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B. C. Min et al.: Fuzzy Logic Path Planner and Motion Controller by Evolutionary Programming for Mobile Robots 155

Such methods as uni-vector field method, geometric calculation method, and heuristics method have been utilized to solve these constraints so far in the robot soc-cer competitions [2,10-13]. In this paper, we propose a fuzzy logic scheme to solve the same problem. The FLC consists of two levels: the planner and the motion con-troller level. Although there is the similar controller that was suggested in [14], we have approached a different way to reduce arrival time to the ball. Firstly, the fuzzy logic path planning is designed by modeling arcs and straight lines using fuzzy logic’s nonlinear model map-ping characteristics in this paper. However, because some path errors can be occurred when geometrical path planning does not consider kinetic characteristic, it is quite challenging to design accurate path controller. In order to solve such problems, fuzzy logic that has robust characteristic of motion controller is used. In addition, the singleton outputs of motion controller level, which are obtained in a heuristic or empirical manner, are op-timized through evolutionary programming.

In Section 2, the overall fuzzy logic posture controller structure that includes a fuzzy logic path planner and a fuzzy motion controller is described. They are explained in Sections 3 and 4, respectively. In Section 5, simula-tions and experimental results are demonstrated. We conclude in Section 6 with some remarks on the results and future research.

2. Overall Fuzzy Control Structure

A. Target System: Robot Soccer System

The system (Figure 1) consists of three parts. The first one is the vision system that locates objects on the field by the global camera that is fixed above the field. Sec-ond component is the host computer which calculates strategies and decides actions for each robot. The last one is robots that follow actions given by the host com-puter through radio frequency (RF) communication. Since the action is represented as a number of sequential motions, sometimes host computer sends each robot's left and right wheel velocities rather than higher linguis-tic commands. In this paper, the fuzzy logic controller implemented in the host computer generates and sends the velocities to the robot through RF communication to achieve shooting behavior of the robot.

B. Modeling of a mobile robot

Differential-drive mobile robots with characteristics of non-slipping and pure rolling are considered. The veloc-ity vector Q = [v w]T consists of the translational velocity of the center of the robot, v, and the rotational velocity, w, defined with respect to the center of the robot. The velocity vector Q and a posture vector P = [x y θ]T are associated with the robot kinematics as follows:

QJv

y

x

P )(

1

0

0

0

sin

cos

θω

θθ

θ=

=

=

(1)

[ ] R

VL

V

L

VR

V L

VR

V wvQ

−=

−+==

2

1

2

12

1

2

1

22

T (2)

where VL is the left wheel velocity and VR is the right wheel velocity. The robot should be controlled to move to any posture by VL and VR. Hence, the fuzzy controller gives the robot a desired direction, θd, at the current po-sition (x, y).

(a)

(b)

Figure 2. (a) Overall fuzzy logic path planner and motion controller structure (b) schematic diagram

Destination

Obstacle

Fuzzy MotionController

Actuator

Fuzzy LogicPath Planner

Fuzzy Controller forShooting BehaviorVision

System

θd + θoff

θd

VL , VR

θ

ρ

φobs

ρobs

φ

obstacle

robot

φobsθ

ρφ

ballgoalρobs

156 International Journal of Fuzzy Systems, Vol. 11, No.3, September2009

C. Overall Fuzzy Controller Because of the constraints mentioned in Section 1, the

three posture variables (ρ,φ,θ) (Figure 2(b)) are required to achieve the shooting action. If all the variables are to be elaborated into one fuzzy rule table, the number of rules could be too large because they will increase by the factor of the number of term sets in each variable. So the controller is decomposed into two sub-controllers (Fig-ure 2) in hierarchical manner that each takes only two variables as inputs. This will significantly reduce the number of rules and make easier to design the controller. In addition, the fuzzy motion controller and the fuzzy logic path planner are included in sub-controllers. This is for generating a global path connecting the present robot position to the ball; however, it can face the constraints. The fuzzy motion controller then commands robot wheel velocities to follow this desired path given the current robot posture.

3. Fuzzy Logic Path Planner

Fuzzy logic path planner is for generating a path glob-

ally that faces the constraints of calculating desired ro-bot's heading angle θd at each relative position (ρ,φ) (Figure 2). It is again divided into two sub-blocks: the destination block that generates a path to lead to the des-tination (the ball) to face the first constraint in Section 1 and the obstacle block that compensate θd for obstacle avoidance to meet the second constraint.

A. Destination Block

This is for obtaining desired heading angle at each robot position. The robot's relative position to the ball is represented in polar coordinates (ρ,φ). Since the lower half plane is symmetric to X-axis, only the upper-half

plane is considered. Figure 3 shows the basic idea of constructing the

planner. The desired path is represented by lines and arcs, and the desired heading angle is defined as θd for the given robot position. θd depicted in Figure 3 can be ob-tained by:

βαπθ −+−=d (3)

−−+−

−+−= −−

min222

1min1

)(tantan

RRyx

R

x

Ry

mincc

min

c

cπ

where xc and yc are the positions of the current robot, and Rmin is the turning radius, which is set to 5 cm consider-ing the size of the ball and the robot. With the Eq. (3), a fuzzy model is formed. In the fuzzy model, ρ and φ are assigned to the inputs based on the position of the robot in polar coordinates. Also, the out-put, θd, has singleton values obtained at the sampled po-sitions shown in Figure 5. As a result, the input, output, and rules of the destination block are defined as follows: 1. Input space (ρ,φ) : relative position of the robot to the ball,

ρ ∈ [0cm, 60cm] φ ∈ [0, 180 deg.]

In accordance with input spaces, the input variable membership functions are depicted in Figure 4. 2. Output (θd): desired heading angle,

θd ∈ [-180 deg., 180 deg.]

3. Rules for destination block 49 rules are obtained using θd at sampled positions as shown in Figure 5. Since input spaces are uniformly di-vided, the rules are sampled at the center of each input region. The resultant rule table for the destination block is in Table 1. Table 1. Rules for destination θd ρ φ NB NM NS ZE PS PM PB

NB 90.0 143.1 157.4 163.7 167.3 169.6 171.2

NM 120.0 158.5 -180.0 -171.0 -166.1 -163.0 -161.0

NS 160.0 172.1 -155.5 -143.6 -137.6 -134.0 -131.6

ZE -170.0 -180.0 -126.9 -120.6 -114.3 -110.7 -108.4

PS -140.0 -135.0 -80.0 -76.9 -71.8 -69.0 -67.4

PM -20.0 -30.0 -34.2 -35.9 -35.5 -40.2 -45.4

PB 0 0 0 0 0 0 0

Figure 3. The desired path using lines and arcs

minR

dθY

XO

minR

dθY

XO

B. C. Min et al.: Fuzzy Logic Path Planner and Motion Controller by Evolutionary Programming for Mobile Robots 157

(a) ρ

(b) φ

Figure 4. Membership functions of ρ and φ according to the

relative position of the robot to the ball

Figure 5. θd, desired heading angle, sampled at each region

B. Obstacle Block This block is to obtain offset angle θoff depicted in

Figure 6, if there are any obstacles nearby. To obtain θoff, four variables such as relative velocity

(Vr), relative direction (Dr), distance (dr), and relative position (Pr, positive if obstacle in front, negative o.w.)

are utilized to get θesc in the presence of obstacles. θesc can be calculated with following the equation:

)(tan 1esc

o

escesc Rf

d

R == −θ (4)

Also, those relative quantities are necessary for obtain-ing the escape radius (Resc) to avoid stationary or moving obstacles.

However, since there are four factors we should con-sider for obtaining Resc, it is difficult to form the FLC by using all of those factors. For this reason, those factors are divided into two FLCs and constructed in hierar-chical structure shown as Figure 7. In Figure 7, Vr and Dr are needed to obtain Resc, while Pr and dr are used to obtain the proportional gain, Wsgn. For example, if the obstacle is located far from the robot, Wsgn gets smaller and becomes 0. In contrast, if the ob-stacle is located nearby the robot, Wsgn gets bigger and reaches 2. Consequently, Wsgn is multiplied with θesc to produce θoff.

θoff = Wsgn × θesc

where, 0 ≤ Wsgn ≤ 2. As a result, the input, output, and rules for the obstacle block are defined as follows: 1. Input space (Vr, Dr, dr, Pr): relative velocity and posi-tion of the obstacle to the robot, Vr ∈ [-0.5, 1.5] Dr ∈ [0 deg., 180 deg.] d r ∈ [0cm, 90cm] Pr ∈ [-0.5, 1.5]

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

phi

Mem

bers

hip

valu

e

-6 0 -4 0 -2 0 0 2 0 4 0 6 00

1 0

2 0

3 0

4 0

5 0

6 0

-6 0 -4 0 -2 0 0 2 0 4 0 6 00

1 0

2 0

3 0

4 0

5 0

6 060

50

40

30

20

10

0-60 -40 -20 0 20 40 60

-10 0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

rho

Mem

bers

hip

valu

e

Figure 6. Obstacle avoidance scheme

θd

do

θesc

θoff

Resc

dr

Figure 7. FLC for obstacle avoidance block

FLC1 f ( · )

FLC2

X

Vr

Dr

Pr

dr

Resc θesc

Wsgn

Wsgn · θesc

158 International Journal of Fuzzy Systems, Vol. 11, No.3, September2009

In accordance with the input space, the input variable membership functions are depicted in Figure 8. 2. Ouput space (θoff): offset angle added to θd,

θoff ∈ [-18Odeg., 180 deg.]

The resultant rule table for the obstacle block divided into FLC1 for Resc and FLC2 for Wsgn is in Table 2.

Table 2. Rule for obstacle block, FLC1(left) and FLC2(right) Resc Dr Wsgn dr Vr NB ZE PB Pr ZE PS PM PB

NB 20 20 20 NE 0.8 0.7 0.6 0.0

ZE 20 25 30 ZE 1.0 1.0 0.9 0.0

PB 20 35 40 PO 1.0 1.0 1.0 0.0

4. Fuzzy Motion Controller

A. Fuzzy Motion Controller Block In the overall structure of Figure 2, the fuzzy motion

controller block receives θd from the fuzzy logic path planner block and part of robot posture information (ρ, θ) through its vision. Then the motion controller block generates appropriate left and right wheel velocities to make θ follow θd at non-zero linear speed before ρ di-minishes. So the motion controller is concerned only for heading angle θ to follow θd with at positive linear ve-locity. For this conventional problem of mobile robots, following heuristics are incorporated: If ρ large →large If | θe | = | (θd + θoff) − θ | large → | VL − VR | large The input, output, and rules for the fuzzy motion con-

(a) Vr (b) Dr

(c) Pr (d) dr

Figure 8. Membership functions of Vr, Dr, Pr and dr for the obstacle block in fuzzy planner

-0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1NB ZE PB

relative velocity

Mem

bers

hip

valu

e

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1NB ZE PB

relative direction

Mem

bers

hip

valu

e

-40 -30 -20 -10 0 10 20 30 400

0.2

0.4

0.6

0.8

1NB ZE PB

relative position

Mem

bers

hip

valu

e

-10 0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1ZE PS PM PB

relative distance

Mem

bers

hip

valu

e

B. C. Min et al.: Fuzzy Logic Path Planner and Motion Controller by Evolutionary Programming for Mobile Robots 159

troller block are defined as follows: 1. Input space (ρ,θe): posture error of the robot to the ball and the path, ρ ∈ [0cm, 60cm]

θe ∈ [-120 deg., 120 deg.]

Depending on the input space, the input variable mem-bership functions are depicted in Figure 9. 2. Output space (VL, VR): desired left and right wheel velocities,

VL, VR ∈ [-54cm/s, 153cm/s] 3. Rules for fuzzy motion controller block According to the above heuristics, 49 rules are acquired for left and right wheel velocities. Table 3 is the rule ta-ble for right wheel speed. Left wheel speed is symmetric with respect to φ = 0. In the table, one unit corresponds to 1.534cm/sec.

(a) ρ

(b) θ

Figure 9. Membership functions of ρ and θe

Table 3. Rules for right wheel VR ρ θe NB NM NS ZE PS PM PB

NB -35 -27 -27 -3 -3 -3 -3

NM -25 8 8 18 31 31 42

NS 15 15 22 35 57 67 67

ZE 30 30 50 60 90 100 100

PS 15 40 44 65 82 92 92

PM 25 51 51 61 68 68 77

PB 35 63 63 67 67 67 67

B. Fuzzy Motion Control Block Tuning based on Evolu-

tionary Programming In Section 4A, the singleton values of the motion con-

troller block were determined with fuzzy control, which has various strong points: intuitive and simple [15]. However, most of the times, using professionals’ knowledge for designing might not be the most suitable system because there will be no professionals’ knowledge for newly introduced subject. In order to make up for the weak points, hybrid systems such as fuzzy system, evolutionary algorithm, and neural net-works, are studied in depth.

Fuzzy system and feed-forward control of neural net-work have similar basic structures. The difference be-tween those two systems is in each node’s number of connections and the function. For instance, fuzzy system uses knowledge related with a plant to decide a network structure by connecting related variables and maintaining inside structure. On the other hand, neural network has a learning ability. In order to use such learning method in fuzzy system as gradient descent, which is back propa-gation, fuzzy system should be expressed mathematical-ly and each of the nodes should be able to differentiate. Therefore, it is impossible to optimize the fuzzy system by using this neural network.

Evolutionary algorithm is suitable to use for optimiz-ing a not differentiable system or a system with local solution. Because of such merits, numerous researchers applied fuzzy system in automation of system design.

For this reason, in this paper, evolutionary fuzzy sys-tem, real variables optimization method based on evolu-tionary algorithm, is used for tuning of fuzzy system in the path motion controller. The condition for optimiza-tion is the minimum time to reach to the target point. As shown in Figure 10, the optimization for fuzzy system uses membership function’s center point and width. In order not to change the order of the membership func-tions, following restrictions are considered in evolution-ary algorithm process.

Ai−1 < Ai (i−1 < i) (5)

-10 0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

rho

Mem

bers

hip

valu

e

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

Theta error

Mem

bers

hip

valu

e

160 International Journal of Fuzzy Systems, Vol. 11, No.3, September2009

Figure 11 is the flow chart that shows the learning scheme of the evolutionary program. An important fea-ture of EPs is that the range of mutations, the stepsize, is not fixed but inherited. Mutation creates a new offspring x’

i from each singleton value, xi by adding it to a Gauss-ian number with mean 0 and standard deviation σi.

x’

i = xi + Ni(0, σi) (6)

where σi is xi's maximum boundary within which the designer allows to change the singleton value. By using probability selection method for selecting offspring, σi is set to one tenth of xi's original value which was obtained heuristically, and total population N is set to 20. To test the tracking performance, 36 test data were used. They are 36 different starting postures. The time consumed for the robot to arrive from each point to the origin (destina-tion) is all summed up and compared for the selection. The initial time using heuristic singleton values was 36.6 seconds. After 10,000 generations, the tuned controller

ρ(a) ρ

(b) θe

Figure 12. Optimized membership functions of ρ and θe

reduced the time from 4.3 seconds to 32.3 seconds. Fig-ure 12 is the results of optimizing the minimum time required to the target point while satisfying the re-striction condition of Eq. (6).

5. Simulation and Experiment A. Simulation

Simulation is performed based on the following robot kinematics:

SLR T

D

kVkVkk ⋅−+=+ )()(

)()1( θθ (7)

SLR T

kkkVkVkxkx ⋅++⋅−+=+

2

)1()(cos

2

)()()()1(

θθ (8)

SLR T

kkkVkVkyky ⋅++⋅

−+=+

2

)1()(sin

2

)()()()1(

θθ (9)

Robot's physical quantities are:

-10 0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

rho

Mem

bers

hip

valu

e

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1NB NM NS ZE PS PM PB

Theta error

Mem

bers

hip

valu

e

Figure 11. The flow chart of the proposed Optimization method using EP

Simulator

Evaluation

Initialize

Reproduction

Mutation

Evaluation

Selection

generation<Max?

FuzzyPosture

controller

Mobilerobot

5 TestSets

Cost Function

Terminate

Figure 10. The optimization for fuzzy system using the tri-angular-shaped membership function’s center point and

width

1iA − iA 1iA +

1id − id 1id +

1iA − iA 1iA +

1id − id 1id +

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C. Min et al.: F

This work wngineering F

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Ching-ChHuang, aHumanoidnational JMarch 200H. -S. ShJ.-H. Kimture for coal-time apAutonomo1997. Y.-J. Kimary PrograFast MobSecond AEvolution Y.-J. Kimary Prografor Fast Min ArtificiaChing-Chand Chi-TGA for Ttional JouMarch 200S. Ishikawbot navigaRobotics, J. S. ChoiMobile Rgent ContElectrical J.-M. Yanfor TrajecMobile Rand AutomNishant Asion GuidSoccer”, BInstitute of

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was supporteFoundation (Kment (MEST

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ang Wong, and Yu-Tingd Robot for Journal of Fu08. him, H.-S. K

m, “Designingooperative mpplication to ous Systems,

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bile Robot NAsia-Pacific and Learnin

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Mobile Roboal Intelligencang Wong,

Tai Cheng, “FTwo-wheeledurnal of Fuz07.

wa, “A methoation by usinVol. 9, NO. , J. S. Joo, “F

Robot with Otrol Class teEngineering

ng and J.-H. ctory TrackinRobots,” IEEmation, Vol. Agrawal, Tamded ReactiveB.S. thesis, D

of Technologym, D.-H, Kim

amming-Basbile Robot NAsia-Pacific and Learnin

m, D.-H. Kim

Path Planner a

wledgment

ed by the KoKOSEF) granT) (R01-2008

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ontrol heeled botics

e Vi-Robot ndian

ution-od for gs of

ulated

ution-

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telemmem

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s currently an w at Universi

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mming-Baseobile Robot l Intelligencer, P. Hannahmple strategoccer,” Robo21, No. 2, ppgs of the 1st an), FederatAssociation , H. -S. Kim,le Extraction

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Byung-Chgree in EleKyung Hein 2008. Hdegree in neering frocurrent resgent contro

Moon-Suin Informfrom the Seoul, KoElectrical AdvancedTechnolog2000. Sin

h division, ET

Dong-Hanand Ph. Dneering frtute of (KAIST), and 2003with the Dneering at

assistant proty of Illinois ient of the fire FIRA robot e in the area o

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ation of Int(FIRA), 199, H. -S. Shimn for Shootit,” 1999 IEEference Pr

Mendel, “Gem Examples

and Cybernet

heol Min receectronics Engee University

He is currentlyElectronics

om Kyung Hesearch interes

rol and naviga

Su Lee receivemation & Con

University oorea in 1998 al Engineeringd Institute gy (KAIST), nce then, he

TRI, Korea, as

n Kim receivD. degrees inrom the Korea

Science a, Daejon, Kor3, respectivelDepartment ot Kyung Hee

ofessor. He waat Urbaba-Chrst award of asoccer in 200

of a multi-agen

obots

or Field Met,” Lecture NVerlag, 1999ier, S. Smithlision-avoidautonomous S997. n Soccer Roternational

99. m and J. -H. King Action CEE Internatioroceedings,

enerating Fus,” IEEE Tratics, Vol. 22,

eived his B.S.gineering fromy, Yongin, Koy pursuing his

and Radio Eee University.sts include intation control.

ed his B.S. dentrol Engineeof Kwang Wand M.S. degreg from the Kof Science Deajon, Kore

e has been s a research se

ved his B.S., n Electrical Ea Advanced Iand Technorea, in 1995, 1ly. He has b

of Electrical EUniversity, was a post doct

hampaign in 2a humanoid r02. His currennt robot system

163

thod Notes

. h, P. ance Sys-

obot-Ro-

Kim, Con-onal

pp.

uzzy ans-, No.

. de-m the orea, M.S.

Engi-. His telli-

egree ering

Woon, ee in

Korea and

ea in with

enior

M.S Engi-Insti-ology 1998 been

Engi-where

toral 2003. robot nt re-m.