Control of Parallel Flexible Five Bar Manipulator
using QFT
Sandeep Karande, P. S. V. Nataraj, P. S. Gandhi and Manoj M. Deshpande
Systems and Control Engineering
Indian Institute of Technology Bombay
Mumbai- 400076, India
Email: [email protected], [email protected]
Abstract—Light weight robotic manipulators have significant
advantages over their heavy counterparts in terms of efficiency
of energy consumption and speed. However, they also have an
inherent problem of vibrations which significantly affect their
performance. Control strategies need to be developed to control
their vibrational behavior. Many different control strategies
have already been developed and tested on the manipulators.
However, not enough work has been done to test the Quantitative
Feedback Theory, a frequency domain based robust control
strategy. Though several nonlinear model/control based methods
have been developed for control of single or serial flexible link
manipulator, the domain of parallel flexible links manipulators
is unexplored. QFT, being a robust control strategy, promises to
exactly tackle these uncertainties in the manipulator. This project
is aimed at analyzing the performance of a QFT based controller
in controlling the vibrations in flexible multilink manipulator
systems.
I. I NTRODUCTION
ROBOTIC manipulators find diverse applications in industries today. Manipulator structures with low stiffness can cause
undesirable structural vibrations during use. One way to avoid
these vibrations is to increase the stiffness of the manipulator
links. This requires the manipulators to be heavy and bulky.
However, this directly reduces the efficiency of power consumption and speed of the manipulators, particularly if the
payload weight is not large enough. Also, the light weight
manipulators which inherently exhibit large vibrations find
wide range of applications today in various fields like space
robots and medical surgeries. Hence, it becomes imperative to
develop control strategies to curb these undesirable vibrations.
Many control strategies like PID, adaptive, optimal, sliding
mode, non-linear feedback, Lyapunov theory based strategies,
h-infinity and other robust strategies have already been developed and tested on the manipulators. Details of these works
can be found in the literature review conducted by Dwivedy
and Eberhard [1]. However, not enough work has been done
to test the effectiveness of the Quantitative Feedback Theory.
From the literature review, it has been found that the control
of a single flexible link with piezoelectric actuator using QFT
has been already achieved by four independent teams [2]–[5].
In all these works, two separate controllers were developed in
order to control the hub position by motor control and the tip
vibrations by piezoelectric actuator control. The control of a
flexible link without a piezoelectric actuator by developing
a single controller has also been achieved [6]; The non
linear model used in this case was obtained by experimental
identification techniques.
In the present work, the control of a flexible parallel five
bar manipulator by using a single rigid body actuation using
QFT has been successfully achieved. The plant model was
obtained theoretically by using the Lagrangian formulation and
the assumed modes approach. Fig. 1 shows the experimental
setup. The links 1 (X) and 3 (Y) are flexible, while the links 2
and 4 are rigid. The rigid angular positions of the links 1 and
3 are sensed by two separate encoders, while their flexural tip
deflections are sensed by two strain gauge connected at the
bases of the flexible links. The actuation to the two links is
provided by two DC motors.
Fig. 1.
Flexible Parallel Five bar Setup
The section 2 deals with the dynamic modeling of the
manipulator system. Section 3 discusses the controller design
and the results obtained. Section 4 concludes the paper.
II. M ODELING THE PARALLEL FIVE BAR MANIPULATOR
We would first obtain the dynamical equation for a single
flexible link with tip mass. Later, a widely accepted assumed
modes method is used to develop equations to develop relevant
system equations for the flexible five bar manipulator.
A. Modeling the Single Flexible Link with point mass at the
tip
The single flexible link was modeled by using the Lagrangian approach along with the assumed modes method for
discretization. Further, the equations were derived by using
the instantaneous structural approach [7]. In this approach, we
freeze the rigid body degrees of freedom of the system and
consider only the flexural dynamics. The forces responsible
for flexural motion are considered to be the inertial forces
in the links caused due to their rigid body motion. Thus in
this approach effect of rigid body motion has been considered
through inertial forces rather than rigid and flexural body combined kinematics. This approximation significantly reduces the
computations involved and works well in deriving the approximate linear dynamical equation for a single flexible link. The
final system equation is obtained between the overall rigid
angular displacement and the flexural angular displacement of
the link.
For both the motors, torque constant, rotor inertia are 0.0176
Nm/Amp and 1.33e-6 Kg m2 . Encoder resolution is 4096
Counts/rev. For flexible link length and mass is 0.23m. and
0.09kg. respectively. For rigid links length and mass is 0.23m.
and 0.08kg. Elbow and tip joint masses are 0.03kg. and 0.04kg
respectively. The gain of strain gages is 2.54cm/v.
We first develop a model to determine flexural deflections
caused in the link given rigid body inertial forces as a force
loading condition. Rigid body motion dynamics of the link
in turn gives these inertial forces. Using these correlations a
linear model between the angular rigid body motion of the
link and its overall tip motion is obtained [7].
We first determine the natural modes of vibration of the
link. The dynamic equation of the flexural link is given by the
Euler-Bernoulli beam equation [8]:
∂ 2 y(x, t)
∂2
[EI(x)
] + f (x, t)
∂x2
∂x2
∂ 2 y(x, t)
= ρ(x)
, 0 < x < L (1)
∂t2
To determine the natural modes, we consider the free
vibration of the beam which happens when f (x, t) = 0 In
this case, the solution becomes separable in space and time.
Thus, for free vibrations, we have:
−
the corresponding equations for bending moment and shearing
force are obtained as:
Y (x) = C1 [sin βx − sinhβx−
sin βL + sinh βL
[cos βx − coshβx]] (9)
cos βL + cosh βL
The mode shape was normalized using the normalization
definition given in Equation 10
d4 Y (x)
ω2 ρ
− β 4 Y (x) = 0, β 4 =
4
dx
EI
The general solution for Equation 3 is of the form:
(3)
Y (x) = C1 sin βx+C2 cos βx+C3 sinh βx+C4 cosh βx (4)
The boundary conditions that the link is subjected to are: 1.
The deflection and the slope of the deflection curve at the fixed
end are zero:
y(0, t) = 0
(5)
∂y(x, t)
|x=0 = 0
(6)
∂x
2. The end of the link at x=L experiences an effective
lumped mass M. No effective inertia is experienced. Thus,
L
Z
[φ(x)]2 dx = 1
(10)
0
The characteristic equation is obtained as:
(1 + cos βL cosh βL)−
Mβ
(sin βL cosh(βL) − cos βL sinh βL) = 0
ρc
(11)
2
where β4 = ωEIρc As per the assumed modes method, the
flexural deflection of the link y(x,t) is assumed to be of the
form:
y(x, t) = φ(x)q(t)
(12)
where, φ(x) is the mode shape and q(t) is the generalized
coordinate of the link, representing the flexural motion. The
expressions for kinetic and potential energies of the link are
given as:
2
Z
1
∂y(x, t)
KE(t) =
ρ(x)
dx
(13)
2
∂t
Z
1
2
(2)
Putting Equation 2 in Equation 1, we get:
(7)
∂
∂ 2 y(x, t)
] |x=L = M ÿ(L, t)
(8)
[EI
∂x
∂x2
Using these boundary conditions, we get the mode shape of
the link as:
P E(t) =
y(x, t) = y(x)F (t)
∂ 2 y(x, t)
|x=L = 0
∂x2
EI
L
EI
0
∂ 2 y(x, t)
∂x2
2
dx
(14)
Inserting Equation 12 in Equations 13 and 14 and simplifying, we get the following expressions for kinetic and potential
energies:
KE(t) =
1
mq̇(t)
2
(15)
1
kq(t)
(16)
2
where, the equivalent mass m and the equivalent stiffness k
are given by:
Z L
2
m=
ρ(x) (φ(x)) dx
(17)
P E(t) =
0
Z
k=
L
EI
0
φ(x)
x2
2
dx
(18)
The Lagrange’s equation of motion in the generalized coordinate q(t) is given as:
d ∂L
∂L
−
=P
(19)
dt ∂ q̇
∂q
where, L = KE − P E. P(t) is the generalized force corresponding to the generalized coordinate q(t) given by:
Z L
P (t) =
f (x, t)φ(x)dx
(20)
Thus, from Equations 22, 23, 24 and 26, the transfer function
between the flexural angular link tip position, θf and the rigid
angular link position, θ is obtained as:
)s2
( −Aφ(L)
θf (s)
L
=
2
θ(s)
ms + cs + k
Now, the link tip experiences an effective point mass M.
Hence, the linear density term ρ(x) becomes:
ρ(x) = ρc + M δ(L)
0
where, f(x,t) is the inertial force acting along the beam length.
Inserting these expressions for kinetic and potential energies
(Equations 15 and 16) in the Lagrange’s equation 19 and
simplifying the expression, we obtain the following system
equation:
mq̈(t) + k¨(t) = P (t)
(21)
Equation 21 includes the mass and stiffness terms, but no
damping term. Thus,in order to introduce damping in the
theoretical model, a damping coefficient term was externally
added:
mq̈(t) + cq̇(t) + kq(t) = P (t)
(22)
ytip (t) = y(L, t) = φ(L)q(t)
Using Equation. 20, the generalized force is obtained as:
Z
A=
(24)
L
ρ(x)xφ(x)dx
(25)
0
Now, the value of the flexural deflection of the link is practically very small. Thus, from Fig. 2, we obtain the flexural
angle as:
Fig. 2.
Deflected Link Position
tan (θf (t)) ≈ θf (t) =
ytip (t)
L
2
= mc + M (φ(L))
(26)
(29)
Similarly, substituting Equation 28 in Equation 25, we get:
L
ρ(x)xφ(x)dx
0
(23)
f (x, t) = −ρxθ̈(t)
P (t) = −A¨(θ)(t)
0
A=
The inertial force per unit length acting along the link length
due to its rotation is given by:
(28)
where, δ(L) is the dirac delta function. Substituting Equation
28 in Equation 17, we get:
Z L
2
m=
ρ(x) (φ(x)) dx
0
Z L
2
=
(ρc + M δ(L)) (φ(x)) dx
Z
The tip deflection of the link is given by:
where,
(27)
Z
=
L
(ρc + M δ(L)) xφ(x)dx = Ac + M Lφ(L) (30)
0
Substituting Equations 29 and 30 in Equation 27, we get the
dynamical equation of the link as:
2
Ac φ(L)
−
+
M
(φ(L))
s2
L
θf (s)
=
(31)
2
θ(s)
m + M (φ(L)) s2 + k
c
Equation 31 is the dynamical equation that relates the flexural
deflection of the link tip to its rigid motion. Using Equation
31, the transfer function between the over all tip deflection
and the rigid motion can be obtained.
B. Dynamical behavior of the parallel five bar manipulator
system
In this section, we will obtain the control relevant dynamical
equations of the parallel five bar manipulator. The equations
will later be used to design and implement a QFT based
controller on the manipulator.
The schematic of the parallel five bar manipulator is as
shown in Fig. 3. In this figure, the links 1 and 2 are flexible,
while the links 3 and 4 are rigid.
The natural frequency of a cantilever always depends on
the load conditions that its free end is subjected to. During
the operation of the manipulator system, as the angle between
the two flexible links changes, the effective lumped masses
that the flexible links experience at their free ends change.
This causes a continuous change in the natural frequency of
the system as the flexible links move relative to each other.
These natural frequencies were experimentally determined for
different angles between the flexible links of the manipulator.
III. QFT BASED CONTROLLER DESIGN AND
IMPLEMENTATION
The design task is to obtain the controller and the prefilter
transfer functions that satisfy the desired performance specifications. Fig. 5 shows the overall block diagram of the closed
loop system that will be developed by using QFT.
Fig. 3.
Parallel Five Bar Manipulator
Fig. 5.
The change in natural frequency causes a continuous change
in the values of the parameters m, k and A as the manipulator
configuration changes; i.e.m = m (θ2 − θ1 ), k = k (θ2 − θ1 )
and A = A (θ2 − θ1 ). To take into consideration, the changing
modal frequency ,1st mode, along with the configuration, it
was decided to develop separate linearized equations, which
would represent the system in its different configurations.
Furthermore, to make the modeling task simple, it was decided
to treat the two flexible links as individual links that do not
interact with each other. In this case, the vibration induced in
one link due to the interaction with the other link would be
considered as a disturbance, which needs to be rejected. As
we see later, inherent robustness property of QFT takes care of
this assumption. At different frequencies, the effective lumped
masses that the link tips experience, the equivalent linearized
model of the system in each configuration were obtained.
The motors that drive the flexible links were found to
contain a lot of friction. A closed loop proportional controller
was designed for the motor so that the friction is overcome and
the desired trajectory is traced by the motor shaft. From experiments, it was found that the motor dynamics was affected
by the flexural motion of the link. Hence, different equivalent
second order models of the motor at different configurations
were obtained.
The overall plant considered for QFT based control design
has the desired rigid displacement, θr,desired as input and the
overall tip position of the link, θt as output as shown in Fig. 4.
Fig. 4.
Closed Loop Plant Model
The performance specifications were given in the form of
the desired step response that the closed loop system should
exhibit.
Following specifications were chosen in the present case:
1) The system should have an over damped response, i.e.,
no overshoot.
2) Rise time (time for the output to reach from 10% to 90%
of the set value) should lie between 0.28 secs and 0.42 secs.
Using these specifications, the controller C(s) and the prefilter F(s) were designed. The plant representing the manipulator when angle between the two links is 90◦ , was selected
as the nominal plant. nominal transfer function was obtained
as mentioned below.
C(s) =
F (s) =
s
5.8
+1
2
,
s
s 50
+1
s
2.176 + 1
s2
182.25 + 0.1047s
(32)
.
+1
(33)
Fig. 6 shows the final loop shape obtained by designing
the controller, while Fig. 7 shows the prefilter design. It
is observed that although the design satisfied the bounds at
lower frequencies, it could not satisfy the bounds at higher
frequencies.
Overall Plant Model
Fig. 6.
The control relevant system equations were thus obtained
for all the configurations. These were used to design and
implement a QFT based controller on the manipulator.
Final Loop Shape for Controller Design
The rise time obtained is 0.32 seconds, which lies well
within the specified limits with zero overshoot.
[6] Ziaei, K., Wang, D. W. L., Design and experimental evaluation of a
single robust position/force controller for a single flexible link (SFL)
manipulator in collision, Proceedings of the 2003 IEEE International
Conference on Robotics and Automation, Taipei, Taiwan, Sept., 14-19,
2003
[7] Subrahmanyan, P. K., Seshu, P., Dynamics of a flexible five bar
manipulator, Computers and Structures, Vol. 63, No. 2, p. 283-294, 1997
[8] Meirovitch, L., Elements of vibration analysis, McGraw Hill, 2nd edition,
1969
Fig. 7.
Prefilter design
Finally, the controller was experimentally tested on the
manipulator system. Since the two flexible links and the
two motors driving them are absolutely identical, the QFT
controller designed above was implemented on both of them
independently. Both the links were given two exactly similar
reference signal inputs. Thus, in this case, the end point C of
the manipulator should trace a straight line. The experimental
plots of the rigid displacements, the flexural displacements,
the over-all displacements and the controller efforts of both
the links are shown in figs. 9 and 10 respectively. The line
trajectory traced by point C is shown if Fig. 11.
Fig. 8.
X link: Rigid Displacement
IV. C ONCLUSION
The experimental modeling of five bar manipulator is done.
The QFT based controller for the obtained model for stability
and performance specification is then designed. The obtained
QFT controller is then successfully tested on the experimental
setup. QFT based controller designed above gave a rise time of
0.37 seconds which lies well within the desired specifications.
No overshooting was observed in the step responses of both
the links. The regulation was obtained with highly reduced
vibrations. It was concluded that a QFT based control design
for a parallel five bar manipulator that delivered the desired
specifications was obtained.
Fig. 9.
X link: Flexural Displacement
R EFERENCES
[1] Dwivedy, S. K., Eberhard, P., Dynamic analysis of flexible manipulators,
a literature review, Mechanism and Machine Theory, 41 (2006), p. 749777
[2] Song, B., Jayasuriya, S., Low order robust controller design and its
implementation on a single flexible robot link, Proceedings of the 1992
Japan - USA Symposium on Flexible Automation, Part I
[3] Choi, S.B., Cho, S.S., Shin, H.C., Kim, H.K., Quantitative feedback
theory control of a single-link flexible manipulator featuring piezoelectric
actuator and sensor, Smart Materials and Structures, June 1999, Vol 8,
Issue 3.
[4] Kerr, M. L., Jayasuriya, S., Asokanthan, S. F., QFT based robust control
of a single-link flexible manipulator, Journal of Vibration and Control,
January 2007, Vol. 13, Issue 1, p. 3-27
[5] Seung, B. C., Seung, S. C., Young,-S. J., Vibration and tracking control
of piezoceramic-based smart structures via quantitative feedback theory,
Proc. SPIE- International Society for Optical Engineering (USA), Vol:
3039, 1997, p. 596-605
Fig. 10.
X link: Total Displacement
Fig. 11.
X link: Controller effort given by voltage input to motor
Fig. 12.
Fig. 13.
Y link: Rigid Displacement
Y link: Flexural Displacement
Fig. 14.
Fig. 15.
Y link: Total Displacement
Y link: Controller effort given by voltage input to motor
Fig. 16.
Line trajectory traced by end point C of the link