1988_pseudolinks and the self-tuning control
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40 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 18 , NO . 1, J A N U A R Y / F € B R I J A R Y 9XX
Pseudolinks and the Self-Tuning Controlof a Nonrigid Link Mechanism
Abstract -The pseudolink concept for representing the tip position of a
nonrigid link is developed. Pseudolink length and angle are determined on
line using joint encoder information together with strain measurements
taken along the link. The advantages to using the approach are that
external sensors are not required and trajectory planning may be carried
out upriori under a rigid link assumption. A self-tuning type control based
on pseudolinks is discussed. Laboratory experiments on a rotating nonrigid
link show the proposed control leads to an improved performance over a
control that ignores compliance.
1. INTRODUCTION
HE PROBLEM of controlling mechanisms having
Tonrigid links has received widespread attention in
the past decade, fueled in large measure by the develop-ment of controls for flexible-space structures. Of the
published treatments on controlling compliant link mecha-
nisms, the majority deal with developing an off-line opti-
mal control for a manipulator having a known endpointload [1]-[3]. Most studies to date are based on simulation
and have developed control schemes that are highly prob-
lem dependent. Of the reported laboratory experiments,
two schools of thought predominate: endpoint sensing and
modal suppression.
Endpoint sensing techniques make use of a sensor which
is external to the manipulator and in a known global
position to determine the global location of the mecha-
nism’s tip [2]. T h s is essentially what is used o n the space
shuttle manipulator where an astronaut uses visual feed-back to position the gripper. The problem with suchschemes is th at in three-dimensional space the sensors are
necessarily complicated an d expensive (e.g., a person o r a
fast hgh-resolution externally based camera).In modal suppression the idea is to detect and com-
pensate for compliant behavior through a combination ofpassive and active damping [4], [5]. Passive damping is
accomplished by adding a so-called constraining layer tothe ou tside su rface of the link. T h s viscoelastic layer
absorbs much of the energy from the second and higher
Manuscript received November 27, 1986; revised September 11, 1987.This work supported in part by the National Science Foundation underGra nt CDR-850 0022. This paper was presented in part at the 25thConference on Decision an d Control, Athens, Greece, December 1986.
D. C. Nemir is with the Electrical Engineering D epartment, Universityof Texas at El Paso, El Paso, TX 79968.
A J. Koivo and R . L. Kashyap are with the School of ElectricalEngineering, Purdue University, West Lafayette, IN 47907.
IEEE Log Number 8717875.
Fig. 1. Two degree-of-freedom manipulator.
modes whle adding only minimally to the link mass. Byusing strain gages mounted along the link, the first mode
vibrations are determined on line. Active damping is then
implemented by adding a correcting torque to the input tooffset the vibration.
This paper develops a pseudolink concept for repre-
senting the tip position of a nonrigid link. The pseudolinkis the line segmen t connecting the proximal and distal ends
of the link. Pseudolink length and angle are estimated on
line using joint encoder readings together with measure-ments of the strain at various points along the link. The
advantages to the approach are that sensors external to themanipulator are not required and trajectory planning may
be carried out a priori under a rigid link assumption. This
latt er feature is of great importance since it means that anyof the many trajectory planning schemes that have been
proposed for rigid link manipulators may be used. Control
may then be implemented by treating the (fictitious) rigid
pseud olinks as if they were the actual links of the manipu-
lator. The feasibility of t h s approach is demonstrated by
implementing a pseudolink based self-tuning control on a
laboratory model.
11. PSEUDOLINK PORTRAYAL OF LINKT I P POSITION
Fig. 1 depicts a two-degree-of-freedom (DOF) manipu-
lator. The revolute and prismatic joint variables are a ( [ )
and r ( ) , respectively. The maximum prismatic extension
is R. Thus the workspace of the manipulator is bounded
by a circle of radius R. Our focus is on the compliant linkassociated with the prismatic joint a nd all subsequentreferences to “the link” will mean the prismatic link.
Fig. 2 portrays a nonrigid link in global ( X Y ) coordi-
nates. If the link w ere undeform ed, it would lie along the
0018-9472/88/0100-0040$01.00 01988 IEEE
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NEMI R el U I . : PSEUDOLINKS AND THE SELF-TUNING CONTROL OF A NONR IGID LINK MECHANISM
YC.r(
0 1.98-
U
0
0. l . % -
L
1
C
1
6 1.53-
~
31
I X
Fig. 2. Compliant link.
dashed line. T h s dashed line is the so-called rigid bodyposition and makes an angle a ( t ) elative to the X axis.
The rigid body ( x y ) coordinate system shares a common
origin with the global coordinate system. In rigid bodycoordinates, at time t a point that is located a distance x
from the hub on the undeformed link is deflected a dis-tance y ( t , x ) away from the x axis. Displacement y ( t , x )
in the deformed link may be represented by
m
Y ( t 7 X ) = c p 1 ( t ) 4 , t x ) (1)
1 =1
where +,(x), i = 1,2, . . . is a set of appropriately chosen
basis functions; p , ( t ) , i =1,2, . . . is the corresponding set
of weighting; and x ranges from 0 to r ( t ) . It is conve-
nient to deal in normalized coordinates:
where ( ( t ) will range from 0 to 1.
A line segment drawn from the hu b of the link to the tip
will be denoted as a pseudolink. It has an angle of y ( t )
away from the X axis and is portrayed as a dotted line in
Fig. 2. Although the pseudolink is itself fictitious, the
control which we propose will be based in part on pseudo-link angle y ( t ) .
Using normalized coordinates the angle of the pseudo-
link relative to the rigid body position is
Using a series representation of the arctan function,
= J ( t , l ) . (3 )
When J ( , 1 ) = 0.2 (as portrayed in Fig. 3), the app roxima-
tion in (3) will differ from the true value by less than 1.5
percent. We make the following assumption.
Assumption A: The deflection of the normalized linkaway from the rigid body position is small.
Knowledge of the pseudolink angle and the pseudolink
length completely describe the tip position of a nonrigid
Fig. 3. Normalized tip deflection of 0.2
1.- 4 I--.o -37.5 a . 0 -1e.s 0.00 ie .5 s . 0 w.s so.0
Anqle Quay From Horizontal
Fig. 4. Pseudolink length as function of loading
link relative to the hub. Pseudolink length is obtained
under the following assumption.Assumption B: The tip of the d eformed link is a distance
r ( t ) from the hub where r ( t ) is the length of the und e-formed link.
Under this assumption, the pseudolink in normalized
coordinates will always have unit length. The validity of
this assumption is assessed via a series of static tests on a
loaded cantilever beam. The beam is aluminum tubing
(type 2024-T6), with an outer diameter of 11.11 mm and
wall thc kne ss 0.889 mm. The beam length is r =1.54 rn (a
constant). The beam is rotated in the vertical plane throughvarious angles a, relative to the horizontal. The loading is
gravitational, and measurements are made after all vibra-
tions have subsided. The use of different angles enables us
to observe not only the transverse displacements, but theeffects of v arious tensile an d compressive loadin gs as well.
Loads corresponding to M/m= 2, 4, and 6 were used,where rn is the mass of the unloaded link and M is the
end poin t load. For each configuration a picture was takenof the beam and digitized. The distance from the hub to
the tip (the pseudolink length) was then calculated. Themaximum observed tip deflection away from the rigid
body position was 0.252 rn (0.164 in normalized deflec-
tion).
Fig. 4 ummarizes the results from 37 experiments. The
means (standard deviations) for the pseudolink length for
M / r n = 2, 4, and 6 are 1.539 (0.0042), 1.534 (0.0042) and
1.533 (0.0040) m, respectively. Since pseudolink length
does not vary greatly over a variety of loadings, assump-
tion B appe ars well founded. It is important to stress that
this result is not generic. We claim it only for relatively
small deformations of a beam or link away from the
undeformed configuration (i.e., deformations satisfying as-
sumption A).
Th e link shape may be represented to any arbitrary
precision by truncating the summation in (1) to a finite k
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42 IEEE TRANSACTIONS ON SYSTEMS, M A N , AND CYBERNETICS, VO L 18. NO 1, J A N [ AKY/rE R K I , \ R Y 10x8
weighted bases, hence tions
k
jw) c P I ( t ) W ) .I = 1
Substituting (4) into (3 ) yields
Measurements of the joint variables a ( t ) an d r ( t ) ar e
available at any given time t . Strain measurements alongthe link are used to determine the weights p , t ) , which in
turn are used (see ( 6 ) ) to determine y ( t ) . The link tip
position is uniquely defined by y ( t ) an d r ( t ) .
111. DETERMININGIN KSHAPE ROMSTRAINMEASUREMENTS
Compliance in the link at a given time may be char-
acterized by the knowledge of the basis weightings
p l ( t ) ; . ., p k ( t ) n (4). For a beam of circular cross sec-tion, the strain at a point x along the beam is given by [ 6 ]
(7 )
where D is the diameter of the beam and ~ ( t ,) is the
strain in m/m . By normalizing to a link of unit length, thest ra in a t point t ( t ) t time t is
J
(10)
Th e weights p l ( t ) ; . ., p k ( t )may be determined a t time t
by solving (10). From (6) the pseudolink angle y ( t ) may
then be constructed. When the prismatic joint is retracted
so that E,( t ) < 0, the ith strain measurement is meaning-
less and must be ignored. At all times, however, the
estimate of k basis weightings requires a minimum of livalid strain measurements,
In normalized coordinates the set of basis functions
which are appropriate for describing link deflection in a
dynamical situation are not obvious. The candidates we
considered and discarded include a polynomial basis and
the set of eigenfunctions that are used to model the free
vibration of a clamped beam with an endpoint mass [ 7 ] .
The best results were obtained using the so-called u i n t i -
lever eigenfunctions (modes) that arise from the solution of
the partial differential equation that characterizes the free
vibration of a clamped beam with no endpoint mass. These
basis functions are given by [7]
q j ) = cosh q , - os q,E - l sinh q , - in q , i ,
The identification of k basis weightings in a link requires
strain measurements at s points along the link, where
coshq, +cos q ,
”= sinhn. +sinn . ’ ( 1 2 )
sa k . The strain measurements are made using strain
gages that are cemented to the link. Due to the prismatic an d q ,> 0 is the i th solution to
joint, the compliant link has a time-varying length. The COS 17, coshq, = -1. (1 3 )distance between the hub and the ith strain gage is thus
time varying. Denote the distance between the ith straingage and the hub by x ,( t , ( t ) ) .Then for arbitrary exten-
sions r ( t ) , the ith strain gage is attached to the link at adistance x , t , r( ) )= r ( t )- R - x , t , R)] from the hub,
where we note that x , ( t , R ) s the location of the ith strain
gage when the link is fully extended ( R is the maximum
prismatic extension). In normalized coordinates, this pointof attachm ent is
Although we find (via simulation and experimentation) the
cantilever eigenfunctions to be an excellent basis for a
compliant link in a dynamical situation, no theoretical
justification exists for their use. This is because ou r proh-
lem considers the forced rotation of a link with a nonzeroendpoint mass, a problem having a different set of
boundary conditions than those for which the cantilever
eigenfunctions are derived.
X I ( & r ( t > +x , t ,R )- R IV. TH ETIMESERIES ODEL. (9)
Two variables are needed to describe the manipulator
tip position at a given time t-the length of the prism atic
joint r ( t ) and the pseudolink angle y ( t ) . The variable t
4 4, ( t ) = r ( t )
Substituting j ( t , ( t ) ) rom (4) into (8) and evaluating the
strain at E l ( t ) , &(t) , . . .,E,(t), yields the system of equa-
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NEMIR el u/.: SEUDOLlNKS AND THE SELF-TUNING CONTROL OF A NONRIGID LINK MECHANISM
~
43
will represent a discrete time, that is, t E {T,2T, 3T- . . }
where T is the sampling time. The two-degree-of-freedom
manip ulator may be modeled in discrete time based on the
vectors
where u l ( t ) an d u 2 ( t ) are, respectively, the voltages ap-plied to the revolute and prismatic joints at time t and
{ wl( ) } , { w2(t)} re noise sequences. An autoregressiveexogeneous ( A M ) type time series for describing the two
DOF manipulator is
z ( r ) = A l z ( t - l ) + . . . + A , z ( t - n )
+ B,u(-
d ) + . . . + B,v( t-
m-
d )+ w ( t ) (17)where A , , i = l , 2 ; . . , n an d B,, = l , 2 ; . - , m are 2 x 2
matrices of c onstan t coefficients. This type of model we
denote as an ARX(n, m , d ) model since it has n autore-
gressive terms, m exogeneous terms and a delay d , where
n , m , an d d are positive integers. Using the delay operator
q - l , where q - l y ( t ) = y ( t - l ) , (17) may be rewritten as
A ( q - l ) z ( t ) = B ( q-').(t) + w ( t )
A ( q P 1 ) = I - A , q - l - . . . - A,q -"
(18)
(19)
where
an d
V. TH EEXPERIMENTALETUP
A single link mechanism has been built for carrying out
control experiments. The device uses joint 1 (a revolutejoint) of a Stanford experimental arm. Our setup has only
a single degree of freedom-no prismatic joi nt is involved.
The link is made from type 2024-T6 aluminum tubing
having circular cross section of diameter 9.5 mm and wall
thickness 0.89 mm. The link is 1.34 m long and rotates in a
horizontal plane in response to a torque applied at the
hub. The experimental setup is pictured in Fig. 5. Strain
measurements are taken at distances of 0.03, 0.64, an d
1.256 m fro m the center of the hub. Each point of s train
measurem ent co nsists of two diametrically opposed strain
gages mounted in the plane of bending. These form theupper two legs of a Wheatstone bridge. The advantage to
using two gages at a measuring point is that temperatureand torsional influences will affect both sides of the bridge
equally and therefore will cancel. From each amplifier the
Fig. 5. Fxperimental setup.
Fig. 6. Tip support
signal goes to a latched sample and hold. All three strain
signals are latched at the same time.
Th e end of the link is attached to a wheeled suppo rt(Fig. 6) that rides on a table. This support has two thin
wheels, one of wh ich is attached to a position encoder. Thepur pos e of the en coder is to determine where the link is on
the table relative to a known reference. It has a resolution
of 0.08 cm. Experimentally, the encoder yields informationon the pseudolink angle. Ths information is used for
verification only and is not used for feedback during
control experiments. Inputs and outputs to the system are
made via a Unimation controller operating in conjunction
with a VA X 11/780. An A/D conversion card in the
Unimation controller converts analog strain signals to
digital values with a resolution of 12 bits. Strain readings
and the position encoders at the revolute joint and the tipsupport are read at each sampling time. The tip support
has a mass of 0.613 kg, and the link has a moment ofinertia of 1.24 kg. m2 referred to the hub.
Th e setup jus t described was used in a series of ex peri-ments. The link tip is uniquely defined by pseudolink
length r ( t )= r =1.34 m and pseudolink angle y ( t ) . For
our problem we have determined [8] that know ledge of the
weighting corresponding to the first cantilever eigenfunc-
tion is sufficient to model compliance for this system.
Consequently, k = 1 in (6), and p l ( t ) is the only basis
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44 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VO L 18 , NO 1, ANI I ARY/ I t BRL ' AKY I O X X
Fig. 7. Pole placement controller
weighting that needs to be estimated. From (lo),
where t1= xl/r = 0.0224, and, similarly, c 2= 0.478 and
t 3 0.937. The function + ] ( E ) is given by (11) where
q1 =1.875. Making use of the Moore-Penrose pseudoin-
verse [lo], from (21),
p l ( t )= [36.25 13.63 0.271 c ( t , t 2 ) . (22)[The pseudoinverse in (22) reveals the relative importanceof strain signals at the three selected measurements points
along the link. The strain measurement at t1 (near thehub) is weighted more heavily than the strain at t 2 nd
much more heavily than the strain at c3 (near the tip).
Since dynamically the strain at t1 was observed to begreater than that at either t2 r t3,he strain measurement
at El is the major d eterminant of the basis weighting p l ( t ) .The rotating link is a single-degree-of-freedom system.
Under assumption B the pseudolink has a constant length
of 1.34 m. Knowledge of the pseudolink angle y ( t ) then
completely defines tip location. An ARX(n, in, d)-typetime series is used to model the system
&-'Mi)B ( q - l ) U l ( t ) + W l ( t ) (23)
A ( 4-1) = I - a,q-' - . . . - nq-n (24)
B( 4 - l ) = bOqpd . + b,,,q-d-m (25)
where
an d { v l ( t ) } an d { wl(t)} re, respectively, the input volt-
age and noise sequences. Given a model and model param-eters characterizing our single-input single-output system,
a control may be designed. The controller has a transferfunction of P (q -l) /L (q P1 ) and is portrayed in Fig. 7. The desired trajectory is y r e f ( t ) . The polynomials P (4 -l )
an d L ( 4 - l ) may be chosen so as to give the closed-loop
system a desired dynamical response. This is done by
choosing L(q- ') an d P ( q - ' ) so that [ l l ]
L ( q - ' ) A ( q - ' ) + B ( q - ' ) P ( q ' ) = A * ( y 1 ) (26)
where A*( 4-l) defines the desired dynamical response.
Since the coefficients of A ( q - ' ) an d B ( q - ' ) are notknown, they must be estimated on line an d the estimates
used to construct the control. Ths is the basi\ for a
self-tuning control scheme.
At each sampling time, strain measurements are takenand used to determine p l ( t ) . The position encoder a t thehub determines a( t ) .Equation (6) is then used to generate
the measured pseudolink angle y ( t ) . The estimation I S
effected via sequential least squares where the parameter
an d regress ion vectors are, respectively,
The algorithm implements the following two update stagesat t ime t :
8,=6,- + P,@( - ) y ( )- T( t - 1)6, , (29)
where P, is an ( n + m + l ) x ( n m + 1) matrix, P,,=
10001, 8,=0, an d A is a forgetting factor [12j. Th epurpose of A is to downweight old measurements i n
constructing a parameter estimate. All experiments used a
value of A = 0.95.
The implementation of self-tuning for controlling pseu-
dolink angle y ( t ) requires the following steps at each
sampling time:1) read the h ub encoder a ( t ) and the strains c ( t , t l ) ,
2) calculate p l ( t ) using (22);
3) calculate y ( t ) using (6);
4) update the parameter estimates 8 ((28) and (29));
5) compute the coefficients in L ( 4 - l ) and P (4 - l ) ( see
6) construct the pole placement control L(q-')ui(!) =
' ( t , 213 ' ( 4 3 ) ;
(26));
p ( q - >Y(t>.
VI. EXPERIMENTAL RESULTS
The experimental setup described in Section V w as used
to m ake a series of tests. All experiments use the reference
trajectory and corresponding velocity profile shown in Fig.
8. The desired pseudolink angle rotates through an angle
of 45", comes to a complete stop, and then returns. Thedesired polynomial in all experiments was chosen so as to
g v e a second-order response
A *@ ' ) =1- a ,q- ' - a z q 2 (30)
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N L M I R t'l d : SLIJDOLINKS AN D THE SELF-TUNING CONTROL OF A NONRIGID LINK MECHANISM
t
tl----A- 1 .00
-2.00
0 0 0 9 . 1 0.64 12.9 17.2 Il.5 25.0 0 . 1 3l .q
TIME ( S K )
Fig. 8. Reference trajectory.
3.50
h D e s i r e d . - . . _ - -Observed-2.00t
0.00 Y . 3 0 0.61 12.9 17.2 21.5 B . 0 m.1 Pi,+
TIRE ( s ec )
Fig. 10. Response of compliant link under benchmark control.
Ibserved -D e s i r e d - - - - - -
3.50
Observed -e s i r e d .~ . .
2.000.00 'I.= 0.61 11.9 17.2 Zl.3 m.0 30.1 W.9
TIRE (sec)
Joint position for compliant link under benchmark control.ig . 11.
3.50 1 I
e.oo 4 I0 . 0 0 Lt.30 8.61 12.9 17.2 21.5 2S.8 30.1 3t.9
TIME ( sec)
Fig. 9. Response of rigid link under benchmar k control.
where a , = 2 e p ! " , ~ r c o s ( T w , / ~ ) ,u 2= - -2s"nr, and
(, a,, re, respectively, the damping factor a nd un damp ed
natural frequency corresponding to a continuous-time sec-
ond-order system sampled every T seconds.As a b enchm ark, we consider the self-tuning control of a
rigid link having a mom ent of inertia referred to the hub of
1.24 kg -m 2 (the same as for the compliant link described
in the previous section). A sampling time of 0.021 s is used
an d A* ( q p l ) is chosen to co rrespond to { = 0.8, 0 =100.
An ARX(2,2,1) model is used leading to the results in Fig.
9. Except for the initial samples (the learning period), the
pseudolink angle y ( t ) ( = a ( t ) for t h s rigid link) closely
tracks the desired trajectory. We stress that the benchmark
control w as based on joint angle only, strain measurements
are irrelevant for this rigid link case. Consequently, the
algorithm of Section V must be modified to omit steps 2)
and 3) and replace y ( t ) by a ( t ) .
When the same self-tuning control (using an AR X(2 ,2,1)model based on a ( t ) )was applied to the compliant link,
the pseudolink angle portrayed in Fig. 10 was obtained.
These data were generated using the encoder at the tip
support. The corresponding hub position is shown in Fig. 11 . The two plots demonstrate the problems that can arise
from igno ring compliance. Since the control is based o n a
Observed-2.00 4 I
0.00 Y . 3 0 8.61 12.9 17.2 21.S 2S.8 30.1 3.9
TIME ( se c )
Fig. 12 . Pseudolink determination using strain measurements.
rigid link assumption, the hub encoder has no means of
detecting errors in tip position which occur due to nonrigid
behavior in the link. The control, consequently, has no
means of correcting for these errors. Fig. 12 portrays the
pseudolink trajectory, as generated from strain measure-
ments, which was obtained. A comparison of Figs. 10 an d
12 reveals a close correspondence between the pseudo link
angles resulting from tip encoder measurements and those
generated from strain measurements. Thus for this particu-
lar experiment the first mode content was sufficient to
describe the tip deflection due to compliance.
The choice of a model for the compliant link is notobvious. A compliant link is a distributed parameter sys-
tem and is thus inherently of infinite order. Furthermore,
there is a significant propagation delay between the time a
torque is applied to the join t to th e time that the tip of the
link responds. Given that the model structure has the form
of (23), appropriate values of d , n , an d m must be
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46 IEEE TRANSACTIONS ON SYSTEMS, MAN, A ND CYBERNETICS, VOL. 18, NO . 1, JANUARYjFEBRUARY 1988
TAB LE I
DETERMINATIONF MODELDELAY
Sampling Time Total Delay System Propagation
T (SI d T X d -1 (s)
0.014 8 0.0980.021 6 0.105
0.028 5 0.1120.035 4 0.105
0.042 3 0.084
0.049 3 0.098
determined. Du e to the amount of time required for up dat -
ing parameter estimates and constructing the control, the
minimum sampling time is 0.014 s. For the model order
determination, six different sampling times ranging from
0.014 s to 0.049 s were used. For each sampling time, sets
of { ul ( f )}, { y ( f )} were experimentally determined by op-
erating the system in closed loop using a self-tuning (and
hence time-varying) control based on { a ( t ) } . For each
data set the first 300 samples were used to fit the model.
The model was then used to predict the next 1000 samples.
The variance of the resulting error sequence between true
and predicted y can be used to make a decision as to thecorrect model order [13]. There is always a unit delay in
the system due to sampling. Thus the delay due to system
propagation is d - 1. For all combinations of n an d m
(with n and m ranging from 1 o 9) the smallest residuals
were obtained for the values of d shown in Table I. In all
cases the delay due to system propagation was found to be
approximately 0.1 s.
As ex pected, the mean square value of residuals that
were obtained in f itting models to the data decreased as
the model order increased. An interesting finding is that
for a given num ber of coefficients n + m in the model, the
residual variance is much more a function of the exoge-
nous order m than of the order of the autoregression. We
hypothesize th at this is due to the distributed nature of the
system-the control at a given time must prop agatethrough the system before it influences the tip. A reaction
torque is then propagated back.
A variety of different models and performance criteria
were investigated in applying pseudolink based self-tuning
to the control of the compliant link. In most instances the
pseudolink-based control yields an improved performance
over a control which ignores compliance (e.g., the
benchmark of Figs. 10 and 11). In some instances it is
worse. Of ou r successes, two basic types of beh avior were
noted. For some controls the hub seems relatively insensi-
tive to vibrations of the link, while for other implementa-
tions the hub moves so as to actively damp out oscillatory
behavior. These two types of b ehavior are characterized by
the following two experiments.
n
U
L"
Y.-A
0
a
3.9
D e s i r e d - - _ - - Observed -k
2.00 J , , , , , , , /
0.00 Y .30 B.61 IC.) 17.2 Bl.5 ;cS.B P .1 P!.V
TIME ( sec)
Fig. 13. Pseudolink angle using control based on AR X(1 ,5,5) model.
3 . 0
1 D e s i r e d - - - - - Observed- 1
2.00 I0 .00 9.30 B.61 12.9 17.2 21,s 2S.S m .1 m.11
TINE ( sec )
Fig. 14. Joint angle using control based on ARX(1,5,5) model.
characterize the system are thus assumed to be
A ( q - 1 ) =1- a,q-1 (31)
and
B( 4 - l ) = b0q-' + blq-6 + . . . + b4q-9. (32)
The corresponding controller is characterized by the poly-
nomials
p ( q - ' ) = p04- l (33)
(34)
and
L (4-1) = 1+ /,q- 1+ . . + , q - 9 .
As in the benchmark, the control objective was pole place-
ment control with the desired closed-loop response given
by (30), where [= 0.8 an d w, =100. Figs. 13 and 14 depict
the pseudolink angle and corresponding hub angles that
were obtained. After a brief learning period, the pseudo-
link angle tracks the reference trajectory with some over-
shoo t at the peaks of the trajectory. Thw overshoot is
much less than that which was experienced with the
benchmark control (Fig. 10).A com parison of Figs. 11 an d
14 reveals the reason for the improved performance. Un-
like in the benchmark, the hub angle in this experiment
behaved sluggishly and did not reinforce vibratory behav-
ior.
Experiment I Experiment 2
The system was modeled as an ARX (1,5,5) time series,
using a sampling time of 0.021 s. This corresponds to a
system delay of 5 X 0.021= 0.104 s. The polynomials that
A n AR X (4,4,3) model was used, and control was imple-
mented using T = 0.035 s. This model assumes a system
delay of 3 X 0.035 = 0.105 s. The controller was designed to
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NLMIR ei o/.: PSEUDOLINKS AND THE SELF-TUNING CONTROL OF A NONRIGID LINK MECHANISM
-47
3.50,. Desired - - - - - - Observed-I
111 1
T I N E (sec)
Fig. 15 . Pseudolink angle using control based on ARX(4,4,3) model.
3.50
Desired - . . Observed-e .00
40 . 0 0 9.30 8.61 12.9 17.2 81.5 25.8 30.1 3 . 9
T I N E (sec)
Fig 16. Join t angle using control based on ARX(4,4,3) model.
give a second-order closed-loop response corresponding to
5 =6, a,, 100. The pseudolink angle and corresponding
hub angles are shown in Figs. 15 and 16. Unllke the
A R X (1,5,5)-based control of experiment 1, the hub posi-
tion for the ARX(4,4,3)-based control undergoes rela-
tively large excursions to position the pseudolink, effec-
tively damping out vibrations.
VII. DISCUSSION
As in all adaptive control techniques the self-tuning
algorithm combines both identification and control. Each
of these two facets of the technique presents its own
problems and possibilities. The issue of an accep table
model for the system is still an open question as evidenced
by the results in experiments 1 an d 2 (both representing an
improvement over the benchmark) using entirely differentmodels. In the identification stage of the self-tuning al-
gorithm, we used a constant forgetting factor and em-
ployed a dead zone.' Use of constant forgetting may lead
to the so-called drift and burst phenomenon, and in our
experiments we fo und the choice of the factor to have a
great impact on the performance. Alternatives are to use a
variable forgetting factor or periodic covariance resetting
[ 111. The sampling time used in expe riments played a role
in the success of the control. Due to system limitations, theminimum sampling time is T = 0.014 s. However, better
1 Updates of the parame ter estimates are made at a given sampling timeonly i f the identification residual exceeds some specified threshold [ll].
This is a reasonable modification since at a given time if the identifica-tion residual is small, the model adequately describes the output and doesnot need to be updated.
performance was achieved by using larger values of T.
There are two possible reasons. With a larger sampling
time a given measurement provides more new information
about the system, enhancing identification. Furthermore,
high-frequency disturbances are effectively filtered out [14].
For our experimental setup we found the first mode to
give a satisfactory description of link com pliance. How -
ever, pseudolink angle y may be constructed using any
arbitrary number of basis weightings and using (10) and
( 5 ) . The only restriction on the number of bases is that
there be a t least as many points of strain measurement as
there are basis functions. Our feeling is that to minimize
errors, it is preferable to use redundant strain measure-
ments in assessing the first mode. For other problems,
particularly for controlling mechanisms exhibiting a higher
degree of link co mpliance, it may be necessary to estimate
the second and perhaps even thrd mode content.
A compliant link manipulator is complicated. The dis-
crete time model for such a mechanism is at best only an
approximation of the true system. Assuming a correct
model, our regulator forces the closed-loop system to have
second-order dynamics with a specified 5 an d w,. As with
any second-order system, there is a trade-off betweenspeed of response and overshoot. In experiments 1 and 2,
an d w, were chosen as a compromise between these fea-
tures. Unless future reference samples are used in con-
structing the present control (a realistic possibility since
the desired trajectory is known u priori) , a lag of at least
the pro pagation delay must be tolerated.
Link compliance in a robotic manipulator impacts not
only the link tip position but the tip orientation as well.
Thi s is of special concern when a g ripper or additional
links are to be considered. For the two-DOF m anipulator
the orientation at the tip, relative to the
tion, is defined by the angle
d? , (1)= c p 1 i 4 7
1 = 1
where we have used an approximation
leading to (3).
rigid body posi-
(35)
similar to that
The-technique in t h s paper is readily applicable to the
control of a m ultilink mechanism where all links are rigid
except for the most distal link. The analysis may be
extended to a nonrigid link operating in three dimensions
by adding additional strain gage pairs in q uadrature with
the existing gages. Bases corresponding to deflections in
two planes of bending could then be determined and used
to assess pseudolink position. When gravity is a factor (as
it is in all terrestrial mechanisms), an additional bias termin the model will have to be identified on line [15].
VIII. CONCLUSI ON
This paper developed a pseudolink concept for de-
termining the tip position of a com pliant link. Experiments
indicate that for a given prismatic extension the pseudo-
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4x IEEE T RANS ACI I ONS ON SYSlLMS, M A N , AND CYBFRNETICS. VO L 18 . NO 1. J A N l l A R Y / F L B K I I A R Y 1988
link length remains constant under a variety of static
loadings. Based on this result, at time t the tip position of
a two-degree-of-freedom compliant link manip ulator is
completely specified by the prismatic extension r ( ) an d
pseudolink angle y ( r ) . A self-tuning control based on r ( t )
an d y ( t ) was proposed for the two-degree-of-freedommanipulator. Laboratory experiments on a rotating com-
pliant link with no prismatic extension show the pseudo-
link based control to have an improved performance overa control that ignores compliance. The self-tuning feature
of the control makes it applicable for different endpoint
masses.
Th e major advantage to basing a control on pseudolinks
is that trajectory planning may be carried out a priori
under a rigid link assumption. It is important to emphasize
tha t the pseud olink idea is not restricted to use with a
self-tuning control. Virtually any technique that has been
proposed for the on-line control of a rigid link robot could
conceivably be applied to controlling nonrigid links by
basing the control on pseudolink angle rather than joint
angle. Such techniques include, but are not limited to,
variable structure control, model reference control and the
classical PID control.
ACKNOW LEDGMENT
The authors gratefully acknowledge Tom Howell, theAluminum Company of America’s CIDMAC representa-
tive at Purdue, for his help in designing the experimental
apparatus .
REFERENCES
Y. Sakawa, R. Ito, and N. Fujii, “Optimal control of a flexiblearm,” in Control Tlieorv for Distrihuted Purunieter Systems i ind
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R. H. C anno n and E. Schmitz, “Initia l experiments on the end-pointcontrol of a flexible one-link robot,” In t . J. Robotics Res.. vol. 3,no . 3, pp. 62-75, Fall 1984.
G. G . Hast ings and W . J. Book, “Experiments in optimal control of
a flexible arm ,” in Proc. 1985 American Control Conf., pp. 728-729,1985.W. J. Book, “Combined approaches to lightweight arm utilization,”Robotics Munuf . , vol. 4. no. 3, pp. 97-107, May-June 1985.T. Alberts, G. Hastings, W. J. Book. and S. Dickerson, “Experi-ments in optimal control of a flexible arm with passive damping,”presented at the 5th VPI&SU /AIAA Symp. Dynamics and Contro lof Large Structures, Blacksburg, VA, June 1 2, 1985.S. Timoshenko and D. H. Y oung, Elmients of Strength of Muteriuls.
Princeton, NJ: Van Nostrand, 1968.D. J. Gorman, F re e V ib ra ti on A n u l ~ wf Beams und Shufts. Ne wYork: Wiley, 1975.D. C. Nemir, “Analysis and self-tuning control of a rotatingcompliant link,” Ph.D. dissertation, Dept. of Electrical Engineering,Purdue Univ., West Lafayette, IN , Dec. 1986.G. R. Widmann, “Control system design of robots with flexiblejoints,” in Recent Trends in Robotics: Modeling, Control unci Eiiucu-
t i on , M . Jamshidi, L. Y. S. Luh, and M. Shahinpoor, Eds. New
York: Elsevier, 1986, pp. 145-150.A. Albert, Regression and the M oore- Penrose Pseudoinirerse. Ne wYork Academic, 1972.G. C . Goodwin and K. S. Sin, Aduptirie Filtering, Prediction und
Cont,ol.
L. Ljung and T. Soderstrom, Theory un d Prucrice of Recursii)e
Identification. Cambridge, MA: MIT Press, 1983.R. L. Kashyap and A. R. Rao, Dynuniic Stochastic Models from
Enipiricul Data.
New Yo rk: Springer-Verlag, 1983, pp. 175-187.
Englewood Cliffs, NJ: Prentice-Hall, 1984.
New York: Academic, 1976.
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IC. J. Astrom and B. Wittenmark. Computer Corrrrolletl ~Y.v.~tert~.\
Englewood Cliffs, NJ: Prentice-Hall, 1984.A . J. Koibo and T. H. Guo. “Adaptive linear controller for roboticmanipulators,” I E E E Truns. Automur. Cotit. , vol. AC-2R. no. 2 , pp .162-171, F eb . 1983.D. C. Nemir, A. J. Koivo. and R. L Kashyap, “Self-tuning controlof a rotating compliant link,” in Proc. 25th Con/ . Decision uti(/
Control. Athens, Greece, 1986.
[16]
David C. Nemir (AX-S’82-M’X5) received theB S E E degree at the Univtrsity of T e u . ,Austin . in 197X. the M S degree from tiannonUnibersity, Ene. PA in 1981. and the Ph D
degree from Purdue Unicersitv, in 1986
From 1978 to 1981 he iwrked for the (ieneralElectn c Company, Erie, We\: L afayette. I N , de -sigmng dc motors and microproLewr-hawdmotor controllers From 1981 to 1986 he wa\
employed by Purdue Unicer>i!y ax a TeachingAssistant, Research Assistant, and Lecturer He
is presently an Assistdnt Professor i n the Electrical Engineering Depart-ment at the University of Texas at El Paso Hi s current rewarch interwtkinclude system identification. variable structure control. and thc modelingand control of distributed parameter system5
A. J. Koivo (S’61-M’63-SM’XO) received the Di-
ploma in electrical engineering at the FinlandInstitute of Technology, Finlantl. the h4.S de -gree at Indiana U niversity, and the Ph.D . degreeat Cornell University. Ithaca. N Y .
Currently, he is a Professor in the School ofElectrical Engineering a! Purdue University.West Lafaye!te, Ii% His research interests in -
clude optimal estimation and identification. an dcomputer control with specific applications to
robotics. HIScurrent research is concentrated in
the sensory feedback control of rigid and flexible-link robotic manipula-tors and intelligent assembly for manufacturing.
Dr. Koivo is a Rotary Fellow and a member of Eta Kappa Nu andSigma Xi. H e received the 1979 D. Ewing Award for Excellence in
Teaching in the School of Electrical Engineering at Purdue UniversityHe has been a member of organizing program committees and programchairperson of several national and international conferences on roboticsand automation. He is a liaison representative of the IEEE ControlSystems Society in the IEEE Robotics and Automation Council. He iscurrently a member of the Administrative Committee ( A D C O M ) and theChai rman of the Technical Committee on Robotics in the IEEE Systems.
Man , and Cybernetics Society.
Rangasanii L. Kashyap (M’70-SM’77- F’RO) re-ceived the D .I.I.Sc. and M.E . degrees from theIndian Institute of Science. Bangalore, India, in1960 and 1962, respectively, and the Ph.D. de-gree in engineering from Harvard Univeriity,Cambridge, MA, in 1965.
He is currently a Professor of Electrical En-gineering and Associate Director of the NationalScience Foundation supported‘Engineering Re-search Center on Intellieent Manufacturine atPurdue University, West Lafayette, IN. He was a
Postdoctoral Research Fellow in Applied Mathematics at Harvard Un i -
versity du ring 1965-1966. During the fall semester of 1974, he was aVisiting Professor at Harvard University. During !he spring semester of1974, he was a Visiting Research Associate at the University of Cali-fornia, Berkeley. His current research interests are random fields in image
processing, applications of artificial intelligence methods in manufactur-ing and robotics.
Dr. Kashyap won the 1967 National Electronic Conference AnnualBest Research Paper Award for his paper “Optimization of stochasticfinite-state machines.” H e was an Associate Editor of the IEEE TRANSAC -TIONS ON AUTOMATIC CO NTR OL. e has been the Program Chairpe rsonfor several meetings of the IEEE Computer Society in the area of patternrecogni tion a nd ima ge processing. He is a member of the Association forComputing Machinery and Sigma Xi.