A Practical Fuzzy Logic Controller for the Path Tracking ofWheeled Mobile Robots By T.H. Lee, H.K. Lam, F.H.F. Leung, and P.K.S. Tam
This article tackles the path-tracking problem ofwheeled mobile robots (WMRs) that are used in theMicro Robot Soccer Tournament (MiroSot). The basic
configuration of MiroSot (Figure 1) comprises a football sta-dium (ground plane) with two teams [1]-[2]. Each team hasthree WMRs (Figure 2), a camera for image capture, a hostcomputer, and an RF data transmitter. The camera capturesthe football stadium’s images that are sent to and processedby the host computer. Based on the real-time locations ofthe robots and the ball and the soccer game strategy, thehost computer determines the actions of its team of robots.The objective is to score points by pushing the ball to theopponent’s goal as many times as possible and to preventthe opponent team from scoring points. To achieve this ob-jective, a path has to be generated by the game strategy forthe home robot to follow—a path-tracking problem. Foreach robot, the host computer generates the correspondingcontrol signals driving the wheel at each side to ensure goodpath-tracking capability. The robot will have both linear andangular displacements until it arrives at the target position.
To tackle the path-tracking problem, some other prob-lems have to be solved. First, owing to the nonlinear dy-namic and nonholonomic characteristics of the WMRs, thecontroller design will be a difficult problem, as we do nothave a systematic and simple controller design methodol-ogy for nonlinear systems. Traditionally, we may derive amathematical system model from which a suitable control-ler is designed [3]. However, the model of the WMRs used inMiroSot is very complicated. The WMR dynamic equationsare obtained by the well-known Lagrange equations [6]
ddt
L L∂∂
− ∂
∂=
&q q�,
(1)
where � represents the torque. The Lagrangian variable L isequal to the difference between the kinetic energy and thepotential energy. Based on (1), the dynamic equation can beexpressed as
M q q V q q q G q B q A q( )&& ( , & ) & ( ) ( ) ( )+ + = −mT� λ, (2)
where M(q) represents the 3 3× inertia matrix (which issymmetric), V q,q qm( )& & represents the 3 1× vector of centrif-ugal and Coriolis torques, G(q) represents the gravitytorques, A(q) is given by the nonholonomic constraints, λ isa Lagrange multiplier associated with the constraints, andB(q) is a 3 2× matrix. In the present case, the variables aredefined as follows:
q M q
V q q q
=
=
=
x
y
m
m
I
c
c
m
θ, ( ) ,
( , & ) &
0 0
0 0
0 0
0
0
0 0
= =−
=
, ( ) ( )
sin
cosG q 0 A q, T
r
l
θθ
ττ
�
=−
, ( )
cos cos
sin sin ,B q1r
R R
θ θθ θ
,
λ θ θ θ= − +m x yc c( & cos & sin )&. xc and yc are the x and y coordi-nates of the WMR in the football stadium, respectively; θ isthe heading angle of the WMR; I is the moment of inertia ofthe WMR about its center (which is difficult to obtain); m isthe mass of the robot; τr and τ l are torque control inputs gen-erated by the right and left motors, respectively; and R and rare the distances between the two wheels and the radius ofthe wheel, respectively. Substituting the above variablesinto (2) and simplifying the equation, we have
Lam ([email protected]), Lee, Leung, and Tam are with the Centre for Multimedia Signal Processing, Department of Electronic and In-formation Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.
This plant model is the basis for designing the position con-troller of the WMR. More details about the modeling ofwheeled mobile robots can be found in [6]. The objective isto drive the robot, of output cordinates xc and yc, to a de-sired position (position control) and follow a desired path(path tracking) in the football stadium by feeding an appro-priate torque (input of the system), τ, to the robot. Unfortu-nately, the model of (3) may not be very useful because ofthe parameter uncertainties in practical robots. This makesthe design of the controller difficult to realize. Even if a non-linear controller can be designed based on the mathemati-cal model, the controller may be too complicated forimplementation in software. The fact that three robots areto be controlled further complicates the problem. More-over, controlling of the robots is just one of the tasks (othersinclude decision making, game strategy, and path planning)performed by the host computer. On the other hand, the po-sition and heading angle of the robot, which are the inputsto the controller, are obtained from the captured image only.Figure 3 shows a functional block diagram of the closed-loop control system. The WMR has no embedded positionsensor, and its RF receiver only listens to commands fromthe host computer. The position information is obtained byrecognizing the color mark on top of each WMR. In view ofthe low resolution of the camera (320 240× pixels), the read-ings of the robots’ positions and heading angles are subjectto tolerances. In practice, for a stationary WMR, the toler-ance of position will be about three pixels, whereas that ofthe heading angle will be even greater (sometimes morethan 20°). Errors of input signals are therefore inevitable.The situation will be further worsened if the designed con-troller requires measured linear and angular velocities as in-puts. This is because the velocity is estimated by dividing a
pixel count by time, which is inherently inaccurate when thepixel reading itself has errors.
Some researchers have proposed the use of sliding modecontrol for the WMR [6]. Although this controller can pro-vide a faster response, the structure of the sliding controlleris quite complex and the computational demand is alsohigh. Moreover, the derivation of the controller in [6] is sub-ject to the following assumptions: 1) the system states usedfor the controller can be measured exactly, and 2) the head-ing angle of the robot and the angle coordinate cannot beperpendicular to each other.
In view of the difficulties mentioned above, a fuzzylogic controller (FLC) is proposed in this article. This FLCis designed based on a simple P-controller (proportional
April 2003 IEEE Control Systems Magazine 61
Figure 2. Wheeled mobile robot.
RFTransmitter
and Interface
WholeSystem
Commands( and )v vL R
Hardware
Software
Database forGame Strategy
and Game Rules
Extraction ofPosition for All
WMRs and the Ball
Host Computer
Controller
Video Cameraand Capture
InterfaceImage
RF Data Link
RFReceiver
MCU(89C52)
Velocity Commands
Motor PIDControllers
Left and RightMotors
Wheeled Mobile Robot 1
Wheeled Mobile Robot 2
Wheeled Mobile Robot 3
Voltages
Velocities
Figure 3. Block diagram of closed-loop control system.
controller) that is able to control the robot practically.The P-control law is given by
u =
=
+−
v
v
k d k
k d kl
r
d e e
d e e
θ
θ
θθ
,(4)
d x x y ye r c r c= − + −( ) ( )2 2 ,
θ θ θe r= − ,
where vl and vr are voltages applied to the left-wheel andright-wheel motors, respectively; kd is the gain for the errordistance de between the current and the desired positions;kθ is the gain for the error angleθe between the current anddesired orientations; and xr , yr , and θr are the referencehorizontal position, vertical position, and heading angle,respectively. Using the control law of (4), the P-controllercontrols the translational and rotational motions of the ro-bot. In particular, when θe = 0, the P-control law becomesv v k dl r d e= = , which results in translational motion only. Onthe other hand, when de = 0, we have v kl e= θθ andv kr e= − θθ , which results in a rotational motion only. The ad-vantages of this P-controller include: 1) the control law issimple, 2) the effect of the input error tolerance is reducedas only two system states are used, and 3) the plant modelneed not be known. However, the values of kd and kθ have tobe determined based on trial and error. They need not beoptimal but are obtained as a tradeoff between speed andstability.
To improve the performance, the proposed FLC [4]-[5] isused. It incorporates expert knowledge into the controllerdesign process using some linguistic rules. Such an FLC is anonlinear controller that retains the advantages of theP-controller but with an adaptive gain for each state variableso that a quick response can be achieved. Since the velocityand acceleration are not used as inputs, error accumulationis not a problem.
Fuzzy controllers [7]-[8] have been developed for WMRpath tracking. The main difference between the previouswork and ours is that in the former, consequent parts of thefuzzy rules are fuzzy terms with triangular-shaped member-ship functions, whereas rule consequents in our case areP-controllers. In [8], simulation results but no experimental
results are provided. Shooting action and obsta-cle avoidance were also achieved in [7]-[8].These two actions can also be achieved by ourproposed fuzzy controller with a planned path.For instance, the shooting action can beachieved if a path that passes through the balland avoids all the obstacles is planned for therobot. The target point of the path is at the posi-tion of the ball.
Fuzzy Logic ControllerA fuzzy controller having the following four rules, which aredesigned based on human knowledge, is proposed to con-trol the WMR.
Rule 1: IF de is small and | |θe is small, THEN
u u=
= =
+−
v
vd
dl
r
e e
e e1
0 25 012
0 25 012
. .
. .
θθ
.
Rule 2: IF de is small and | |θe is large, THEN
u u=
= =
+−
v
vd
dl
r
e e
e e2
0 25 0 25
0 25 0 25
. .
. .
θθ
.
Rule 3: IF de is large and | |θe is small, THEN
u u=
= =
+−
v
v
d
dl
r
e e
e e3
0 8 012
0 8 012
. .
. .
θθ
.
Rule 4: IF de is large and | |θe is large, THEN
u u=
= =
+−
v
vd
dl
r
e e
e e4
0 8 0 25
0 8 0 25
. .
. ..
θθ (5)
The gains of this fuzzy controller are manually fine-tunedbased on performance with the real system. The input mem-bership functions are defined in Figure 4. They are defined as
m dd
ee
11( ) min max , ,= −−
080
1601 ,
(6)
m d m de e12 111( ) ( )= − , (7)
| |( ) | |m e
e21
180
180θ
θ=
− +,
(8)
| |( )m me e22 211θ θ= − ( ). (9)
The max( )⋅ and min( )⋅ functions in (6) are to restrict the valueof m de11( ) to lie between zero and one even when the value ofde is larger than 80. As the maximum value of | |θe is 180, the
62 IEEE Control Systems Magazine April 2003
1.0
Input (| |)θe0 180
1.0m11 m12m21 m22
0 80Input ( )de
Small Large Small Large
–80
Figure 4. Input membership functions.
value of m e21( )θ will lie between zero and one, andthe max( )⋅ and min( )⋅ functions are not neces-sary. The grades of memberships,w1 tow4 , of rule1 to rule 4, respectively, are defined as follows:
w m m1 11 21= × , (10)
w m m2 11 22= × , (11)
w m m3 12 21= × , (12)
w m m4 12 22= × . (13)
The output of the FLC is given by
uu
= =
=
∑
∑
w
w
i ii
ii
1
4
1
4 ,
(14)
which shows that the controller has adaptive gains with re-spect to different operational regions.
Experimental ResultsWe developed a fuzzy control program using Visual C++ ver-sion 1.52 and implemented it on a Pentium II 450-MHz com-puter. The dimensions of each robot are 7.5 cm × 7.5 cm × 7.5cm. It has two driving wheels on two sides and an omni-directional passive wheel (rolling ball) at the bottom. TheCCD camera has a resolution of320 240× pixels and a rate of30 frames per second. Hence, the sampling period for the ro-bot to get the position and heading angle information is 33ms. The parameters of the WMR used in this article are asfollows: m = 0.45 kg, r = 0.02 m, R = 0.07 m. Because the pro-posed FLC is designed based on a model-free approach, thevalue of I is not needed. The maximum linear velocity of theWMR is 1.48 m/s. The maximum torque is 2.1 N-m. The di-mensions of the football stadium are 150 cm × 130 cm.
The experiment has two parts. The first part is positioncontrol, and we compare the P-controller of (4) with the pro-posed fuzzy controller of (14) for controlling the sameWMR. The second part is path-tracking control, and wecompare the performance of the proposed FLC with that of apublished sliding mode controller [6], which used a robot ofthe same specification.
Position ControlTo test the position control, the initial position of the WMRis (260, 200) (indicated by the white mark in Figure 5), andwe move it to the target position (xr , yr ) = (40, 40). (Positions
are given in units of pixels throughout this article.) The ini-tial heading angle of the WMR is about −90° (i.e., it faces thebottom side of the platform of Figure 5(a)). The fuzzy con-troller of (14) is employed to perform the position controltask. The error distance de used by the fuzzy controller is thedistance between the current position of the WMR and thetarget position, and the error angleθe is the angle of the cen-ter of the WMR toward the target point measured horizon-tally. For comparison purposes, a P-controller is employedto perform the same task. The gain for the error distance kd
is 0.25, and the gain for the error angle kθ is 0.12 for the tradi-tional P-controller of (4). The gains of the traditional P-con-troller are obtained by trial and error. If the gains are set toolarge for the P-controller, the system will be unstable.
The trajectories of the WMR are captured. Figure 5(a) and(b) shows the actual images of the WMR under the control ofthe traditional P-controller and the proposed fuzzy control-ler, respectively. The trajectories are plotted in Figure 6. Thesolid line corresponds to the fuzzy-logic-controlled WMR,while the dotted line is from the P-controlled WMR. The plot-ted data are extracted from the image frames. The action
April 2003 IEEE Control Systems Magazine 63
(a) (b)
Figure 5. Position control of the WMR: (a) trajectory of P-controlled WMR and(b) trajectory of fuzzy-logic-controlled WMR.
0 50 100 150 200 250 3000
50
100
150
200
Trajectory of Coordinate [Pixels]x
Traj
ecto
ryof
Coo
rdin
ate
[Pix
els]
y
Figure 6. Trajectory plot of the WMR controlled by the FLC (solidline) and the P-controller (dotted line).
time of the fuzzy-logic-controlled WMR from starting to sendcommands to arriving at the desired point is about 2 s. It istwice as fast as the P-controlled WMR.
Tracking ControlTo test the path-tracking control, we use the reference pathsstated in [6], which employed a sliding mode controller to real-ize path tracking. First, we test the tracking performance for astraight-line reference path at an angle of about 45°. The initialposition of the WMR is (183, 31), and the initial reference pointis (30, 6). The reference path is a straight line governed byy t x tr r( ) ( )= −24. In practice, the reference path is generatedas discrete points depending on t. In this example, we putx t tr ( ) = , and the reference path is a sequence of 201 points of( ( ), ( ))x t y tr r = [(30, 6), (31, 7), ... (230, 206)] that are generatedone by one at each sampling period (33 ms). The initial head-ing angle is 135° (i.e., the WMR is perpendicular to the refer-ence path). The fuzzy controller (14) is employed to performthe path-tracking task. Effectively, every generated point is anew destination of the WMR after a sampling period. Figure7(a) shows the image of the trajectory. Figure 7(b) shows theplot of the reference (dotted line with “+”) and the actual (solidline) trajectories. Figure 7(c) and (d) shows the trajectory er-rors of the WMR’s x-coordinate and y-coordinate with respectto time, respectively. Arrows in Figure 7(b) indicate the move-
ment direction of the WMR. Figure7(b) shows that the WMR can reachand follow the reference path even ifthe initial position of the WMR is faraway. It can be seen that the trackingtime is about 3 s, whereas the pub-lished controller’s tracking time isabout 8 s.
Next, we implement path trackingfor a curved trajectory. The initial po-sition of the WMR is (39, 89), and theinitial reference point is (30, 28). Theinitial heading angle is −45°. The ref-erence path is approximately gov-erned by the following equations:
y t x t
xr r
r
( ) . ( . ( ))
. ( .
= −−223 947 0 0208
31 7517 0 0218
exp
exp ( ))
. ( . ( ))
. ( .
t
x t
xr
r
+− −
321982 0 0222
203 9485 0 0259
exp
exp ( ))
. ( . ( ))
t
x tr− −0 4451 67 8449exp
and
x t tr ( ) = .
This equation is derived from acurve-fitting function based on the
data published in [6]. The fuzzy controller (14) is again em-ployed to perform the path-tracking task. Experimental re-sults are shown in Figure 8. The tracking time is about 3 s,whereas the published controller’s tracking time is about 10s. It can been seen from the results of the straight and curvedpath tracking that steady-state errors are present. This is be-cause no integral action is taken in the proposed FLC for sim-plicity of its structure and low computational demand.Referring to Figures 7 and 8, the steady-state errors of the xand y trajectories are about five pixels, respectively. As theresolution of the camera is 320 240× pixels and the dimen-sion of the football stadium is 150 cm × 130 cm, thesteady-state errors of the x and y trajectories are about( )5 150 320× = 2.3 cm and( / )5 130 240× = 2.7 cm, respectively.These steady-state errors are acceptable in a soccer game.The trajectories correspond to the x-y position of the centerof the WMR, which has a dimension of 7.5 cm × 7.5 cm × 7.5cm. The steady-state errors will not cause the WMR to missthe ball, as its side dimension is large enough.
ConclusionA fuzzy logic controller has been proposed to control WMRsin a robot soccer game. A heuristic fuzzy logic controller hasbeen designed based on a model-free approach. Hardware ex-perimental results have been presented to verify that the FLC
64 IEEE Control Systems Magazine April 2003
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Figure 7. (a) Straight-line path tracking; (b) actual (solid line) and reference (dotted linewith “+”) paths (unit of x-axis and y-axis are in pixel); (c) trajectory error of x-coordinateagainst time; (d) trajectory error of y-coordinate against time.
can control a WMR. The performance of a fine-tuned P-con-troller has been compared with that of the proposed fuzzylogic controller. The response time of the fuzzy-logic-con-trolled WMR is two times faster. Good tracking control perfor-mance was also obtained from the proposed controller.
AcknowledgmentThe work described in this article was substantially sup-ported by a research grant from the Centre for MultimediaSignal Processing, Department of Electronic and Informa-tion Engineering, The Hong Kong Polytechnic University(Project A420).
References[1] J.H. Kim, A Booklet on MIROSOT’96 (Micro Robot World Cup Soccer Tourna-ment). MIROSOT Organizing Committee, Apr. 1996.[2] H. Kitano, Ed., RoboCup-97: Robot Soccer World Cup I. Berlin: Springer, 1998.[3] C. de Wit, B. Siciliano, and G. Bastin, Theory of Robot Control. London:Springer-Verlag, 1996.[4] C.C. Lee. “Fuzzy logic in control systems: Fuzzy logic control—Part I,” IEEETrans. Syst., Man, Cybernet., vol., 20, pp. 404-418, Mar./Apr. 1990.[5] C.C. Lee. “Fuzzy logic in control systems: Fuzzy logic control—Part II,”IEEE Trans. Syst., Man, Cybernet., vol., 20, no. 2, pp. 419-435, Mar./Apr. 1990.[6] J.M Yang and J.H. Kim, “Sliding mode control for trajectory tracking ofnonholonomic wheeled mobile robots,” IEEE Trans. Robot. Automat., vol. 15,pp. 578-587, June 1999.[7] M.J. Jung, H.S. Kim, H.S. Shim, and J.H. Kim, “Fuzzy rule extraction forshooting action controller of soccer robot,” in Proc. 8th IEEE Int. Conf. FuzzySystems (FUZZ-IEEE’99), Seoul, 1999, pp. 556-561.
[8] S. Ishikawa, “A method of autonomous mo-bile robot navigation by using fuzzy control,”Advanced Robot. J., vol. 9, no. 1, pp. 29-52, 1995.
T.H. Lee received the B.Eng. andM.Phil. degrees from the Depart-ment of Electronic and InformationEngineering, Hong Kong Polytech-nic University, in 1998 and 2002, re-spectively. He is currently workingas an electronics engineer for ChinaResources Semiconductor Co., Ltd.,Hong Kong. His current research in-terests include electronic circuitdesign, power electronics, and ro-bot soccer.
H.K. Lam received the B.Eng. andPh.D. degrees from the Depart-ment of Electronic and Informa-tion Engineering, Hong KongPolytechnic University, in 1995and 2000, respectively. He is cur-rently a research fellow in the De-par tment of E lectronic andInformation Engineering, HongKong Polytechnic University. Hiscurrent research interests includeintelligent control and systems,computational intelligence, ro-
bust control, and robot soccer.
Frank H. Leung received the B.Eng. and Ph.D. degrees inelectronic engineering from Hong Kong Polytechnic Univer-sity in 1988 and 1992, respectively. He joined Hong KongPolytechnic University in 1992 and is now an associate pro-fessor in the Department of Electronic and Information Engi-neering. He has published more than 100 research papers oncomputational intelligence, control, and power electronics.At present, he is actively involved in research on the intelli-gent multimedia home and eBook. He is a reviewer for manyinternational journals and has helped organize many inter-national conferences. He is a member of the IEEE, a Char-tered Engineer, and a corporate member of IEE.
Peter K.S. Tam received the B.E., M.E., and Ph.D. degreesfrom the University of Newcastle, Newcastle, Australia, in1971, 1973, and 1976, respectively, all in electrical engineer-ing. From 1967 to 1980, he held a number of industrial and ac-ademic positions in Australia. In 1980, he joined Hong KongPolytechnic University as a senior lecturer. He is currentlyan associate professor in the Department of Electronic andInformation Engineering. He has participated in the organi-zation of a number of symposiums and conferences. His re-search interests include signal processing, automaticcontrol, fuzzy systems, and neural networks.
April 2003 IEEE Control Systems Magazine 65
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