. The gyroscope measures angular velocity.

The integration of this signal provides the relative

yaw angle. The gyroscope has a drift in

measurement. The compass sensor provides the

ready-to-use absolute yaw angle.

The robot’s nonlinear dynamic model realization

and the robot localization using discrete extended

kalman filter is proposed. The comparison of the real

robot position and heading errors by using odometer,

gyroscope, and compass sensor are presented.

This paper is organized as follows: Section 2

explains the robot modelling, section 3 describes the

extended kalman filter, section 4 shows the

calculation of robot estimated positions and

headings, section 5 presents the experimental results

while section 6 contains the conclusion.

2 ROBOT MODELLING

The first step in model based localization is to

develop a suitable mathematical model for the robot.

When considering the structure of the mobile robot,

it is a four wheel vehicle with the driving principle

of a car, steering the front wheels and propelling the

rear wheels. The nonlinear dynamic model

367

Ruangpayoongsak N., Roth H. and Mayr R. (2005).

LOCALIZATION FOR A CAR-LIKE MOBILE ROBOT USING NONLINEAR DYNAMIC MODEL.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 367-372

DOI: 10.5220/0001176803670372

 SciTePress

represents the dynamic movement of the robot with

the nonlinear characteristics of the side force at the

wheels. This property has majority effect when the

car moves along curvature path. In the following, the

nonlinear dynamic model is introduced and

subsequently the application on the mobile robot is

described.

Figure 1: Dynamical variables of the vehicle

2.1 Nonlinear dynamic model

The nonlinear dynamic mathematical model for a

four wheel vehicle was determined as a single track

model describing transverse and longitudinal

dynamics, neglecting roll and pitch angles and

comprising front and rear wheels to one fictitious

wheel (Riekert and Schunck, 1940, Mayr, 1991).

Fig. 1 shows the dynamic variables of a vehicle

as follows:

• the yaw angle

, vehicle orientation,

• the yaw velocity

’, the first derivative of the

yaw angle,

• the longitudinal velocity v,

• the sideslip angle

• the actual position X and Y of the center of

gravity in Cartesian coordinates,

• the front side force S

and the rear side force

• the rear longitudinal forces H resulting from

the driving motor,

• the steering angle

In addition, there are the following constants:

Vehicle mass m, moment of inertia

and the

distances l

) between the front (rear) wheels and

the longitudinal axes of the car.

Based on the balance of the forces acting on the

vehicle in the longitudinal and lateral directions, the

torques and the kinematic conditions, the nonlinear

dynamic vehicle model is presented as follows:

(){}

ββψβ

cossin

STH

+−−=

(){}

δβ

+− cos

(1)

′

(2)

()

hhvv

lSlS −=

cos

(3)

(){}

ββ

sincos

STH

v −−=

(){}

δβ

+− sin

(4)

The front and rear side forces S

and S

of the

vehicle depend on the slip angles

and

, while

depends itself also on the steering angle

. The

nonlinear functions Γ

and Γ

determine the

dynamics in the tires as follows:

)(

vvv

(5)

)(

hhh

(6)

By these functions representing a characteristic

line, the behavior of the wheels and the tires is taken

into consideration. This characteristic line includes

limitations and descending behavior for high values

in the argument. As shown in Fig. 2, here the

functions are approximated by three straight lines

describing the dependence of side force values S on

their argument

. For low arguments, a nearly

proportional ascend of the side force can be

recognized, while beyond the value

max

the side

force is descending. At

max

the value for the

corresponding side force reaches its maximum. The

area below

max

is called the ascending part, while

the area, where the side force descends is called the

descending part of the characteristic line. As a

Figure 2: Characteristic line Γ of the wheels and tyres

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368

consequence of the gradual inverse dynamics, the

car would begin to skid when driving in the

descending part. During a normal maneuver without

skidding, every wheel of the vehicle is working in

the ascending area of the characteristic line. This

nonlinear characteristic coming from automotive

technology is also applicable to a car-like mobile

robot in indoor environment over dry and flat floor.

As here, high absolute values for

and

will

never come up, the functions Γ

and Γ

can be

simplified to the amplification factors c

and c

Then, (5) and (6) result in

⎟

⎠

⎞

⎜

⎝

⎛

+−==

δψβα

ccS

vvvv

(7)

⎟

⎠

⎞

⎜

⎝

⎛

+==

ψβα

ccS

hhhh

(8)

2.2 Model realization

The mobile experimental robot for locomotion and

intelligent navigation (MERLIN) has been designed

and developed for both, indoor and outdoor

environments. The sensors onboard are odometer,

gyroscope, 3D compass, ultrasonic sensor, infrared

sensor, and bumper as shown in Fig. 3. The semi-

autonomous tele-control using joystick and path

commands are available via internet (Kuhle et. al.,

2004).

The moment of inertia from calculation is equal

to 0.169167 kg·m

. The side force constants c

and

are from experiments. By keeping a constant

steering angle

and a constant driving motor force

H, the robot drives on a circular path with a constant

speed and the robots position is externally recorded

by a V-scope

positioning sensor. The data are

exploited with the model equations to obtain the side

force constant. The results are c

= 168450 and c

152290. The length from the center of gravity to the

front wheels is l

= 0.15 meter and to the rear wheels

= 0.15 meter. The mass of the robot is m = 7 kg

including the batteries. Due to the surface of the

robot is small and as the robot drives with low

speed, the air resistance is neglected.

3 EXTENDED KALMAN FILTER

(EKF)

The kalman filter is a model-based sensor fusion

technique. Two main processes are time update and

measurement update. For a nonlinear dynamic

system, the discrete extended kalman filter is applied

(

Brown, 1983, Welch and Bishop, 2002). Assume

that a nonlinear process has a state vector

and

is governed by the non-linear stochastic differential

equation with a measurement

that is

x ℜ∈

z ℜ∈

= f (x

k-1

), (9)

= h(x

, v

), (10)

where w

and v

represent the process and

measurement noise. After linearization, the

linearized state transition matrix is represented

without noises. The Jacobian matrix of the partial

derivatives of f with respect to x is

[]

)0,,

(

1, kk

−

∂

(11)

where the state vector x is

vx ][

ψψβ

′

(12)

and the Jacobian matrix of partial derivatives of h

with respect to x is

Figure 3: MERLIN

[]

)0,

(

, k

∂

(13)

where h is

xxxxh ][

4321

(14)

The kalman filtering process starts from

initialization of all state variables and matrices.

Assume that the process and measurement noises are

Gaussian with zero mean and are constant

throughout the process. The priori estimate state

variable

and the priori estimate error covariance

at time step k are

−

LOCALIZATION FOR A CAR-LIKE MOBILE ROBOT USING NONLINEAR DYNAMIC MODEL

369

Figure 4: Architecture of the robot localization system

)0,,

(

1 kkk

uxfx

−

(15)

QAPAP

kkkk

−

(16)

where Q is the process noise constant matrix. After

that the kalman gain K

, the posteriori estimate state

variable

, and the posteriori estimate error

covariance P

are calculated as follows:

)(

−−−

+= RHPHHPK

kkk

(17)

))0,

((

ˆˆ

−−

−+=

kkkkk

xhzKxx

(18)

−

−=

kkkk

PHKIP )(

(19)

where R is the measurement noise covariance.

The inputs of the robot model are the steering

angle

and the driving motor force H. Let the input

vector at time step k be

= [

]

(20)

and let the measurement vector be

= [z

β,

ψ,

’,k

v,k

]

(21)

where z

β,

, z

ψ,

, z

’,k

, and z

v,k

are the measured values

of the state variables at time step k. As the robot

drives very slow and, thus, the sideslip angle, which

is difficult to sense, is of minor importance, the

measurement value of the sideslip is set to zero for

all time steps k. Regarding the system model in (1–

4), the discrete system is first obtained using Euler’s

method (

Franklin, 1998). The process and

measurement noise covariance matrices Q and R are

tuned off-line in the simulation for appropriate value

and kept constant during the iterative process. The

selected values are 1x10

-6

for all diagonal matrix

elements.

Figure 5: Variables based on odometer (cont.)

4 POSITION AND HEADING

CALCULATIONS

As shown in Fig. 4, three different estimated paths

are from using measurement of odometer,

gyroscope, and compass estimation. The odometer

estimated position and heading is the non-model

based whereas the other two estimations are based

on kalman filter and the nonlinear dynamic model.

4.1 Odometer position and heading

estimation

The position x

odo,k

, y

odo,k

and heading

odo,k

at step k

are calculated from

kodo

kLkR

kodokodo

1,,

cos

⎟

⎠

⎞

⎜

⎝

⎛

∆+∆

−

(22)

kodo

kLkR

kodokodo

1,,

sin

⎟

⎠

⎞

⎜

⎝

⎛

∆+∆

−

(23)

⎟

⎠

⎞

⎜

⎝

⎛

∆−∆

−

kLkR

kodokodo

1,,

tan

(24)

1,,, −

−

∆

kRkRkR

ddd

(25)

1,,, −

−

∆

kLkLkL

ddd

(26)

where ∆d

R,k

is the difference of driven distances of

the right wheel between the time steps k and k-1 as

shown in Fig. 5. Note that similar calculation is

applied to the difference of distances left ∆d

L,k

. Here,

L,k

represents the already driven distance of the left

wheel at time step k, while d

R,k

is the driven distance

of the right wheel. These driven distances since the

beginning of the ride can be computed from the

amount of pulses from the odometer on each wheel.

The wheel base W is the distance between the center

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370

of the left wheel to the center of the right wheel and

is equal to 23.5 cm.

4.2 Position and heading estimation

using gyroscope and compass

The estimated positions of the relative yaw angle of

the gyroscope and absolute yaw angle of the

compass are calculated by using

()

cos

−

⋅=

kkkkk

xvstx

(27)

()

sin

−

⋅=

kkkkk

yvsty

(28)

where v

, and

are posterior estimated state

variables as in (18). The estimated robot heading is

. st is the time step. Here also the kalman filtering

is applied.

5 EXPERIMENTAL RESULTS

The experiments were performed in indoor and the

sensor data were collected for off-line plots using

MATLAB. All graphs scale has unit in meter and

the data sampling rate is 0.2 sec. The test is designed

for testing the performance of localization when the

robot moves straight on, in a curve path, and the

combination of both.

5.1 Test results

As already mentioned, in order to improve the

quality of the measurement signals, the data coming

from the gyroscope as well as the data from the

compass was subsequently processed by the kalman

filter using the measured velocity signal from the

odometer.

Odometer: Please note that for comparison, the

data coming from the odometer itself (dot line) is

plotted directly using (22-26) and, thus, not updated

by the kalman filter. Due to (24), (25) and (26),

relevant odometer data is not only the average of the

wheel movements but also the difference of the

movements of the left and right wheel in order to get

information about the change of orientation of the

robot.

Gyroscope: The measured yaw angle by

gyroscope are applied as inputs to kalman filter

measurement updates and the positions are obtained

from (27) and (28) shown as solid line. Please note,

that here the odometer only provides the

measurement value for the velocity.

Compass: Similarly, by using the compass yaw

angle with kalman filter, the position estimation

from (27) and (28) are represented by dash-dot line.

Please note, that here also the odometer only

provides the measurement value for the velocity.

Figure 6: Wall-path estimated positions

The wall-path is shown in Fig. 6. At the wall

corners, the robot turns with minimum radius curve.

The estimated final position using odometer has

sometimes large error as shown in the figure. The

gyroscope has drift in some trials whereas the

compass result is quite static for all trials with error

of deviation in measurement as in line path.

For every path type, several driving tests were

performed. Regarding each driving test, the real final

position determined was compared to the position

value measured by the on-board sensors and the

resulting errors were calculated. Subsequently, the

Table 1: Average position errors (e

pos

)

Odometer

(meters)

Compass

(meters)

Gyroscope

(meters)

Rectangular 0.71 0.33

0.17

U 0.88 0.38

0.12

Arcs

0.26

0.32 0.29

Line1 0.53

0.27

1.02

Line2

0.19

0.78 0.27

Wall 15.09

3.35

5.44

Table 2: Average heading errors (e

heading

)

Odometer

(degrees)

Compass

(degrees)

Gyroscope

(degrees)

Rectangular -23 14

-1

U 7 12

Arcs 6 110

Line1

61 6

Line2 5 -91

Wall -58 -52

-10

LOCALIZATION FOR A CAR-LIKE MOBILE ROBOT USING NONLINEAR DYNAMIC MODEL

371

average error values result from the averages of the

respective errors of all similar driving tests.

The average errors in position and heading are

summarized in Tables 1 and 2, respectively. The

position error e

pos

is the error between the actual

final position and the estimated final position. Based

on the actual and estimated values for the yaw angle,

the heading error e

heading

is the error of robot heading

at the final position. These errors are calculated by

yxpos

eee +=

(29)

()

iactualiestimate

heading

∑

−

ψψ

(30)

Where

and

are the mean of error in x and y

direction from several driving tests.

estimate,i

and

actual,i

are the estimated and actual final heading of

test number i, respectively.

In Table 1, the average of final position errors of

gyroscope are not the largest in all trajectories. Due

to the localization based on model, the position

errors of compass and gyro are not much different.

The gyro and compass with discrete EKF

outperforms the odometer positioning for wall path.

As in Table 2, the average of final heading errors of

the well calibrated gyroscope present the excellent

performance over compass and odometer.

5.2 Discussions

For the line1 path, both average position and

heading errors of gyroscope are larger than those of

odometer and compass. This presents the minority

performance of the gyroscope over the odometer in

line1 path by the mechanical structure of the robot.

For the trajectories containing arcs path that are

rectangular, U, and arcs path, the yaw angle

measurement dominates in the position estimation

process. The well calibrated gyroscope presents

good performance without drift. Nevertheless, for

the large scale scenario as for wall path, the

gyroscope provides good estimation, because here

the drift is not relevant.

The compass yaw angle deviation occurs in all

trajectories. This error shows up in most of all the

driving tests. The average errors of final heading for

all trajectories in Table 2 are in most cases larger.

However, there are some cases that the deviation

doesn’t exist as shown in Fig. 6.

For the odometer estimated position, the large

error is presented in the wall path. This error is

caused by the slippage in the wheels and does not

always occur in all driving tests. However, the final

position and heading errors are larger than those of

the gyroscope, except in line1 and line2 trajectories

regarding to the previous discussion of the error in

the estimation using gyroscope.

6 CONCLUSIONS

The discrete extended kalman filter is applied to a

car-like mobile robot for improvement of

localization using the nonlinear dynamic model. The

experiments are performed on six path types and the

final position error and final heading error using

odometer, gyroscope, and compass data are

compared. The well calibrated gyroscope provides

minority performance in robot final position for

line1 path. For the other five path types, the

gyroscope position errors are smaller than those of

odometer and the in performance in final heading of

the gyroscope dominates the compass and odometer

for all trajectories.

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