One simple way to decrease the sensitivity of sliding mode controller to the parametric uncertainties and external disturbances is using of high control gain which decrease also the reac
Trang 11 INTRODUCTION
A robot manipulator is a nonlinear system with high
coupling term whose dynamics consists of uncertainty and
encountered with payload changes, friction and disturbance
[1] On the other hand, sliding mode control (SMC) as a
nonlinear technique with the capabilities of robustness
against the model uncertainties and ability of the disturbance
rejection has been considered in many researches [2-4]
Although the robustness of the SMC is one of its main
characteristics, this is achieved only in the sliding phase and
the system is sensitive to the structured uncertainties and
external disturbances in the reaching phase to the sliding
surface Therefore, different approaches for improving the
performance of the SMC has been proposed which one of
them is intelligent control method such as fuzzy control
system [5-7] Because of the relations between SMC and
fuzzy control, [8], the combination of these two approaches
has been considered as a research topic in last years [9-13]
such that the advantages of both approaches can be used
One simple way to decrease the sensitivity of sliding mode
controller to the parametric uncertainties and external
disturbances is using of high control gain which decrease
also the reaching time and tracking error However, high
control gain increases the oscillations in the control signal
that may lead to the excitation of high frequency unmodeled
dynamics which is an undesired phenomenon To overcome
this drawback, the fuzzy logic can be used for tuning of this
gain In this regard, in [14], a nonlinear sliding surface and
fuzzy logic have been used in the design of a fuzzy terminal
SMC for a robot manipulator Also, a fuzzy variable sliding
surface based method has been proposed in [15] in order to
improve the tracking performance In [16], in addition to
using variable sliding surface, the idea of fuzzy gain tuning
and boundary layer has been presented to achieve more improvements
In this paper, in addition to using the integral term in the sliding surface, [17, 18], at first, the SMC witch is including PID part is designed and its stability guarantee is proved in a lemma Then, in order to improve the controller performance, a fuzzy system is used to tune the gain of reaching phase and also PID part gain Thus, a chattering free SMC is achieved in which the tracking error and reaching time to sliding surface has been reduced without need to variable sliding surface
The reminder of the paper is organized as follows In the section 2, the mathematical model of the robot manipulator
is given The SMC including the PID loop to which is denoted as SMC-PID is presented in section 3 The design
of fuzzy SMC-PID is described in section 4 In section 5, the simulation results are provided and finally, summary and some conclusions are presented
2 THE SYSTEM MATHEMATICAL MODEL The dynamical equation of an n-link robot manipulator in the standard form is as follows [1]:
τ
+ + +C q q q G q d q
q
where M(q)∈Rn×n is a symmetry and bounded positive definite matrix which is called inertial matrix Moreover,
n
R q q
q , , ∈ are the position, velocity, and angular acceleration of the robot joint, respectively The matrix
n n
R q q
C( , )∈ × is the matrix of Coriolis and centrifugal forces such that the matrix H(q)−2C(q,q)is asymmetry, i.e., for a nonzero n×1 vector x we will have:
Sliding Mode PID-Controller Design for Robot Manipulators by Using
Fuzzy Tuning Approach
Mohammad Ataei1, S Ehsan Shafiei2
1 Electronic Department- Engineering Faculty, University of Isfahan, Isfahan, Iran
E-mail: mataei1971@yahoo.com
2 Electrical and Robotic Engineering Faculty, Shahrood University of Technology, Shahrood, Iran
E-mail: sehshf@yahoo.com
Abstract: In this paper, a chattering free sliding mode control (SMC) for a robot manipulator including PID part with a
fuzzy tunable gain is designed The main idea is that the robustness property of SMC and good response characteristics
of PID are combined with fuzzy tuning gain approach to achieve more acceptable performance For this purpose, in the
first stage, a PID sliding surface is considered such that the robot dynamic equations can be rewritten in terms of sliding
surface and its derivative and the related control law of the SMC design will contain a PID part The stability guarantee
of this sliding mode PID-controller is proved by a lemma using direct Lyapunov method Then, in the second stage, in
order to decrease the reaching time to the sliding surface and deleting the oscillations of the response, a fuzzy tuning
system is used for adjusting both controller gains including sliding controller gain parameter and PID coefficient
Finally, the proposed methodology is applied to a two-link robot manipulator including model uncertainty and external
disturbances as a case study The simulation results show the improvements of the results in the case of using the
proposed method in comparison with the conventional SMC
Key Words: Sliding Mode Control, Robot manipulator, PID control, Fuzzy control, Lyapunov theory
Trang 20 )]
, (
2
)
(
[H q − C q q x=
gravity vector, τd∈Rn is the bounded disturbance vector
such that τd ≤TD andτ∈Rn is the control input vector
In the following,H(q), C(q,q) and G(q) are shown by
H, C, and G, respectively
3 SLIDING MODE CONTROL WITH PID
The objective of tracking control is design a control law for
obtaining the suitable input torque τ such the position
vector q can track the desired trajectoryqd In this regard,
the tracking error vector is defined as follows:
q q
e= d − (2)
In order to apply the SMC, the sliding surface is considered
as the relation (3) which contains the integral part in
addition to the derivative term:
+ +
=e e tedt s
0 2
1 λ
λ (3) whereλi is diagonal positive definite matrix Therefore,
0
=
s is a stable sliding surface and e→0as t→∞ The
robot dynamic equations can be rewritten based on the
sliding surface (in term of filtered error) as follows:
τ
+ +
−
= Cs f d s
Where
G edt e
q C e e q
M
f = ( d + + )+ ( d + + t )+
0 2 1 2
(5) Now, the control input can be considered as follows:
) sgn(
f + v +
=
τ (6)
Where
G edt e
q C e e q
M
0 2 1 2
+
(7)
+ +
K
0
λ
tracking loop, and Kv,K are diagonal positive definite
matrices and are defined such that the stability conditions
are guaranteed The sgn(s) is also the sign function
We have also:
F G edt e
q C e e q
M
~
0 2 1 2
λ
(8) where f~= f − fˆ , M~ =M −Mˆ , C~=C−Cˆ ,and
G
G
G~= − ˆ F can also be selected as the following
relation:
G edt e
q C e e q
M
(
~ ) (
~
0 2 1 2
+
(9)
In order to reach the system states ( ee, )to the sliding
surface s=0in a limited time and remain there, the control
law should be designed such that the following sliding condition is satisfied [2]:
2
1
s s Ms
s dt
η
−
< for s≠0 (10) This subject is proved in the following lemma
Lemma- In the SMC design of a system with dynamic equation (1) and sliding surface (3), if the control input τ is selected as (6), by considering F as (9) and
) , , , (K11 K22 Knn diag
components:
K
i D v
Then, the sliding condition (10) is satisfied by equation (4) Proof: Consider the following Lyapunov function candidate:
Ms s
2
1
= (12) Since M is positive definite, for s≠0we have V >0and
by differentiating of the relation (12) and regarding the symmetric property of M, it can be written:
s M s s M s
V = T + T 2
1
(13)
By substituting (4) in (13) and considering that
0 ) 2 (M − C s=
) (
) (
2 1
τ τ
τ τ
− +
=
− + +
−
=
d T
d T
T T
f s
f s Cs s s M s V
(14)
By replacing the relation (6) in (14), Vcan be rewritten as:
=
−
− +
=
−
−
− +
=
n i
i ii v
d T
v d
T
s K s
K f
s
s K s K f f
s V
1 )
~ (
)) sgn(
ˆ (
τ
τ
(15)
Since the following inequality (16) is valid and regarding the relation (11), we have:
s K f
T s K
F+ v + D ≥ ~+τd − v (16)
i i v d
K ≥ [~+τ − ] +η (17) Finally, it can be concluded that:
=
−
≤ n i i
i s V
1
η (18)
This indicates that V is a Lyapunov function and the sliding condition (10) has been satisfied
The use of sign function in the control law leads to high oscillations in control torque which is undesired phenomenon and is called chattering To overcome this drawback, there are some solutions which one of them is using the following saturation function instead of sign function in the discontinuous part of the control law:
Trang 3≤
−
<
<
−
≥
=
ϕ
ϕ ϕ ϕ
ϕ
ϕ
s
s s
s s
sat
1
1
(19)
By this, there is a boundary layer ϕ around the sliding
surface such that when the state trajectory reach to this layer
will be remaining there
4 THE DESIGN OF FUZZY SMC-PID
As it was mentioned before, by using a high gain in SMC
(K), the sensitivity of the controller to the model
uncertainties and external disturbances can be reduced
Moreover, a high gain in PID part of the control system
)
(Kv can reduce the reaching time to sliding surface and
tracking error However, increasing the gain causes the
increment of the oscillations in the input torque around the
sliding surface Therefore, if this gain can be tuned based on
the distance of the states to the sliding surface, a more
acceptable performance can be achieved In other words, the
value of gain should be selected high when the state
trajectory is far from the sliding surface and when the
distance is decreasing, its value should be decreased This
idea can be accomplished by using fuzzy logic in
combination with SMC to tune the gain adaptively
For this purpose, two-input one-output fuzzy system is
designed whose inputs are s and swhich are the distances
of the state trajectories to the sliding surface and its
derivative, respectively The membership functions of these
two inputs are shown in figure (1) The output of the fuzzy
system is denoted by Kfuzzand has been shown in figure
(2) For applying these gains to the control input, the
normalization factors N and Nvas the following relations
are used:
fuzz K N
K= ⋅ (20)
fuzz v
v N K
K = ⋅ (21) These factors can be selected by trial and error such that the
stability condition (17) is satisfied
0
0.2
0.4
0.6
0.8
1
input variable "s"
(a) Fig 1: The membership functions for a) input s
0 0.2 0.4 0.6 0.8 1
inpu variable sd
(b) Continue Fig 1: The membership functions for b) input s
0 0.2 0.4 0.6 0.8 1 0
0.2 0.4 0.6 0.8 1
output variable K
Fig 2: The membership functions of the output Kfuzz
Tab 1: The fuzzy rule base for tuning Kfuzz
PB
PS
Z
NS
NB
s s
B
S
M
B
B
N
B
M
S
M
B
Z
B
B
M
S
B
P
The maximum values of K and Kv are limited according to the system actuators power, and the minimum value of K should not be less than the provided amount in relation (17) The fuzzy base rule has been given in table (1) in which the following abbreviations have been used: NB: Negative Big; NS: Negative Small; Z: Zero; PS: Positive Small; PB: Positive Big; M: Medium For example, when s is negative small (NS) and sis positive (P), then Kfuzzis small (S)
5 THE CASE STUDY AND SIMULATION RESULTS
In order to show the effectiveness of the proposed control law, it is applied to two-links robot with the following parameters:
+
+ +
+
=
β γ
β
γ β γ
β α
2
2 2
cos
cos cos
2 )
(
q
q q
q
+
−
−
=
0 sin
sin ) (
sin )
, (
2 1
2 2 1 2
2 q q
q q q q
q q
q C
γ
γ γ
(23)
Trang 4+ +
=
) cos(
cos(
cos )
(
2 1 1
) 2 1 1 1 1
q q
q q q
q
G
γδ
γδ αδ
(24) where α =(m1+m2)a12 , β=m2a22 , γ =m2a1a2 ,
1
a
g
=
δ , and m1, m2, a1=.7, a2 =.5are the masses
and lengths of the first and second links, respectively The
masses are assumed to be in the end of the arms and the
gravity acceleration is consideredg=9.8 Moreover, the
masses are considered with 10% uncertainty as follow:
2 ,
4 ,
2 2
2 2
1 1
1 1
0
0
≤
∆
∆ +
=
≤
∆
∆ +
=
m m
m m
m m
m m
(25)
0
0
m , and Mˆ , Cˆ , and Gˆ are estimated The desired state trajectory is:
−
=
t
t
qd
π
π
cos 2
cos 1
(26) and the disturbance torque is considered as follows:
=
t
t
d
π
π τ
2 sin 5 0
2 sin 5 0
(27)
which leads to =
5 0
5 0
D
The design parameters are determined as follow:
=
15 0
0 15
1
40 0
0 40
2
λ (28) The values of the ϕ and η are selected as
167
0
=
1 0 1 0
=
η Moreover, the factors N
and Nvare selected as follow:
=
5 0
0 50
10 0
0 5
v
N (29)
In order to show the improvement due to the proposed
method of this paper (Fuzzy SMC-PID), the simulation
results of applying this method are compared with the
related results of the conventional SMC The tracking error
and control law in the case of conventional SMC have been
shown in figures (3) and (4), respectively The
corresponding graphs for the case of applying fuzzy
SMC-PID are also provided in figures (5), and (6)
-0.05
0
0.05
0.1
0.15
time(sec)
-0.5
0
0.5
1
1.5
2
time(sec)
Fig 3: The tracking errors in the case of using conventional SMC
-50 0 50 100 150
time(sec)
-50 0 50 100
time(sec)
Fig 4: The control inputs in the case of using conventional SMC
-0.05 0 0.05 0.1 0.15
time(sec)
-0.5 0 0.5 1 1.5 2
time(sec)
Fig 5: The tracking errors in the case of using Fuzzy SMC-PID
-100 0 100 200
time(sec)
-100 -50 0 50 100
time(sec)
Fig 6: The control inputs in the case of using Fuzzy SMC-PID
As it is seen in these figures, the proposed fuzzy SMC-PID has faster response and less tracking error in comparison with conventional SMC In order to show more clearly the difference between the tracking errors in two cases, the enlarged graphs have been provided in figures (7) and (8)
Trang 50 2 4 6 8 10
-0.01
-0.005
0
0.005
0.01
time(sec)
-5
0
5x 10
-3
time(sec)
Fig 7: The enlargement of the tracking errors in the case of using
conventional SMC
-5
0
-4
time(sec)
-1
-0.5
0
0.5
-3
time(sec)
Fig 8: The enlargement of the tracking errors in the case of using
Fuzzy SMC-PID
6 CONCLUSION
In this paper, design of a sliding mode control with a PID
loop for robot manipulator was presented in which the gain
of both SMC and PID was tuned on-line by using fuzzy
approach Then the stability guarantee of the system was
proved by direct Lyapunov method The proposed
methodology in fact tries to use the advantages of the SMC,
PID and Fuzzy controllers simultaneously, i e., the
robustness against the model uncertainty and external
disturbances, quick response, and on-line automatic gain
tuning, respectively Finally, the simulation results of
applying the proposed methodology to a two-link robot
were provided and compared with corresponding results of
the conventional SMC which show the improvements of
results in the case of using the proposed method
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