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



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



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-



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



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)



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



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



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-



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

Applications.

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.

1141

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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.

Documents

1988_Pseudolinks and the Self-Tuning Control