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Fuzzy controller for flexible-link robot arm by reduced-order
techniques
J. Lin and F.L. Lewis
Abstract: The design and analysis of a large-scale control
system should be based on the best available knowledge instead of
the simplest available model when trcating uncertainties in the
system. Therefore, a large-scale system is better treated by
knowledge-based methods such as fuzzy logic, neural networks,
expert systems, etc. This paper concentrates on fuzzy logic using
the singular perturbation approach for flexible-link robot arm
control. To reduce the spillovcr effect, we will introduce a
singular perturbation approach to derive the slow and fast
subsystems. A composite control design is adopted. Therefore, a
two-time scale fuzzy logic controller will be applied to the
system. The fast-subsystem controller will damp out the vibration
of the flexible structure by an optimal control method. Hence, the
slow-subsystem fuzzy controller dominates the trajectory tracking.
We guarantee the stability of the internal dynamics by adding a
boundary-layer correction based on singular perturbations. Various
case studies are given to verify the control algorithm. It appears
that the fuzzy control method is quite useful in terms of
reliability and robustness.
1 Introduction
The control of robot arms with link flexibility belongs to a
class of problems that includes robots with joint flexibility,
large-scale space structures, and some industrial processes (e.g.
overhead gantry cranes). Control of mechanical manipulators for
tracking a desired trajectory is an extre- mely important problem
that becomes more complex when the robot possesses the flexibility
of multiple links. A flexible-link arm is a distributed parameter
system of infinite order, but due to onboard computer limitations,
sensor inaccuracy and system noise, it must be approxi- mated by a
lower-order model and controlled by a finite- order controller. The
so-called control spillover and observation spillover effects will
then occur, which under certain conditions can lead to instability
[ 1-31.
A survey of flexible arms is given in [4], where it is shown
that the flexibility effects impose serious limitations on what can
be achieved using standard rigid-arm control schemes. Many control
schemes have been proposed for flexible robot arms [5-SI.
To maintain reasonable computational loading, a control- ler
based on a reduced-order model has been proposed [9, 101. In recent
years, singular perturbation theory has been shown to be a
convenient strategy for reduced-order modelling. It is well known
that the dynamics of singularly perturbed systems can be
approximated by the dynamics of
0 IEE, 2002 IEE Pmceeding,s online no. 20020338 DUI:
10.1049/ip-cta:20020338 Paper received 30th October 2001 J. Lin is
with the Department of hkchanical Engineering, Ching Yun Institute
of Technology, 229 Chien-Hsin Road, Jung-Li City, Taiwan 320,
Republic of China F.L. Lewis is with the Automation and Robotics
Research Institute, The University of Texas at Arlington, Texas,
USA
IEE Proc.-Control Theory A&, Vol. 147, No. 3, May 2002
the corresponding rcduced-order and boundary-layer subsystems
for sufficiently small values of the singular perturbation
parameter. The aim is to simplify the software and hardware
implementation of the control algorithms while improving their
robustness. A composite control approach based on a two-time scale
model of the flexible- link arm has been derived [8, 111. This
allows the definition of a slow subsystem corresponding to the
rigid body, and a fast subsystem describing the flexible motion. A
slow control is designed for the slow subsystem and a fast control
is designed to stabilise the fast subsystem.
For many years, classical control engineers began their efforts
with a mathematical model and did not go any further in acquiring
knowledge about the system, Today, control engineers can use all of
the above sources of inforination. Aside from a mathematical model
whose utilisation is clear, numerical (input/output) data can be
used to develop an approximate model as well as a controller, based
on the acquired fuzzy IF-THEN rules
Recently, there has been increased interest in applying the
concepts of fuzzy set theory to flexible structural control. Fuzzy
controllers afford a simple and robust .framework to specify
nonlinear control laws that accom- modate uncertainty and
imprecision. Such controllers may be implemented using a fuzzy
mathematical model of the plant and controller. If linguistic
descriptions of the control are available or can be formulated,
filzzy controllers may be determined without a mathematical model.
Implemen- tations using a linguistic synthesis approach have been
proposed [12-151 and demonstrated to be applicable in theory and in
practice. A genetic algorithm-based approach and a neural network
approach have been suggested for adaptive or optimal tuning of a
fiizzy control systeni [ 16- 181. An approach that combines a
neural network and a fuzzy logic element to address actuator
dynamics, time delay, and higher modes of response has been
evaluated numerically
[121.
177
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The investigation examined the application of control algorithms
based on fuzzy logic to a class of hybrid structural control
systems [6, 19-21]. It included both analytical and experimental
verification of the fuzzy control algorithm. The guidelines for
implementing a fuzzy active control strategy for civil engineering
struc- tures are discussed in [22-241. This paper focuses attention
on the gap between a successful numerical example and the technical
design of the device. We show a rigorous approach to the
position/velocity tracking control of a general nonlinear multilink
flexible arm. This fuzzy logic/singular perturbation approach
brings together and employs the concepts of several of the papers
mentioned above.
DSP chip, AID, DIA,
2 Robot dynamics and sensor system design
2. I Robot dynamics A very convenient form for the approximate
dynamics of a flexible robot arm can be derived using the assumed
mode shape method in Fig. 1. Then, assuming a Bernoulli-Euler beam
model, the deflection of the elastic beam w(y, t ) can be expressed
as a summation of the infinite series terms
00
W(Y9 t ) = c 41(t)$,(rl) (1) r = l
qL(t) are generalised modal co-ordinates and (bl (y) are mode
shape functions that are dependent upon the bound- ary-value
problem (i.e. pinned-pinned, clamped-free, clamped-loaded, etc.),
where y represents the displacement along the neutral axis of the
link.
The components of the dynamic model should be explicitly
separated into matrix form to exhibit the inertia (M),
centrifugal/Coriolis/damping (D), stiffness (9, friction F(x, i),
and gravitational (G) forces.
(2) where B is the input matrix that depends on the clamped link
assumptions, and T is the control input torque. Here we define x=
[qo , 41,. . . , qn]T where qo is the rigid mode, 41,. . . , qI2
are the flexible modes, and n is the number of retained modes in
(1).
The definition of the D(x, k) coefficient matrix is not unique,
although it may be selected to yield an important skew-symmetric
property which is useful in robust control design. Because manual
symbolic expansion of the robot dynamic equations is tedious, time
consuming and error prone, an automated derivation process is
highly desirable. Therefore, a symbolic program [25] has been
written in MATHEMATICA (or MATLAB) to generate the dynamic equation
for a planar robot with arbitrarily assigned rigid
M(x)X + D(x, X)X + K(x)x + F(x, X) + G(x) = BT
DSP chip, AID, DIA, Dig. 110,
encoder interface
I tip mass
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decrementing a counter, as appropriate, with every pulse edge.
The measurement of the joint angle is expressed as
L qn _I
2.2.2 Flexible mode measurement: Many types of sensors have been
used to measure the vibration of a flexible beam. The most popular
has been the strain gauge, which measures the deflection of the
beam. The advantages of the strain gauges are: isolation of beam
variables from rigid rotations, no restrictions on work
positioning, high compatibility with harsh industrial envir-
onments, and low cost. The relationship between the strain 6 and
the generalised co-ordinates q,(t) is
where c is the curvature of the beam. We can expand (6) to
relate each flexible mode to the
measurement of strain at each location a, b, . . . , m. This
relationship can be presented in matrix form as
dx2 dx2 dx2
dx2
We can rewrite (7) as
(7)
The discussion of the flexible modes and strain gauge numbers is
indicated below.
(a) if n = m, [E] is a square matrix, then
1 [ q l q 2 . . . q,lT =-[c]--'6
(b) if 12 # m, [Z] is not a square matrix. From the pseudo-
inverse matrix we obtain
3 Trajectory generator
A typical generation scheme involves: generation of a set of
pass points in the task space of the robot, spline fitting
polynomials to the pass points in position plotted against
IEE Proc.-Control Theory Appl., Val. 147, No. 3, May 2002
the time state space, etc. However, this has some draw- backs.
Instead, we will use a model-based filter to generate a smooth time
history of position, velocity, and accelera- tion given the desired
end-point [26]. The poles of the filter are placed below the
resonant frequencies of the neglected higher modes so that the
joint trajectories do not cause unexpected resonance in the arm by
exciting the unmodelled dynamics. On the other hand, the filter
cut-off is selected above the frequencies of the flexible modes
retained in the robot model, to demonstrate that our controller
effectively controls these modes.
The trajectory generator is of the form
where c is the integral of the position error, e = q d - r. The
desired position set points are fed in through the reference input
r. Defining u(t) as
= -Kl,,,e- KP,,,qd - KV,,Jqd (10)
yields
= c[ &] + [ 81. (11) where KI ,,,, Kp,)! and Kv,,, are the
model proportional, integral, derivative (PID) gains, which are
selected for the desired tracking performance.
The output is defined as
4 Link-tip position control using 1/0 feedback linearisation
In this Section we will use input/output (I/O) feedback
linearisation to design a controller for the link-tip positions y E
R"'. We will complete the design in Section 5 by using singular
perturbation theory to design a controller that provides a
boundary-layer correction to stabilise the flex- ible modes (e.g.
the internal dynamics).
4.7 Input-output feedback linearisation Using standard 1/0
feedback linearisation techniques we differentiate y(t) in (4)
twice, then substitute for ij using ( 2 ) to obtain the
reduced-order system
y = u (13) The auxiliary control input u(t) is defined according
to the reduced-order computed torque (ROCT) control law
(14) System (13) consists of n, subsystems, each with two
integrators in series, and is said to be in Brunovsky canonical
form.
When a finite number of flexible modes is retained in the
dynamic model, matrix CM-'B is nonsingular. It has been shown that
as the number of modes approaches infinity (as
z = (CM-lB)-'[CM-'(Dq + Kq + F + G) + U]
179
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I integrator 1 integrator 2 reference integrator gain input
gain 1
Fig. 3 Block diagram of trajectory generator
in the exact model), this matrix becomes singular. In fact,
retention of a finite number of modes corresponds to 'approximate
feedback linearisation' of systems with an ill-defined relative
degree, and is sufficient for the design of controllers which will
perform adequately.
We define the tracking error e( t ) =yd(t) - y(t) , with yd(t)
the desired trajectory and choose K d > 0, Kp > 0. Then an
outer-loop PD control law for a trajectory following is
u = y d + K d e + K p e (15)
Unfortunately, this obvious selection for u(t) dooms the control
scheme. This is because, even though the ROCT control, consisting
of (15), and (14), decouples the e(t) subsystem from the remainder
of the plant and stabilises it, it almost certainly fails to
stabilise the remainder of the plant as the zero dynamics are
unstable [8, 261.
4.2 Inverse system dynamics To examine the detailed structure of
the closed-loop system after ROCT, we use partitioned matrices as
follows. We rewrite (2), neglecting friction for simplicity, as
Let us define the inverse of the mass matrix
We multiply (16) by (17) from the left, rearrange terms, and
write
i jT = -D$qr - D>kf - K;qf - G," + B:z i j f = -D$q, - D$qf -
K$qf - G; + B?z
( 1 8 ~ )
(18b) with
D:,. = H,D, + HrfDj., 0;i = HfiD, + Hff Dj.9
D> = HrPrf + H f D f D; = HfiDrf + *ffDrr
K; = H r f K f ,
G: = H,.,.G,, B: H,$, + HrfBf, B;! = HfiB, -I- Hj-B'
K; = H f K f Gf" = HpG,
For the general flexible-link robot arm case, BY is small
compared with B,. It is easy to show that a sufficient condition
guaranteeing the existence of (%)-I in terms of
180
the matrix-induced norm (e.g. maximum singular value [71) is
IlBfll < (llB;lH,lHr/ll)-' (19)
Performing a feedback linearisation on (1 8) amounts to
selecting
(20) z = (5;)-'(D;,.4, + D24f + K;qf + G," + U ) to obtain the
dynamics
q r = U (21)
i j f = -D$& - D$qf - K t q f - Gj +B$u (22) where u(t) is
an auxiliary input and the Schur complements are defined as
Db - DU -Ba 5" - 1 p - fi ,f( r ) DR a - 1 a
D; = 0; - B;(B,) Drf K; = K; - B;(B;)-*K;
~ f " = G; - B;(B;)-*G; 5$ = B;(B;)-'
A state-space representation of the dynamics is
I O 0
0 -D$ - K j - D j
The control law (20) is a refined expression for the ROCT
control (14), and system (23) is the complete closed-loop system
after ROCT. This is the internal dynamics after ROCT; the zero
dynamics are the internal dynamics evaluated along y(t)=O (i.e. in
this case q,(t)=O, &(t) = 0, u( t ) = 0).
The given rigid-mode trajectory yd(t) in (21) can be achieved by
appropriate selection of u(t) . Therefore, the function of the
inverse system is to construct the unique flexible-mode trajectory
associated with that rigid-mode trajectory. Selecting u(t) based
only on the desired rigid subsystem performance does not guarantee
a stable inverse system as the zero dynamics are not stable for the
flexible- link robot arm. Hence, we will show how to select u(t) to
achieve the desired rigid-mode performance as well as to
IEE Proc.-Control Theoty Appl., VOI. 147, No. 3, May 2002
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stabilise the inverse dynamics using a time-scale separation in
Section 5 .
5 Internal dynamics stabilisation using boundary-layer
correction
We now use singular perturbation theory to stabilise the zero
dynamics after 1/0 feedback linearisation. As will be seen, this
affords greater accuracy of the time-scale separa- tion assumption.
A typical procedure for singular perturba- tion theory is reviewed
in [lo]. To singularly perturb the feedback-linearised system, we
rewrite (2 1) and (22) as
q, = u (24)
ijf + D$q, + Dgqf + Kjq f + G j = B ~ u (25) For the singular
perturbation analysis to work, we must put (24) and (25) in the
standard form and solve for some fast variables. Since the flexible
subsystem is faster than the rigid one, we must extract an
appropriate parameter from the fast subsystem first.
We define
u = Hf - [HpB,. + HJpfl[H,,.B, + H,fBfl-lffrf (26)
$: = K"' = UKf (27) where K"' indicates the closed-loop
stiffness and K s ~ is the open-loop stiffness appearing in (16).
From [26], it can be seen that U = Hj,- - [Hp B, + Hfl Bf][H,.,.
B,. + H,.. B,-]-'H,.f is invertible.
In general, A,!@' is large, thus IC" UK, is of larger scale than
Kfl (numerical experiments will confirm this). There- fore, it is
more accurate to introduce singular perturbation based on IC'
rather than on Kjj. Therefore, we introduce a positive scaling
factor IC and factor K"' as
Kc' = I +Bf(q,, E ~ O U (30)
where for simplicity, we have dropped the superscript 'b ' . In
the sequel, we shall design a control u to let ( y , j) track ( y d
, >d) sufficiently closely and to stabilise the systems (29) and
(30).
We define the control input
u = i i + u j (3 1)
IEE Proc.-Control Theory Appl., Vol. 147, No. 3, May 2002
where a(t) is the slow control component and udt) is a fast
control component. Setting E = 0 yields the slow dynamics
j = G (32) and an algebraic relation in 5, j j , and 4
(33) = cl-
0 = -8pj - K < - (?f +BfU It should be noted that we use an
overbar to denote the evaluation of nonlinear functions-with c =
0.
Setting E = 0 and solving for r in (33) yields the slow manifold
equation
(34) =cL - 1
= (K ) (-Bj,.j - Gf + BfU) The dynamics (32) are valid on this
manifold.
We now select q1 r - 4 , q2 = E ( and write (30) as ~ i 2 =
-Dj,y - D p 2 - + 4 ) - Gf + B f u (35)
where a time-scale change of z = t / E results in
Setting E = 0 and substituting for 2 from (30) gives the fast
dynamics part of the following slow subsystem description of the
feedback-linearised arm:
Moreover, the fast subsystem is found to be
= 4-2 dS, d z
(39)
which is a linear system parameterised in the slow vari-
ables.
5.7 Tracking requirement In accordance with (37)-(39), uht) cgn
be chosen to give stable zero gynamics provided that (K"', Bj) can
be stabi- lised. If (K", Br> is controllable, there are no zero
dynamics, as uht) can be chosen to control [cy q;]?
As shown by the two reduced-order subsystems (37)- (39), a
composite control strategy can be pursued. The design of a feedback
control for the full system (29)-(30) can be split into two
separate designs of feedback controls i i and uffor the two
reduced-order systems. Formally, it can be given as
u = U + u f (40)
A modified tracking outputy has been used elsewhere [26]. Using
Tikhonov's theorem, we have
y = 7 + O(E) (41)
with t ( t ) given by (34) and O(E) denoting terms of order E.
Hence, we select a(t) for suitable tracking behaviour in
the slow subsystem (37). This corresponds to a relaxed tracking
requirement on the link-tip positions, since q,.(t) is not exactly
specified. This also makes a fast control term available to
stabilise the flexible modes.
181
- Then, udt) is selected to stabilise the fast part (38)-(39).
Since the fast subsystem is only a linearisation of the flexible
(internal system) dynamics induced by the slow manifold
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Table 1: Consequent table for fuzzy logic controller design
e
I 1 I I I I I I I I I -0.5 -0.4 -0.3 -0.2 4 . 1 0 0.1 0.2 0.3
0.4 0.5 -
U
Fig. 7 Membership ,function ,for Q(t)
change an existing fuzzy controller's architecture, i.e.
membership functions and/or rules. The fuzzy member- ship functions
for the e, and its derivative k are shown in Figs. 5 and 6. There
are seven fuzzy membership func- tions, that correspond to large
negative (LN), medium negative (MN), small negative (SN), zero
(ZE), small positive (SP), medium positive (MP), and large positive
(LP) values of the error e, and the error rate B. One can see that
the support vectors e and k are between -0.1 and 0.1. However, e
and k are not restricted to belonging to this region if we extend
the fuzzy membership function to fm. This means that, any e or B
larger than 0.1 will belong to the LP group.
The fuzzy set of the slow subsystem controller output il is
shown in Fig. 7. There are also seven fuzzy membership functions
that are the same as e and k .
After assigning the input and output values to define the fuzzy
sets, we must map each possible input condition to an output
condition. The common expression of such mapping in this research
is defined as
IF (CONDITION OR ANTECEDENT) THEN (ACTION OR CONSEQUENCE)
The fuzzy rule base can be illustrated as a look-up table. A
representation of a table-lookup corresponding to the fuzzy sets
depicted in Fig. 8 is presented in Table 1. This look-up table is a
matrix of seven rows (the number of membership function of e
fuzzy-set) and seven columns (the number of membership function of
k fuzzy-set).
6.2 Defuzzification process Defuzzification is the process of
conversion of a fuzzy quantity represented by a membership function
to a precise
0 LN MN SN ZE SP MP LP
LN LN MN LN LN ZE MP SP MN MN MN MN MN ZE MP MP
SN LN MN SN SN ZE SP SP e ZE SN ZE ZE ZE ZE ZE SP
SP SN SN ZE SP SP MP LP
MP MN MN ZE MP MP MP MP LP SN SN ZE LP LP LP LP
or crisp value. In this study the centroid and the singleton
methods will be used to combine and defuzzify the outputs into
crisp values. First, we use the centroid method to determine each
fired output value. Then, using a singleton or average weight
method, we combine the output values to produce an executable
single value.
7 Experimental design
An experimental setup was designed and constructed to verify
both model development and controller design. The physical setup is
shown in Fig. 10. It consisted of a highly flexible aluminium beam
attached to a rigid hub by a threaded screw to adjust the distance
from the motor shaft. The beam 'was especially designed to be
highly flexible so that the oscillatory behaviour can be clearly
observed.
Hub actuation is accomplished by a direct-drive DC servomotor
fitted with an optical encoder for measuring the position of the
motor shaft. An optical encoder is used as the position feedback
device for sensing the angular displacement of the motor shaft.
In addition to this collocated sensor/actuator arrange- ment,
strain gauges are mounted along the beam to reconstruct the link
deflection. Two sets of strain gauges (Kyowa products) are used to
sense the strains along the beam due to bending. The strain gauges
are of the electrical resistance type and are in foil form. The
strain gauge signals from the bridge are sent to the strain gauge
signal conditioning modules where the strain signals are ampli-
fied and filtered.
The model 200PCT instruNet controller is used to convert the
analogue data from the sensors into digital
0.3-
0.2
-0.1
-0.2 4 , -0.3 0.10 ,
Fig. 8
IEE Proc.
-0.10 -0.10
Viewer surface for system
-Control Theoiy Appl., Vol. 147, No. 3, May 2002 183
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I
desired rcjdyid$ trajecto
I I
(slow control)
fuzzy logic controller
,I *El actual
(fast control) ~ --qp I ~
Fig. 9 Block dirrgu.am jbr fiizzy logic control with
boundary-layer stabiliser
data and to convert the digital output to analogue signals via a
zero-order hold. The Model 200 instruNet controller board is
attached to personal computers via an expansion slot to drive an
instruNet network, and to provide several digital timer 1/0
channels at a 34-pin connector. The real- time control programs are
coded in Visual Basic.
8 Results and discussion
8.1 Desired trajectory We generate the desired trajectory by the
method described in Section 3, and thus, choose PID gains so that
the filter has a cut-off frequency below the neglected mode
resonant frequency. We obtain yet, j o , y, by integrating (11) and
(12). The desired trajectory should be obtained from Fig. 3.
8.2 Open-loop responses The open-loop responses were obtained by
giving the system a step input torque at initial time. The results
of the tip position, and the first and second mode responses are
shown in Figs. I 1 and 12, respectively. It is clear from Fig. 11
that the actual trajectory of the tip position is very bad.
Moreover, Fig. 12 indicates that the first- and second-
Fig. 10
184
Phj.skal setup fol-jexible-link arm
trajectory
mode oscillations are of significant magnitude and exces- sive
duration. In this uncontrolled case, oscillations appear and do not
vanish.
8.3 ~=0.05 Fig. 13 shows the tip position response of the
flexible-link arm with outer-loop PD control and a boundary-layer
stabiliser when c = 0.05 Similarly, the tip position tracking error
of the flexible-link robot arm with outer-loop PD control and
boundary-layer stabiliser is shown in Fig. 14. Additionally, Figs,
15 and 16 demonstrate the tip position and tip position-tracking
error of the flexible-link robot arm with fuzzy logic control and a
boundary-layer stabi- h e r . It is clear that y(t)Ifi,,,, tracks
the desired trajectory yXt) very closely. Figs. 17 and 18 show that
the flexible mode deflections using the fuzzy logic controller are
damped out more quickly than those using outer-loop PD controller.
For the fuzzy logic controlled case, the elastic vibrations
practically vanish after a short period of time and remain
zero.
8.4 ~ = 0 . 0 1 2 Figs. 19 and 20 illustrate the response of the
tip position and tip position tracking response when E =0.012.
Figs. 21 and 22 show the response for tip position and tip position
tracking error in using fuzzy logic controller. From these figures,
the fuzzy logic control used in this paper is better than that
using outer-loop PD control in tip position tracking as well as in
flexible modes deflections.
8.5 Varied E Next we show that the performance of this composite
controller improves when a more appropriate value of E is used.
Figs. 13 and 19 exhibit the trajectory responses of the flexible
modes when c = O . O 5 and t:=O.O12 in using classical outer-loop
PD control law, respectively. Compar- isons of the performance of
the system in using E = 0.05 and c=O.O12, show that the vibration
is damped out fast when c is smaller. However, there is no
difference in using the fuzzy logic controller to obtain the
vibration responses. It appears that the fuzzy control method is
quite useful as
IEE Proc -Control Tlieory Appl., Vol 147, No 3, Mu)) 2002
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301 1.21
10
5 :: 0 -5 1 I I I I I
0 0.2 0.4 0.6 0.8 1 .o time, s
Fig. 11 loop control
Joint angle response for a step command input in open-
1 st mode I_ j-ll I 2nd mode - 0.06
desired trajectory actual trajectory
i i i , t 0 0.5 1.0 1.5 2.0 2.5 3.0
time, s
Fig. 15 fuzzy logic control and boundary-layer stabiliser E =
0.05
Tip position response of flexible-link robot arm with
-0.01 j -0.02 ! 1 1 1 1 1
0 0.5 1.0 1.5 2.0 2.5 3.0
time. s 0 0.5 1.0 1.5 2.0 2.5 3.0
- 0 . 0 2 1
time, s Fig. 16 fuzzy logic control and c = 0.05
Tip position tracking error offlexible-link robot arm with Fig.
12 open-loop control
Flexible modes response for a step command input in
stabiliser
1.21
Q 0.4
0.2 0
0 0.5 1.0 1.5 2.0 2.5 3.0
0.004 0.0034 1 st mode - . ~ ~ 2nd mode - 0 002 0.001
0 -0.001 -0.002 -0 003
J
-0.004 1 1 , 1 1 0 0.5 1.0 1.5 2.0 2.5 3.0
time, s time, s
Fig. 13 outer-loop PD control and boundary-layer stabiliser c =
0.05
Tip position response of jexible-link robot arm with Fig. 17
outer-loop control and boundary-layer stabiliser E = 0.05
Flexible mode response of,flexible-link robot arm with
0.063
-0.04 -O.O21 v v W -0.06 1 I I 1 I I I
0 0.5 1.0 1.5 2.0 2.5 3.0
time, s
Fig. 14 outer-loop PD control and boundary-layer stabiliser E =
0.05
IEE Proc.-Control Theory Appl., Vol. 147, No. 3, May 2002
Tip position tracking error ofjexible-link robot arm with
0.0051
0.004
0.003
0.002 0.001
0 -0.001
1st mode I 2nd mode -
-0.002 ! 1 1 1 1 1 1 0 0.5 1.0 1.5 2.0 2.5 3.0
time, s
Fig. 18 Flexible mode response ofjlexible-link robot arm with
fuzzy logic and boundary-layer stabiliser E = 0.05
185
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regards reliability and robustness. Fuzzy logic controllers can
be used to make very robust controllers for nonlinear systems
because they only act on rules applied to measured outputs and thus
can handle variations. A standard control- ler might not explicitly
consider rules when doing this.
1 st mode 2nd mode
9 Conclusions
Y
a 0 4- 0 2 -
c
This paper has concentrated on fuzzy logic using the singular
perturbation approach for flexible-link robot arm control. To
reduce spillover effect, we introduced a singu-
i - desired trajectory actual trajectory >
0 . I I I
- 1st mode 2nd mode
0 003 0.002
-0 O.OO,j 001 \ f l - -0.002 W -0.003 -0.004 I I I I I I I
0 0.5 1.0 1.5 2.0 2.5 3.0 time, s
Fig. 23 outer-bop PD control and boundary-layer stabiliser
r:=0.012
Flexible-mode response ojjlexible-link robot arm with
1.2,
- desired trajectory actual trajectory
I I I I I I 0 0 5 1 0 1 5 2 0 2 5 3 0
time, s
Fig. 19 outer-loop PD control and boundary-la,ver stabili Fer L
= 0.012
Tzp position response of flexble-link robot arin with
0.061
-0.02
-0.04
-0.06 I I I I I I 0 0.5 1.0 1.5 2.0 2.5 3.0
time, s Fig. 20 outer-loop PD control and boundary-layer
stabiliser c = 0.012
Tip positiorz tracking error ofjlexible-link i-obot aiw with
1.2,
0.001 O . O o 2 Y 7 0 001 O f ~
-0.002 I I I I 1 I I 0 0.5 1.0 1.5 2.0 2.5 3.0
time, s Fig. 24 fuzzy logic control and boundary-layer
stabiliser E = 0.012
Flexible-mode response offlexible-link robot arm with
lar perturbation approach to derive the slow and fast
subsystems. A composite control design was adopted. Therefore, a
twice-time scale fuzzy logic controller was applied. The
fast-subsystem controller will damp out the vibration of the
flexible structure using an optimal control method. Hence, the
slow-subsystem fuzzy controller domi- nates the trajectory
traclting. We can guarantee the stability of the internal dynamics
by adding a boundary-layer correction based on singular
perturbations. In addition, various case studies are used to verify
the control algo- rithm. It appears that the fuzzy controllers are
potential candidates for deriving the control in the presence of
these structural nonlinearities. Future work will focus on the
twice-time scale hierarchical fuzzy logic controller for the
multilink flexible arm.
0.06- 0.05- 0.04- 0.03- 0.02- 0.01 -
0- -0.01 - -0.02- I I I I I
0 0.5 1.0 1.5 2.0 2.5 3.0 time, s
Fig. 22 with jiizzy logic control and boundary-layer stabiliser
c = 0.012
186
@position tracking error of flexible-link robot arm
10 Acknowledgments
This work is supported by the National Science Council research
grant No. NSC 89-22 18-E-23 1-001.
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