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COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas
COMPARISON OF TWO DISTRIBUTED FUZZY LOGICCONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. ShiUniversity of Nevada, Las VegasDepartment of Mechanical EngineeringLas Vegas, NV 89154-4027
Mohamed TrabiaUniversity of Nevada, Las VegasDepartment of Mechanical EngineeringLas Vegas, NV 89154-4027
Based on Shi, L. Z., and M. Trabia, “Comparison of Two Distributed Fuzzy Logic Controllers for Flexible-Link Manipulators,”Proceedings of the ASME Dynamic Systems and Control Division-2000”, DSC-Vol.69-1, ASME, New York,2000, pp. 443-451.Presented at the 2000 ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida, November2000.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 2
Control methodology:• Fuzzy Logic Control (FLC) presents a computationally efficient and
robust alternative to conventional controllers• This paper uses FLC in the single-link flexible manipulator.
Structure design:• The first structure is based on the expert observation of the system;• The second structure uses the importance information of the inputs to
the system output.
Learning ability :Both approaches are tuned using one of nonlinear programming methods,Simplex.
Comparison:The paper compares the two FLC structures with a linear quadraticregulator method to prove the effectiveness of FLCs.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 3
Literature survey:1 Lin, Y. and T. Lee, “An Investigation of fuzzy logic control of flexible
robots,” Robotica, vol. 11, pp. 363-371, 1993.2 Arciniegas, J., A. Eltimsahy, and K. Cios, “Fuzzy inference and the
control of flexible robotic manipulators,” IEEE InternationalSymposium on Intelligent Control, pp. 250-254. 1993.
3 Moudgal, V. G., W. A. Kwong, K. M. Passino, and S. Yurkovich,“Fuzzy learning control for a flexible-link robot,” IEEE Transactions onFuzzy Systems, vol. 3, pp. 199-210, 1995.
4 Yoo, B., S. Jeong, and W. Ham, “Hybrid control of flexible manipulatorbased on fuzzy relations,” IEEE International Conference on FuzzySystems, pp. 817-823, 1996.
5 Kubica, E. and Wang, D., “A two-stage fuzzy logic controller for aflexible single link robot,” International Journal of Robotics andAutomation, Vol. 14, No., pp. 9-14, 1999.
6 Trabia, M., 1998, "Tuning of Distributed Fuzzy Logic Controller for aFlexible-Link Robot," Vibration and Noise Control, DE-Vol. 97,ASME, New York, pp. 57-66.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 4
1 Schematic of a Flexible Manipulator
mt,Jt
Jm
x
y
θ
T(t)
L0
vx P
• Modeling techniques: Finite element approach and Lagrangian• Input variable:T, the motor torque.• Output variables: joint angle, )t(θ , joint angular velocity, )t(θ& ,
displacement of the tip point, )t(vn and velocity of tip point, )t(vn& .
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 5
2 PD-like Distributed Fuzzy Logic Controller
(1) Structure of PD-like distributed fuzzy logic controller
FlexibleManipulator
Joint Angle FuzzyController
θ (t)
v(L,t)
Tip FuzzyController
θd(t)
_ +
+
_ +
_ +
0
_ +
0
eθ
edθ
edtip
etip
Tθ
Ttip
+
T
)(tdθ&)(tθ&
),( tLv&
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 6
(2) Rule of PD-like fuzzy logic controller
Table I Rules for the Joint Angle Fuzzy logic controllerDisplacement Error⇒
Velocity Error⇓ N Z PN N N ZZ N Z PP Z P P
Table II Rules for the Tip Fuzzy logic controllerDisplacement Error⇒
Velocity Error⇓ N Z PN P P ZZ P Z NP Z N N
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 7
4 Importance-based Distributed Fuzzy Logic Controller
(1) Taylor Series Expansion for the system )x,,x,x(fy nL21=• Normalizing to the form of such that, nT
n ],[]y,x,,x,x[ 1021 ∈L
• Collect p sample data p,j]y,x,,x,x[ Tjn,j,j,j LL 121 =∀ ,
• Taylor Series Expansionon a fixed point Tn ],,,[ χχχ L21 :
jii,j
x
n
i inj r)x(
x
f),,(fy
ii
+−∑+===
χ∂∂χχχ
χ121 L
kii,k
x
n
i ink r)x(
x
f),,(fy
ii
+−∑+===
χ∂∂χχχ
χ121 L
• Linear approximation: subtracting the above two equations
∑ −=−=
n
ii,ki,jikj )xx(byy
1where
iixi
i x
fb
χ=∂∂=
• Importance information: eachbi represents the ratio of the varianceof the output variabley with the variance of each input variablexi.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 8
(2) Least Square Error Algorithm:• randomly choosing a subsetq sample data pairs in the form of,
T
kjn,kn,j,k,j]yy,xx,,xx[ −−− L
11from the original data set such thatq is
greater thann.• Rewriting in a matrix form:
XBY ∆∆ =• using the pseudo-inverse formula: YX)XX(B TT* ∆∆∆∆ 1−=• If XX T ∆∆ is a singular matrix, let thei th row vector of matrix∆X be
Tix∆ and the i th element of Y∆ be T
iy∆ , B can be calculated by the
following sequential formulations:)Bxy(xDBB i
Ti
Tiiiii 11111 +++++ −+= ∆∆∆ (8)
101 11
111 −=
+−=
++
+++ q,,i
uDu
DuuDDD
iiTi
iTiii
ii L (9)
• thedegree of importanceof xi IMP(xi) is )x(IMP i = ∑=
n
jji|b||b|
1
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 9
(3) Analysis the Importance of Variables for Flexible Manipulator
-20 -15 -10 -5 0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
IMP
Torque(Nm)
θ(t)v(L,t)dθ(t)dv(L,t)
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 10
(4) The structure of importance-based distributed fuzzy logic controller
FlexibleManipulator
Velocity FuzzyController
θ(t)
v(L,t)
Displacement FuzzyController
θd(t)
_ +
+
_ +
_ +
0
_ +
0
etip
Tv
T d
+
T
)(tdθ&)(tθ&
),( tLv&
edtip
eθ
edθ
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 11
(5) The rules of importance-based distributed fuzzy logic controller
Table III Rules for the Velocity Fuzzy logic controller
Tip Velocity Error⇒Joint Velocity Error⇓
N Z P
N N N ZZ N Z ZP Z P P
Table IV Rules for the Displacement Fuzzy logic controller
Tip Displacement Error⇒Joint Displacement Error⇓
N Z P
N P P ZZ P Z NP Z N N
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 12
5 Tuning of the distributed fuzzy logic controllers usingnonlinear programming
(1) Reason for tuning the fuzzy logic controller:
A good estimate of the location and shape of each input/outputvariable may be unavailable or can be only obtained by operating thesystem extensively.
(2) Tuning algorithms:
Simplex Method of Nelder and Mead
(3) Performance index:
( )∑
−+−+∑
+
++
==
−
−
nt
i
dtipdtipdd
nt
i
dtiptipd L
e
L
eee
L
e
L
eee=PI ii
ii
ii
ii
2
22
1
2222 1
1θθθθ
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 13
(4) Optimization Parameters to tune:
Choose three membership functions for each input/output. Then there arethree parameters to tune for each variable:
Table V Parameters Describing the Membership Functions of a Fuzzy Variable
Membership Function Parameters⇒Membership Function⇓
c σ
N -cP σP
Z 0 σZ
P cP σP
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 14
6 Tuning the PD-like distributed fuzzy logic controller
(1) Physical parameters and mechanical properties of the simulatedflexible manipulator
Parameter ValueLink length,L 1.0 mDensity,ρ 0.1 kg/mBending stiffness,EI 2.0 Nm2
Moment of inertia of the hub,Jm 0.05 Kgm2
Radius of the hub,L0 0.01 mTip mass,mt 1.0 kgTip mass moment of inertia,Jt 10-5 Kgm2
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 15
(2) Number of nodes: Eight elements of equal length are used to modelthe beam.
(3) Simulation trajectory:
• The initial angle of the manipulator is equal to zero• The desired final angle is equal to one radian at 1 second.• The desired joint angle motion is a bang-bang acceleration profile.• The frequency of sampling is hundred samples per second.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 16
(4) Choose the initial Response for the Initial Guess of PD-like fuzzylogic controller.
Variable Parameter ValueCP (rad.) 0.8σP 0.24Eθ
σZ 0.12CP (rad./s) 5.0σP 1.5Edθ
σZ 0.75CP (Nm) 10.0σP 3
Tθ
σZ 1.5CP (m) 0.3σP 0.09etip
σZ 0.045CP (m/s) 5.0σP 1.5edtip
σZ 0.75CP (Nm) 10.0σP 3Ttip
σZ 1.5
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 17
(5) Response of the initial guess of PD-like fuzzy logic controller
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
time(s)
join
tang
le(r
ad)
0 5 10 15 20 25 30 35 40 45 50-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
time(s)
tipde
flect
ion(
m)
Figure 5 Joint Angle Response Figure 6 Tip Point Displacement
• The controller, based on these membership functions, needs a longtime to stabilize. Its performance is clearly unacceptable.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 18
(6) Response for PD-like fuzzy logic controller after Simplex tuning
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
time(s)
join
tang
le(r
ad)
0 1 2 3 4 5 6 7 8 9 10
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
time(s)
tipde
flect
ion(
m)
Figure 4 Joint Angle Response Figure 5 Tip Point Displacement Response
• The algorithm reaches a performance index value of 1.429.• The performance of the tuned controller shows a marked improvement.• The tuned controller reaches a steady state at less than seven seconds.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 19
7 Tuning the importance-based distributed fuzzy logiccontroller
(1) Choose the initial Response for the Initial Guess of Importance-based fuzzy logic controller:
• Results of using the same initial values in PD-like fuzzy logiccontroller: extremely unacceptable magnitudes of the errors and highfrequency responses.
• Choose of initial values for the output of less important fuzzy logiccontroller:Td is scaled down by a factor of twenty-five to indicate thatthe Displacement Controlleris less important than theVelocityController.
• All other variables have members similar to that for PD-like fuzzylogic controller.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 20
(2) Response for the Initial Guess of Importance-base fuzzy logiccontroller
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
time(s)
join
tang
le(r
ad)
0 5 10 15 20 25 30 35 40 45 50-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
time(s)
tipde
flect
ion(
m)
Figure 6 Joint Angle Response Figure 7 Tip Point Displacement
• The manipulator is under-powered since joint angles are not dampedby the end of fifty seconds.
• As an initial guess, this controller is however better than the one usedin the PD-Like distributed controller. This observation is generallycorrect for Importance-Based controllers.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 21
(2) Response for Importance-base fuzzy logic controller after Simplextuning
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
Time(s)
Join
tang
le(r
ad)
0 1 2 3 4 5 6 7 8 9 10
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time(s)
Tip
defle
ctio
n(m
)
Figure 8 Joint Angle Response Figure 9 Tip Point Displacement Response
• The search converges to a point with a performance index of 1.483.• The steady state is reached faster for the tuned Importance-Based
controller that for the PD-Like controller with less overshoot.• It also produces slightly less tip vibration as can be seen in its
performance index.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 22
8 Comparing to linear quadratic regulator
(1) Linear quadratic controller (LQR) theoryA LQR can be defined as finding the appropriate state feedback
controller that minimizes the following cost functional:
∫ ++=f
t
t
TTT dt)uNxuRuxQx(J0
2
The above equation is subject to the state dynamic constraint,BuxAx +=&
The optimal control is obtained through feedback with a control lawdefined as,
u = -KxIn this example identity matrices are used for both Q and R. N matrix isnull. To properly compare the results to those of the fuzzy logiccontrollers, a different LQR is created for every time step.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 23
(2) The performance of the LQR controller
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
Time(s)
Join
tang
le(r
ad)
0 1 2 3 4 5 6 7 8 9 10
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time(s)
Tip
defle
ctio
n(m
)
Figure 9 Joint Angle Response Figure 10 Tip Point Displacement Response
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 24
(3) Comparison with fuzzy logic controller:
• The manipulator with LQR exhibits similar behaviour to the two tunedfuzzy logic controllers.
• The joint angle in LQRs case has a larger error. Settling time is close tonine seconds, which is also larger than settling time for the two tunedfuzzy logic controllers.
• The range of the tip deflection is close to the results of the twoprevious examples.
• LQR controller is a full feedback controller. The error information onthe eighteen variables (or more depended on the number of nodes torepresent the flexible link) are needed to produce the feedback, whichlimit the possibilities of implementing this controller.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 25
9 Conclusions:
(1) We explored two possible structures for the distributed fuzzy logiccontroller of a single-link flexible manipulator.
• PD-Like distributed fuzzy logic controller:Joint Angle and Tipfuzzy logic controllers.
• Importance-Baseddistributed fuzzy logic controller:Velocitycontroller andDisplacementfuzzy logic controller.
(2) Importance-Basedcontroller usually provides better performancebased on an arbitrary initial guess.
(3) Tuning the locations and shapes of the membership functions usingNelder and Mead Simplexnonlinear programming technique.
COMPARISON OF TWO DISTRIBUTED FUZZY LOGIC CONTROLLERS FOR FLEXIBLE-LINK MANIPULATORS
Linda Z. Shi & M. Trabia, University of Nevada, Las Vegas 26
(4) The algorithm is satisfactorily applied to both distributed controllers.Importance-Based distributed controller gives a slightly betterperformance in terms of better settling time, steady state error.
(5) Linear Quadratic Regulator (LQR) is used to compare with the abovetwo distributed fuzzy logic controllers. Tuned fuzzy logic controllersare superior in terms of overshoot, settling time,
(6) The proposed structures of the distributed controllers will be extendedto multiple-link flexible manipulators. The procedures presented inthis paper can be applied to other systems that are difficult tocharacterize.