Intelligent Control and Automation, 2011, 2, 364-370
doi:10.4236/ica.2011.24041 Published Online November 2011 (http://www.SciRP.org/journal/ica)
An Adaptive Fuzzy Controller for Trajectory Tracking of
Robot Manipulator
Amol A. Khalate, Gopinathan Leena, Goshaidas Ray
Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, India
Received June 24, 2011; revised July 15, 2011; accepted August 20, 2011
Abstract
In this paper, an adaptive fuzzy control algorithm is proposed for trajectory tracking of an n-DOF robot ma-
nipulator subjected to parametric uncertainty and it is advantageous compared to the conventional nonlinear
saturation controller. The asymptotic stability of the proposed controller has been derived based on
Lyapunaov energy function. The design procedure is straightforward due to its simple fuzzy rules and con-
trol strategies. The simulation results show that the present control strategy effectively reduces the control
effort with negligible chattering in control torque signals in comparison to the existing nonlinear saturation
controller.
Keywords: Parametric Uncertainty, Lyapunov’s Stability, Adaptive Control, Fuzzy Control, Robotic
Manipulator
1. Introduction
The mathematical model of a dynamic robot manipulator
involves highly nonlinearities, strong coupling and un-
certain system dynamics [1-4]. To overcome this prob-
lem, various adaptive control strategies have been
adopted for such uncertain systems including SISO and
MIMO systems with guaranteed performance. The pa-
rametric uncertainty mainly occurs in robotic manipula-
tor due to the variation in payloads and mass of links.
Many efforts have been made in developing control
scheme to achieve precise tracking controller of a robot
manipulator in presence of parametric uncertainty. The
adaptive control and nonlinear saturation control are the
two major approaches in dealing the parametric uncer-
tainty of a robot manipulator [5,6].
Fuzzy control has shown a great potential since it is
able to handle uncertainty using the programming capa-
bility of human control behavior. Many results have been
published in area of design and stability of fuzzy control
systems [3,7,8]. Efforts have been made to improve
tracking performance of fuzzy controllers with adaptive
algorithms, which are well known as adaptive fuzzy ap-
proach [9]. A vast amount of research work on adaptive
fuzzy control of nonlinear dynamic systems can be found
in [10] and references cited therein.
In this paper, the possibility of replacing the additional
control law given in [2] by an equivalent control law
based on fuzzy-logic approach has been explored. Sub-
sequently, an equivalent adaptive fuzzy controller is
proposed to design a robust control law for trajectory
tracking of a two-link robot manipulator with reduced
control effort as well as reduced chattering in control
force. Initially fuzzy controller is designed from a simple
fuzzy IF-THEN rule and then an adaptive control law is
adopted to update the parameters of the fuzzy controller
during the adaptation procedure. Now, this adaptive
fuzzy controller along with simple PD control term is
used to construct the final control torque for tracking
control of robot manipulators joints.
2. Dynamic Model and Robust Control
Design
The dynamic equation of n-DOF robot manipulator is
governed by [11]






, with ,
nn
Mqq Cqqqgqq 

 
  (1)
where


M
q is nn

inertia matrix, the vector


qq

,Cq represents centrifugal and coriolis forces, and


g
q is the vector of gravitational forces and the matrix






2q qq

,q M
,

NqC is skew-symmetric. The
above dynamic Equation (1) is linearly parameterizable
as

A. A. KHALATE ET AL.365
,,
 
,
M
qq CqqgqYqqq


  (2)
where

is a constant p-dimensional parameter vector
and is an matrix of known functions of the
generalized coordinates and their higher derivatives.
Ynp
It is assumed that there exists an unknown bound on
parametric uncertainty such that
0



 
 (3)
For any specific trajectory, are the desired
positions, velocities, accelerations whereas the position
and velocity errors are defined as d, d
,,
ddd
qqq

qq
qqqq




.
The adaptive algorithm [1] is very useful for dealing the
parametric uncertainties involve in a robot manipulator.
The reference velocity and acceleration of the following
quantities are defined as
,
rd rd
qq qqq 

 q 
(4)
and reference velocity error is given as
r
qq qq   

 
(5)
The nominal control law is given as
 

00 00
0
,
,,,
rr
rr
M
qqC qqqg qK
Y qqqqK



 

 
  (6)
where 0

represents fixed robot parameter, these pa-
rameters are not updated or changed with time. The gain
matrices
K
and  are positive definite diagonal ma-
trices.
Now, we define the control torque

in terms of
nominal control vector 0

as



0
0
,, ,
,,,
rr
rr
Yqqqq u
Yqqqqu K





 
 
(7)
where is an additional control input that will be de-
signed to achieve robustness to parametric uncertainty
represented by
u

 . Substituting the control law (7) into
(1) and after simplification we have,
 

,,,,
rr

M
qC qqKYqqqqu    

 (8)
3. Simple Fuzzy Logic Control
In [2], the additional input is defined as
if
if
TT
T
TT
YY
Y
uYY















(9)
where 0

 and the upper uncertainty bound is defined
as
12
2
0
1
p
i
i
 





 (10)
The parameter

indicates the measure of uncer-
tainty that may leads to a higher controller gain. To
overcome this problem different weights or gains can be
assigned to each component of and it is assumed
there exists an upper bound on each parametric uncer-
tainty as

ut
, 1,2,,.
ii
i  
p (11)
The control law (9) can be modified following the dis-
cussion in [2] as
if
;1,2,,
if
i
iii
i
i
i
iii
i
ui














.p (12)
where i

denotes the element of the vector
th
iT Y

,
i

is element of
th
i

and is element of
(see (9)).
i
uth iu
It can be seen if i

is chosen very small then the
nonlinear saturation control (12) will act as an ON-OFF
controller. It is possible to avoid hard switching and also
to achieve a better performance if we grade i

in num-
ber of linguistic terms.
Therefore, we need to construct additional control in-
put with the help of “r” numbers of IF-THEN fuzzy
rules to achieve robustness against the parametric uncer-
tainty. The jth rule is given as:
u
Rule j:
IF i

is
j
i
M
THEN is
i
u
j
i

(13)
where, i

and
j
i
M
,1,2,,, 1,2,,ipjr


are the
premise variables and the fuzzy terms, respectively; r is
the number of IF-THEN rule;
j
i

is the singleton fuzzy
inference. The crisp value of i can be obtained after
weighted average deffuzzification,
u
 
1
rjj T
iiiiii
j
uw W
i
 


 (14)
and
 


1
j
i
k
i
i
M
j
ii r
i
M
k
w







;
  
1
1,0 1
rjj
ii ii
j
ww 




for all j

12
,,T
r
iiii i
Wwww 





, .
12
,,T
r
iii i
 




where
j
i
M

is the grade of membership of i

in the

A. A. KHALATE ET AL. 366
fuzzy set
j
i
M
witht j
ii

.
4. Design of Adaptive Fuzzy Controller
In this section, we have proposed an adaptive fuzzy logic
controller as an additional control input to achieve ro-
bustness against the parametric uncertainty. We have
summarized the results of proposed work in following
theorem.
Theorem: Let the ith element of the additional control
input be defined as

T
iioi
u
i
W
W
(15)
where, iii is adaptive and
is fuzzy compensation (14) for

T
W

T
oi i
 T
ii
W
i

 with an adaptive law
as
ii
Wi



 (16)
and the gain matrix
K
in (8) is chosen such that
max
K
I

, (17)
where

max
2
max ,
if
if
i
T
ii oii
i
iT
ii oii
i
i
W
W



















(18)
Then for the trajectory tracking error of a robot ma-
nipulator is uniformly ultimately bounded along the tra-
jectory i.e. the tracking errors and its derivatives
converge to zero with application of additional control
law (15) in presence of parametric uncertainities.
q
q



ut
Proof: Let us assume that, the parameter vector oi

is selected in an adaptive way so that is the
adaptive compensation for
T
oioi i
W  
i

.
Let us choose Lyapunov function as


1
11
,22
p
TT
i
i
VMq 

 

i
i

(19)
The time derivative of V along the trajectories is
 
1
1
2
p
TT i
i
VMq Mq 

 


T
i


(20)
Using the property


2,
TMq Cqq




0






, we
have

1
p
TT T
ii
i
VKYu 

 


 (21)
Note that 12p12p
,
TY





 
 


11
pp
TT
ii ii
ii
VK u  

i

 


 


(22)
We know that 0iiii





, and i

can be
negative also so (23) reduces to



1
p
TT
iii ii
i
VK u  


 


(23)
Substituting the adaptive fuzzy control from (15)


1
T
pTT T
iiii oiiii
i
VK
WW






 





(24)

 
1
T
pTTT
ii oiiii iii
i
VK
WW

 






  (25)
Using the adaptive law (16)

 
1
T
pTTT
ii oiiii iii
i
VK
W








 

(26)

1
p
TT
ii oii
i
VK W  

 



(27)
There exists a small positive scalar 0
i

 satisfying
following inequality [12],
1,2,,.
T
ioiii
Wi   p (28)
Similarly, with 0
i

 we can write,
2
1,2,,.
T
ioiiii
i
Wi



 p (29)
Substituting (29) in (27),
2
1
p
Ti
i
VK



 
 (30)
Using max

for i

with


,,,
max the above Equation (30)
can be written as
max i  1, 2ip


max max
TT T
VKI KI


   

From (17), max 0KI


, hence and the
closed-loop system is asymptotically stable and the posi-
tion and velocity tracking errors and converge to
zero. Moreover, from (29) one can get (18). □
0V

q


q

The design procedure of adaptive fuzzy controller for
robot manipulator can be summarized in following steps:


,
(21) can be written as
Step 1: Obtain the parameter vector ,
according to fuzzy IF-THEN Rules.
, 1,2,,6
ii
Step 2: Using the adaptive law (16) obtain
to compute optimal parameter vector , 1,2,,6
ii


A. A. KHALATE ET AL.367
0i
Step 3: Compute the additional control input (15),
,
ii



T
iio


. 1, 2,, 6i
ii
uW1, 2,i
i
, . , 6
Step 4: Obtain

from (18) and get


max max i

 
for .
1, 2,,ip
Step 5: Choose the gain matrix such that
K
max
K
I


,, ,
rr
Yqqqq
.
Step 6: Obtain the final control torque according to (7)
 
0u K

 
22
1211
12
1
mlI
l




2
 
11
32
52
ml
ml
ml
,mm
12
,ll
,
cc
ll
.
5. Description of Two-Ink Robot
Manipulator
In order to show effectiveness of the proposed controller,
computer simulations have been performed with the
two-link planar manipulator model [11]. Figure 1 shows
this two link planar manipulator with the following
parameterizations are considered as
2
2 22
411
622
cc
cc
c
mlI
ml
ml



 


(31)
where 12
the masses of links 1 and 2 respectively,
are the lengths of link 1 and 2 respectively and
12
are the lengths where the masses of link 1 and 2
are assumed to be concentrated respectively. Using the
above parameters, the matrices

M
q, and
the vectors

,Cqq 


g
q,

in (1) are given as

Mq



,sin Cq
q



 
2 232
2
cos qq 



12 3
23 2
2cos
cos q









322
321
sin
0
qq
qq








(32)
 
3212
sin qq q  







(33)
Figure 1. Two-link planar robot.






451 612
612
cos cos
cos

g
qg qq
gq gqq
 


 








(34)


6
,,; Yqqq 


 (35)
where the component of are given as
ij
y

,,Yqqq




qq

r


gqq
 



 
111121 2
13212
2
22 12141
151161 2
21
cos 2
sin2 cos
cos cos
0
yq yqq
yqqq
qqqqy gq
yg qyg qq
y


 


 
 

 

1
221 2
2
232 1224
25261 2
cossin 0
0 cos
yqq
yqqqqy
yyg

 

 
 
(36)
The each component of in (7) is given
as

,, ,
rr
Yqqqq
 
 
 
 
1111212
1321 2
222 12141
151161 2
21
cos 2
sin2 cos
cos cos
0
rr
rr
rr
yq yqq
yqqq
qqqqqygq
yg qyg qq
y


 


 
 
 
 

2212
232121 124
25261 2
cossin 0
0 cos
rr
rr
yqq
yqqqqqy
yy

 

 
 
(37)
The unloaded manipulator parameters used in simula-
tion are given in Table 1. Using the values from Table 1
in (31) we can get i

as shown in Table 2.
We assume that due to unknown load carried by the
robot as part of second link, the parameters , 2c
and
2
m l
2
I
will change to 22 , 22cc mm ll

 and
22
I
I

, respectively. We will design an adaptive con-
troller that will ensure the asymptotic stability in pres-
ence of parameter uncertainties in the intervals
22 2
15
010; 00.5; 0
12
c
ml I. (38)
The nominal parameter vector 0

is shown in Table
3, is obtained after choosing the mean value for the range
of possible i

. The uncertainty bound for each parame-
ter i

is shown in Table 4. The design steps of the
proposed adaptive fuzzy controller are discussed below:
5.1. Controller Design
Step 1: The parameter vector , is the
fuzzy controller consequent can be obtained according to
fuzzy IF-THEN rules (13). Each premises variable i
, 1,2,,6
ii

is
described with help of five fuzzy terms NB, NS, Z, PS,

A. A. KHALATE ET AL. 368
Table 1. Parameters of the unloaded arm.
1
m 2 m 1 l 2 l 1c l 2c l 1
I
2
I
10 5 1 1 0.5 0.5 10/125/12
Table 2. θi for the unloaded arm.
1

2

3

4

5

6

8.33 1.67 2.5 5 5 2.5
Table 3. Nominal parameter vector θ0.
01

02

03

04

05

06

13.33 8.96 8.75 5 10 8.75
Table 4. Uncertainty bounds ρi.
1

2

3

4

5

6

5 6.25 6.25 0 5 6.25
and PB. Therefore, it requires five rules to describe for
each . These fuzzy IF-THEN rules are
as follows
,
i
u1, 2,, 6i
1) IF i

is NB THEN is NB;
i
u
2) IF i

is NS THEN is NS;
i
3) IF
u
i

is Z THEN is Z;
i
u
4) IF i

is PS THEN is PS;
i
u
5) IF i

is PB THEN u is PB.
i
These fuzzy rules are generated using the control
structure (12) of [2]. The membership functions for input
and output variables are shown in Figure 2.
Table 5 shows the parameters of membership func-
tions for input variables and Table 6 shows the parame-
ters of membership functions for output variables. Note
that there is no uncertainty in parameter 4

, i.e. 40


.
Therefore, the component of is zero.
4
u
,6
u
Step 2: i are computed using the
adaptive law (16). Now we can get the optimal parameter
vector as .
, 1,2,i

,
iii

i
0
Step 3: The additional control input is obtained based
on adaptive fuzzy control law (15), ,
.
1, 2,, 6

T
iioi
uW
i
1, 2,, 6i
Step 4: Now we will compute i

using (18) and
maximum value of it chosen as

max max i

 .
Step 5: The gain matrix is now updated so as to
meet the stability condition (17)
K
max
K
I

.
Step 6: The final control torque is obtained according
to (6)
 
0
,, ,
rr
Yqqqqu K

 
 
Figure 2. Membership functions for input and output vari-
ables.
Table 5. Parameters of membership functions for input
variable.
1i
a 2i a 3i a 4i a 5i a
1

−0.040 −0.020 0.000 0.020 0.040
2

−0.200 −0.100 0.000 0.100 0.200
3

−0.150 −0.075 0.000 0.075 0.150
5

−0.200 −0.100 0.000 0.100 0.200
6

−0.400 −0.200 0.000 0.200 0.400
Table 6. Parameters of membership functions for output
variable.
1i
b 2i b 3i b 4i b 5i b
1
u −5.000 −2.500 0.000 2.500 5.000
2
u −7.290 −3.645 0.000 3.645 7.290
3
u −6.250 −3.125 0.000 3.125 6.250
5
u −5.000 −2.500 0.000 2.500 5.000
6
u −6.250 −3.125 0.000 3.125 6.250
5.2. Simulation Results
Desired trajectories for both the joints are given as


1cos, sin
dd
qatqa


at (39)
 . Note that the
proposed control algorithm updates the gain
K
of PD
term unlike the previous algorithm presented in [2].
where 2π5a

and simulation results are obtained
with a sampling time T = 0.005 sec. Figures 3-6 show

A. A. KHALATE ET AL.369
01234
-150
0
150
300
5
(f)
Time (sec)
012345
-300
0
300
600
01234
-5
0
5
10
15
5
Nm Nm X10
- 3
(e)
(d)
(c)
012345
0
5
10
X10
- 3
012345
0
1
2
(b)
01234
0
1
2
5
(a)
Figure 3. System response base d on [2] using controller (12)
with ε = 0.1.
012345
0
10
20
30 (f)
012345
-75
0
75
150
Nm
(e)
01234
-150
0
150
300
5
(d)
Nm
012345
-5
0
5
10
15
X10
- 3
012345
-4
-2
0
2
X10
- 3
012345
0
1
2
(c)
(b)
01234
0
1
2
5
012345
0
10
20
30
(h)
(g)
(a)
Time (sec)
Figure 4. System response based on proposed adaptive
fuzzy controller (15) (manipulator is unloaded).
012345
-150
0
150
300
Time (sec)
(f)
012345
-300
0
300
600
(e)
012345
-5
0
5
10
15
(d)
(c)
012345
0
5
10
012345
0
1
2
X10
- 3
X10
- 3
(b)
(a)
012345
0
1
2
Figure 5. System response base d on [2] using controller (12)
with ε = 0.1 (Manipulator is loaded at time 2 sec).
012345
0
10
20
30
(h)
(g)
Time (sec)
012345
0
10
20
30 (f)
012345
-75
0
75
150
X10
- 3
Nm
(e)
012345
-150
0
150
300
Nm
(d)
012345
-5
0
5
10
15
X10
- 3
(c)
012345
-4
-2
0
2
012345
0
1
2
(b)
(a)
012345
0
1
2
Figure 6. System response based on proposed adaptive
fuzzy controller (15) (manipulator is loaded at time 2 sec).

A. A. KHALATE ET AL. 370
trol scheme is effective in reducing chattering in torque
control signal and simultaneously control effort is less in
comparison to the results in [2].
the system response of a two-link robot manipulator based
on nonlinear saturation controller [2] and proposed adap-
tive fuzzy controller (15) for two cases while the robot
manipulator is (i) unloaded (ii) loaded. Results are pre-
sented in the following sequences in the figures: joint
positions (joint-1 (1), joint-2 (2)), tracking errors (1
and 2), and input torques (1
q qq
q

and 2

) respectively. It
may be mentioned that the robot manipulator is loaded at
time 2 sec. i.e. the second link lifts some mass due to
which the parameter of the robot changes as given in
(38).
7. References
[1] J. J. E. Slotine and W. Li, “On the Adaptive Control of
Robotic Manipulators,” International Journal of Robotics
Research, Vol. 6, No. 3, 1987, pp. 49-59.
doi:10.1177/027836498700600303
[2] M. W. Spong, “On the Robust Control of Robot Manipu-
lators,” IEEE Transactions on Automatic Control, Vol.
37, No. 11, 1992, pp. 1782-1786. doi:10.1109/9.173151
The results of the proposed adaptive fuzzy controller
are compared with that of [2] (see (12)) where fixed PD
controller gains are considered as and
. In both the methods, it is observed that
the tracking error responses remain almost the same or-
der and insensitive irrespective of the payload variation.
Further it has been observed through simulation studies
that the proposed technique effectively alleviates or re-
duces the chattering effect in control signals. A signifi-
cant chattering in the control signal is noticed while a
robot arm takes a sharp turn under loaded condition.
Simulation result shows that the proposed controller ef-
fectively reduces the magnitude of input torque or in
other words effectively reduces the control effort com-
pared to the method discussed in [2]. It may be further
observed from the figures ((see Figures 4(g)-(h) and
Figures 6(g)-(h)) how the diagonal elements of gain
matrix

diag75 50K[3] A. B. Sharkawy, M. M. Othman and A. M. A. Khalil, “A
Robust Fuzzy Tracking Control Scheme for Robotic Ma-
nipulators with Experimental Verification,” Intelligent Con-
trol and Automation, Vol. 2, No. 2, 2011, pp. 100-111.
doi:10.4236/ica.2011.22012


diag40 15
K
(11 22 ,
K
K) of PD controller are updated adap-
tively.
[4] M. Galicki, “An Adaptive Regulator of Robotic Manipu-
lators in the Task Space,” IEEE Transactions on Auto-
matic Control, Vol. 53, No. 4, 2008, pp. 1058-1062.
doi:10.1109/TAC.2008.921022
[5] M. W. Spong, S. Hutchinson and M. Vidyasagar, “Robot
Modeling and Control,” John Wiley & Sons Inc., New
York, 2006.
[6] C. C. Cheah, C. Liu and J. J. E. Soltine, “Adaptive Track-
ing Control for Robots with Unknown Kinematics and
Dynamic Uncertainty,” International Journal of Robotics
Research, Vol. 25, No. 3, 2006, pp. 283-296.
doi:10.1177/0278364906063830
[7] T. H. S. Li and Y. C. Huang, “MIMO Adaptive Fuzzy
Terminal Sliding-Mode Controller for Robotic Manipu-
lators,” Information Sciences, Vol. 180, No. 23, 2010, pp.
4641-4660. doi:10.1016/j.ins.2010.08.009
6. Conclusions
[8] Z. Bingul and O. karahan, “A Fuzzy Logic Controller
Tuned with PSO for 2 DOF Robot Trajectory Control,”
Expert Systems with Applications, Vol. 38, No. 1, 2011,
pp. 1017-1031. doi:10.1016/j.eswa.2010.07.131
In this paper, an adaptive fuzzy control law is proposed
for trajectory tracking of robot manipulator with a view
to reduce the chattering effect in torque control signal.
The advantages of fuzzy and adaptive control strategies
are combined and subsequently the stability condition of
robot manipulator is derived based on Lyapunov theorem.
The implementation of proposed controller is very
straightforward due to the use of simple fuzzy rules and
control strategies. The gain of PD term is updated
with time and hence proposed adaptive control scheme
removes the disadvantage of using fixed large gain val-
ues in [2]. Simulation results show that the present con-
K
[9] L.-X. Wang, “Stable Adaptive Fuzzy Control of Nonlin-
ear Systems,” IEEE Transactions on Fuzzy Systems, Vol.
1, No. 2, 1993, pp. 146-155.
[10] G. Feng, “A Survey on Analysis and Design of Model-
Based Fuzzy Control Systems,” IEEE Transactions on
Fuzzy Systems, Vol. 14, No. 5, 2006, pp. 676-697.
[11] M. W. Spong and M. Vidyasagar, “Robot Dynamics and
Control,” Wiely, New York, 1989.
[12] L. X. Wang, “A Course in Fuzzy Systems and Control,”
Prentice-Hall, Englewood Cliffs, 1997.