This document discusses the theoretical and experimental study of dynamics and control of a two-link flexible robot arm. It presents the following: 1) A linear dynamic model of the robot arm is constructed using Ritz's approach and validated experimentally. 2) Three active control schemes are designed using pole placement technique for path tracking and vibration suppression, and tested via simulation. 3) A discrete-time state variable feedback control scheme is presented and designed to control the robot arm's motion using three sensors for position and curvature feedback.

1. Theoretical and Experimental Study of Dynamics and Control
of a Two-link Flexible Robot Manipulator of Revolute Joints
Chih-Wu JEN, Salvatore NICOSIA & Paolo VALIGI
Department of Computer Science, Systems and Production
University of Rome “Tor Vergata”, Rome, ITALY 00133
Abstract
The linearized dynamic model of a prototype robot
arm of two flexible links is constructed using Ritz’s
approach and is validated by experimental frequency
response functions. Some active control schemes are
then designed to manipulate this robot arm on a small
manoeuvre base, using pole placement technique, for
the purpose of path tracking as well as vibration sup-
pression, and are tested by computer simulations.
1 Introduction
This paper discusses some issues of dynamic mod-
elling and controller design of the two link flexible
robot arm of the University of Rome, Tor Vergata.
A different modelling and control approach has been
described in [1,2]. The linear dynamic model of the
robot arm about a certain configuration is obtained
directly via structural dynamics in combination with
a polynomial type Ritz’s spatial discretization scheme
and substructural synthesis.
The state equations describing the reduced-order
linear model are then determined formally from the
dynamic analysis and the system parameters are enu-
merated according to the dimensional or material pa-
rameters of the robot structures as well as the physical
constants of the other system components. This sys-
tem model is then validated by comparing the theoret-
ical frequency response functions calculated based on
the model with those obtained experimentally. Based
IinkA I link B
I
Figure 1: A 2-link flexible robot arm
on a reduced model of the flexible robot obtained in
0-8186-7352-4/96 $05.00 0 1996 IEEE
dynamic analysis, a controller for the purpose of vibra-
tion suppression of the robot arm during its manipula-
tion is designed with three different schemes. The first
scheme is a classical output feedback control, which is
based on the assumption of weak coupling between the
modes to be controlled. The second one uses a decou-
pler for modal decomposition of the system, and an
observer to estimate the modal state variables accord-
ing to the accessible outputs. This is a general scheme,
applicable in case the coupling between modes are not
negligible, usually as there is a considerably heavy
payload. The third scheme differs from the second one,
a continuous-time design, in its discrete-time realiza-
tion. Since the dynamic characteristics of the plant
to be controlled are well determined, the controller
parameters of all the three schemes can be explicitly
evaluated using the pole placement technique. The
performances of the different control laws obtained
are then tested by computer simulation. The three
proposed controller designs give all quite good com-
parable simulation results, showing the validity and
similar performances of these algorithms. The discrete
time design with decoupling technique and state esti-
mation has as advantages of less required sensors, low
sampling frequency, appears most simple and reliable,
and is therefore recommended for practical implemen-
tation on the test bench robot arm.
2 Dynamic Modelling
A schematic diagram of the 2-link experimental
flexible robot arm may be seen in Fig. 1 (a list of
the design parameters of this robot constructed in the
laboratory can be seen in Table 1).The robot is com-
posed of two slender links made of hardened steel,
modelled by engineering beams, and two DC directly
driving motors, modelled by two revolute frictionless
joints with certain mass and moment of inertia due to
the mass distributions of the stators and rotors which
can exert an actuating torque. All friction forces as
well as gravity are neglected (the robot arms move in
an horizontal plane). The positions of the hub angles
or the two motor shaft angles are measured by two
optical encoders. The hub speed of the first motor
is obtained by an analog tachometer incorporated in
the motor. The measurements of the deformation of
beams are done by use of a set of strain gauges placed
in pairs on the opposite sides of the links.
237

2. Table 1: Design parameters of the robot structure
Wv m 2 1
L I m 0.317 0.403
P Kgm-3
E "-2
Mh Ka
If the torques TA and TB of the DC motors are
chosen as the inputs to the robot system, the modal
analysis has shown that the first and the second nat-
ural frequencies non-rigid modes) are closely located
same for the third and the fourth non-rigid modes; in
fact, for the experimental robot in a fully stretched
position fi = 21.52, f2 = 25.58, f3 = 67.4 and
f4 = 69.09 Hertz), hence a stiffer system requiring
a relatively high sampling frequency. This means that
while one tries to adopt a reduced order model at
the stage of controller design, one is supposed to take
into consideration at the same time an even number
of modes, hence a more complex modelling. A rem-
to each other an6are relatively high frequencies (the
8253 8253
2.05E +11
1.2 0.14'15
2.05E +11
%%
Jh Kgm2
J+ Kqm2
Mt Kg
frameaccelerated2 'w, 6 accelerated
2.34E - 3
1.0525 0
2.57E-4 0
2.55E - 5
motor 0
flexible robotarms ' /
W
Figure 2: Coordinate systems of the 2-link robot
edy for this problem is to choose another excitation
as input, namely the angular acceleration of the first
hub qhA. This is made possible by letting the first DC
servo-motor to work in speed mode, and this actuator
may be considered as an ideal speed source which is
independent of the load applied to the motor shaft or
of the generated torque.
and TB being adopted, the dynamic
analysis of the robot structure may be carried out in
an accelerated frame which is attached to the hub, or
The inputs
the rotor of the first motor since the hub is directly
driven by the rotor (see Fig. 2). With respect to this
frame, the description of the end conditions and the
excitations to the beam become somewhat different:
the first link becomes a cantilevered beam rigidly fixed
to the hub of the first motor, and the whole robot
structure are subjected to a field of inertial body force
introduced by the accelerated coordinate frames. The
modal analysis of the structure based on these con-
ditions has shown that the first and the second non-
rigid modes have natural frequencies far away sepa-
rated, and the first one is relatively low in fact, for
the experimental robot in a fully stretche6configura-
tion fi = 3.845 and f2 = 21.54 Hertz). This allows
therefore non-negligible advantages of using a slower
sampling and adopting a simpler reduced-order model
for controller design. Therefore, this preliminary anal-
ysis justifies the choice of the system inputs, and the
dynamic analysis and modelling in the sequel will be
carried on this basic assumption.
Euler-Bernoulli beam equation is adopted as the
differential equations governing the dynamic behav-
ior of the robot structure components with respect to
certain coordinate systems. In order to obtain a dy-
namic model of finite order, one has to discretize the
structure. The lateral displacement is expanded into
a power series in the spirit of Ritz
k
Wi(Zi,t)= 2(5)aki(t), i = A , B (1)
Lik=O
and the longitudinal displacements being assumed to
be uniform throughout the beams since the longitu-
dinal deformations are ne ligible, are represented by
two constant fields, i.e. uif~i,t)= bi(t), i = A, B.
Since the actual motions of the beams are re-
strained by two revolute joints hinges), in order for
a solution of the displacement fieds to be compatible,
the constraints on displacement and slopes at these
restraining points must also be considered.
Integrating the differential dynamic equations for
the component beams in the spirit of Principle of
Virtual Power with properly discretized displacement
fields, and taking into account the compatibility con-
ditions occurring at the joints, one obtains the follow-
ing linear ordinary differential equations for flexible
structures under excitation [3,4]
where K M ) is the stiffness (mass) matrix, F t ) is
the generaized force vector, and q is the generaized
coordinates vector. By changing the generalized co-
ordinates to the normal coordinates, one obtains the
following decoupled dynamic equation
where (U,fl are modal solutions, and T is the the
normal coold.inates vector. The displacement fields
for the beams can then be exressed as finite linear
238

3. combinations of some approximate modal shape func-
tions:
i = A, B (4)
where @ i ( z i ) and ! P i ( x i ) are respectively the modal
shapes in the transverse and longitudinal directions
generated by the modal analysis on the structure.
The acceleration of the first hub & & Abeing consid-
ered as one of the plant inputs, the velocity qhA and
the position qhA become naturally two state variables
of the plant. These state variables once chosen, one
can write the following state equation characterizing
the robot system
w i = @ T ( x i ) ~ ( t ) ui = ! P T ( x i ) ~ ( t ) ,
d
dt
- X = AX+BU (5)
where the state vector z = [ qhA
the input vector U = [ TB
matrices A(O,*),B(6;)have the following forms:
qhA TT i ’ ~lT,
qhA IT, and the system
0 1 0
O O O F T
A = [ :0 :0 -a2: ~]0 B = [ O O O E T
i being the time derivative of the normal coordinate
vector. Some accessible measurements are used as
modeshape No. 1 modeshape No. 2
Figure 3: Mode shapes of the 2-link flexible robot un-
der fixed-hinged-free boundary conditions
output variables: the position and velocity of the
two hubs qhi and qhi, the curvature ^/a of a point
x i = xsgd (i = A, B ) on the i-th link. These out-
put variables being chosen, the output equation of the
system can be written as follows
y = C z + D u (7)
where Y = [ qhA qhA ^/A ^/A qhB qhB IT is the output
vector, the output matrix C(6,*)is defined as
C =
- 7
B
s z + w:
state
flexiblelinks estimationdecouplercompensatormotors
Figure 4:Block diagram of the open loop system
3 Theoretical and Experimental Re-
Some experimental transfer functions of the robot
structure in its fully stretched configuration (i.e. 6; =
0) and with respect to different kinds of excitation
modes (hence different boundary conditions) have
been obtained using the harmonic excitation tech-
nique. The experimental results corresponding to a
specific testing condition are shown in Fig. 5 (the
positions of the strain gauges are x S g ~= -0.20LA,
XsgB = -0.092L~ and XsgBb = -0.95L~) and are
compared with the theoretical results based on dy-
namic analysis by eight truncated tenth degree poly-
nomial mode shape functions (the theoretical curves
are drawn in broken lines and the experimental curves
in solid line). The excellent agreements between these
experimental and theoretical frequency response func-
tions confirm the validity of the dynamic modelling
using the proposed approach.
4 Controller Design
sults of Dynamic Analysis
Since the first two characteristic modes have rel-
atively low associated characteristic frequencies and
these ones are separated far enough with the higher
eigen-frequencies, one can reasonably choose only two
modes for the purpose of controller design.
The dynamics of the complete manipulator must
include that of the actuators. For the two-link flexible
robot considered above the actuators are two direct
current permanent magnet motors driven by their own
servo-amplifier. For a reason been described above,
one has intentionally let the first motor, which ac-
tuates the first joint (or joint A), to work in “speed
mode”, while the second one, which actuates the sec-
ond joint (or joint B),in “current mode”. The math-
ematical descriptions for these ideal actuators are
qhA = K ~ A U ~ ATB = KmB’&B (9)
where U,A and U,B are the servo-amplifier input volt-
ages (see Fig. 4).
At the input of the first servo-motor (motor A),one
includes in series an analogue integrator compensator
for zero steady state position error and also for sim-
plification of the decoupling of the open loop system,
whose mathematical form is U,A = &u,A.
For the purpose of decoupling between the input
variables and the state variables, one can try to put in
series a decoupler before the input of the compensators
(8)
and D = 0.
239

4. of the following form (only two modes are considered) proposed
The state variables composed of two normal coor-
dinates and their time derivatives ( T I , T Z , +I and +z)
may be estimated as linear combinations of the output
variables (assumed four as well: q h B , Y A , 4~ and ? A ) .
It is obvious that the best estimation of the two nor-
mal coordinates can be achieved by multiplying these
outputs with the inverse of the output matrix, which
is non-singular for this case.
The dynamic model derived above is only a simpli-
fied model, and is only valid at low frequencies, while
the actual physical model is much more complicated
and its characteristic at higher frequencies may be ig-
nored by this over-simplified model. This unmodeled
characteristics may cause the problem of controller
spill-over. A remedy for this problem is to remove
high frequencies by inserting a low-pass filter in the
feedback paths. This low-pass filter can also avoid
the phenomenon of aliasing associated to the sampled
data system design.
Since the end-effector position control is our design
purposer the first tip angle instead of the hub angle
qhA is considered in the the tracking controller design.
The contribution of the rigid mode to the deflection of
the first beam tip being negligibly small (i.e. 4 ~ 1 ( 0=
the hub angle qhA by the following transfer function
O), the tip angle of the first link qtA can be related to
As there is a negligible payload, the tip angle of the
second link qtB is practically identical to the hub an-
gle qhB, since the bending deformation of the second
link is much less significant than that of the first link.
So, the transfer function between these two angles are
assumed to be an identity function.
Although three control schemes have been elabo-
rated and tested by simulation with success, due to
the limit of page number, only the design of a state
variable feedback control scheme in the discrete-time
domain for the two-link flexible robot arm is presented
here. In this scheme, only three state variables r1, rz
and qhA are used, hence only three sensors (two posi-
tions and one curvature) are required.
A discrete-time domain system model can be ob-
tained from the continuous-time model by adding a
zero-order hold at the plant input and a sampler at its
output. The transfer function of the plant in discrete-
time domain corresponding to the first (rigid) mode
motion has the following form for a sampled-data dou-
ble integrator
where T, is the sampling time. The following control
law based on sampled-data state variable feedback is
In order to ensure an appropriate response speed and
stability margin, the four closed loop poles will in
practice be allocated altogether to the neighbourhood
of z p l , resulting in a characteristic polynomial of the
form D ( 2 - l ) = ( ~ - , z ~ ~ z - ~ ) ~ ,where zpl is a value to be
determined. Identifying this characteristic polynomial
in 2-l to the denominator of the closed loop transfer
function, which can be found from Eqs. (12) and (131,
and applying the identity theorem yields five algebraic
equations with the four controller parameters hol, hll,
h21, Kcl,and the multiple pole zpl as unknowns to be
determined. The controller parameters of the feedfor-
ward block can be determined by the assumption of
zero steady state tracking errors for the first mode, or
equivalently zero error constants.
The same methodology can be applied for the de-
sign of a second control loop which aims to suppress
the undesirable vibrations mainly due to the funda-
mental mode of the structure. A hierarchical struc-
ture is adopted in designing the controller for the path
tracking of the first flexible robot link. The suppres-
sion of the fundamental vibration mode of the robot
arm being achieved by the second control loop, the
path tracking of the tip of the first link with respect
to a desired trajectory can be made possible by adding
upon the second loop the third set of feedforward-
feedback blocks properly chosen.
The discrete-time transfer function between the
control signal u3 and the hub angle qhA is found to
be
T," K , A P ~ ~z - l + z-' 1- u~.z-'+ z-'
qhA = -- U 3
where KO= l - c O s ~ l T ' e and a1 = 2 coswlT,. One pro-
poses the following control law based on discrete-time
state variable feedback
2 KoZ (1- z - ~ ) ~(1- ~ ~ ~ z - 1 ) ~
(14)
U 1
U3 = Ke3 [ ( f @ 3+f13.t-1 +fZ3Z-')qdA
-(ha3 +h13z-1 -k hZ3z-')qhA] (15)
The technique of pole placement allows the determi-
nation of the four controller parameters ho3, h13, h23,
Ke3,and all the poles of the closed loop system, these
being found to be stable poles. The conditions for zero
error constants can help the determination of the feed-
forward controller parameters fo3, fi3 and fZ3 present
for a better tip path tracking.
5 Simulation Results of State Feed-
The control laws obtained by the discrete-time
state variable variable feeforward-feedback scheme are
tested by computer simulation on SIMULINK. The
first and the second link are moved by the actua-
tors according to two trapezoidal velocity programs
back Control Scheme
240

5. tain configuration) within 0.6 sec. Under these small
(or bang-bang trajectories) characterized by an angle
increase of about loo from an initial position (i.e. cer-
amplitude motions, the open loop system can be de-
scribed by a simple dynamic model linearized about
this configuration, and the control scheme designed on
this dynamic model can therefore be applied. These
two bang-bang signals are given at two different in-
stants, so that one can observe the effect of coupling
between the modes. Two pulse type perturbations are
also applied to the inputs of the decoupler at two dif-
ferent instants after the bang-bangs, for the purpose of
testing the disturbance rejection ability and the stabil-
ity of the closed lop system. The simulation results for
a specific configuration (6’; = a) are shown in Fig. 6.
In this figure, the desired signals are drawn in dash-
dotted lines, the signals relating to link A are drawn
in dashed line, while the signals relating to link B are
u2 and u3 shown in these figures are smoothed by a
zero hold hold. In practice, these signals should be in
the second hub angle QhB (assumed equivalent to the
tip angle q t s ) show slight overshoots (less than 10 %)
at the edges of the trajectories, good damping prop-
erty (dying out in less than two cycles), zero steady
state tracking errors with respect to the steps, ramps,
as well as parabolic reference signals, and excellent
disturbance rejection ability. The performance of this
control scheme is nearly as good as that of the designs
in the continuous-time domain, although less sensors
of 10 msec, nearly 10 times longer than that used for
are required and a much longer sampling time (order
the continuous-time case) is adopted.
m,
B
E -
U
UU
* N O T
z z z z - k
(ap) u ! ~ (ap) WEE)
drawn in solid lines. Note that the control signals u1, 2 -1:- I
9
5discrete form while fed to the decoupler.
--The response curves of the first tip angle f&A and u s
(ap) WE)(ap)U!SD
Figure 5: FRF with TB as excitation ($hA = 0)
Acknowledgements N
This paper is based on a work supported by AS1
(Italian Space Agency).
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Figure 6: Simulation results of control Scheme
241