A general control scheme to control flexible arms with friction in the joints is proposed in this paper. This scheme presents the advantage of being robust in the sense that it minimizes the effects of the Coulomb friction existing in the motor and the effects of changes in the dynamic friction coefficient. A justification of the robustness properties of the scheme is given in terms of the sensitivity analysis

Brown, H. Benjamin, Jr.

Feliu, Vicente

Rattan, Kuldip S.

English

NASA Technical Reports Server

N89- 19815 
A ROBUST CONTROL SCHEME FOR FLEXIBLE ARMS 
WITH FRICTION IN THE JOINTS 
Kuldip S. Rattan' , Vicente Feliu2, and  H. Benjamin Brown Jr. 
Robotics Institute 
Carnegie Mellon University 
Pittsburgh, Pennsylvania 15213 
ABSTRACT 
A general control scheme to control flexible 
arms with friction in the joints is proposed in 
this paper. This scheme presents the advantage 
of being robust in the sense that it minimizes the 
effects of the Coulomb friction existing in the mo- 
tor and the effects of changes in the dynamic fric- 
tion coefficient. A justification of the robustness 
properties of the scheme is given in terms of the 
sensitivity analysis. 
1 INTRODUCTION 
In the last few years, considerable research effort 
has been devoted in controlling flexible structures 
and, in particular, flexible arms [l-61. Very little 
effort has been made in controlling flexible arms 
when Coulomb and dynamic friction are present 
in the joints, in spite of this being common in 
practice. In fact we have not found any paper 
that specifically deals with and tries to solve this 
problem. 
This paper is devoted to solving the above prob- 
lem. In order to do this, a new general control 
scheme is proposed. Existing methods to control 
flexible arms are based on the explicit control of 
the tip position, where the controller generates 
the current for the D.C. motor of the joint as a 
control signal. The proposed method is based on 
the simultaneous control of the joint motor posi- 
tion and tip position, and the implementation of 
two nested closed loops : the inner loop that con- 
trols the motor position and the outer loop that 
controls the tip position. 
The general control scheme is presented in Sec- 
tion 2. Section 3 compares the sensitivity of our 
general control scheme with the sensitivity of other 
existing methods. This shows our scheme to be 
more robust to changes in friction than the others. 
Finally, some conclusions are drawn in Section 4. 
2 GENERAL CONTROL 
SCHEME 
As was mentioned in the introduction, there are 
many applications in which friction must be taken 
into account when controlling a flexible arm. Only 
in the case when the coupling torque between the 
beam and the motor is many times larger than 
the friction torque it can be neglected. This may 
be true for very large structures, but it is not true 
in many industrial applications. 
In order to reduce the effects of friction, the 
control scheme of figure 1 is proposed. In this 
scheme two variables are controlled: motor and 
tip position ( e ,  and et respectively ). These two 
variables are controlled by two nested closed loops 
and two different controllers (Rl(s) and Rz(s ) )  
a.re used. These controllers are each designed sep- 
arately and according to different criteria. In fig- 
ure 1 ,  the open-loop transfer function of the mo- 
tor G M ( s )  has all its poles and zeroes in the left 
half-plane. The open-loop transfer function of the 
flexible beam G B ( s )  has its poles in the left half- 
plane but may have ( for arms with more than 
one vibrational mode ) some zeroes in the right 
half-plane (non-minimum phase system). F(s) is 
a filter for the reference and is normally designed - -  
'Visiting Professor, Department of Electrical System Engineering, Wright State University, Dayton, OH. 45435. 
'Visiting Professor, Dpto Ingenieria Electrica y Control, UNED, Ciudad Universitaria, Madrid-28040, Spain. 
429 
https://ntrs.nasa.gov/search.jsp?R=19890010504 2020-03-20T03:49:44+00:00Z
I in conjuction with Rl(s). 
The use of this control scheme has been moti- 
vated by the well known property that the sen- 
sitivity of a closed-loop system to perturbations 
can be made arbitrarily small by increasing the 
gain of the open-loop, provided that the system 
remains stable ( Kuo [8] ). Therefore considering 
the friction (in general terms) as a perturbation, 
we can reduce its effects by increasing the gains 
of the controller. 
If we try to do this using the existing control 
arms, the gains cannot be arbitrarily increased 
because of the right half-plane zeroes. In the pro- 
posed scheme, because G M ( s )  is minimum phase, 
the gains of the inner loop can be arbitrarily in- 
creased ( using an appropiate controller Rz (s)) 
without making the system unstable. Intuitively, 
the high gain inner loop to control the motor posi- 
tion makes the system insensitive to friction and, 
a second outer loop may then be designed to con- 
trol the tip position. This second loop cannot 
have a high gain because G g  (s) is non-minimum 
phase. However, the friction effects have been 
largely removed by the high-gain inner loop con- 
trolling the motor position. 
I schemes (such as shown in figure 2) for flexible 
3 SENSITIVITY ANALYSIS 
The previous ideas intuitively justify the reason 
for using our control scheme. This section is de- 
voted to  giving an analytical proof. The analysis 
carried out here is quite straightforward and will 
give a quantitative idea of how much the robust- 
ness is increased using the nested loop scheme. 
In order to do this comparison, a typical con- 
trol scheme like the one shown in figure 2 will 
be used ( Cannon 111 ). The sensitivity char- 
acteristics of this system will be taken as rep- 
resentative of the existing methods because they 
are based on controlling the tip position using a 
controller that generates a command for the cur- 
rent of the D.C. motor. The sensitivities of these 
methods are thus of the same order of magni- 
tude. Two comparative analyses will be done : 
one checking the signal-to-noise ratio (consider- 
ing the Coulomb friction as the noise), and the 
other checking the sensitivity to variations in the 
dynamic friction coefficient. 
In order to do this comparative analysis, the 
state-space control scheme of figure 2 is first ex- 
pressed in terms of its equivalent transfer func- 
tions. 
Assume that the plant is represented by the 
state space-equations 
where C = [ 1 0 0 0 . . - 0 ] is of dimension n. The 
controller is a row vector of dimension n of the 
form 
where K is a vector of dimension n-1, and IC1 is 
the coefficient corresponding to the tip position 
error (error in the first state). The states x of the 
system can be reconstructed from the state-space 
equation (1) as follows 
0 0 
C B  0 0 
C A B  C B  0 . . o  
. . .  
. . .  
C A n - 2 B  CAn-3B CAn-4B 
(3) 
where [C CA C.4' CA"-l] is the observabil- 
ity matrix. If the system is observable (which is 
true in all the models of flexible arms), then the 
states g of the system may be reconstructed from 
the measurements of the input i and the  output 
using a linear law of the form 
430 
where P and Q are polynomial column vectors 
in s. Equation (4) can be easily obtained from 
equation (3) by inverting the observability ma- 
trix. Adding equation (4) in the scheme of figure 
2 and substituting the state-space equation (1) of 
the plant by G M ( s ) G B ( s ) ,  we obtain the system 
shown in figure 3. Figure 3 can be simplified and 
an equivalent transfer function scheme can be ob- 
tained, as shown in figure 4, where 
and 
The comparative analysis may now be done 
between the schemes of figures 1 and 4. 
3.1 Signal-to-Noise Ratio Analysis 
The signal-to-noise ratio of a system is defined 
( Kuo [8] ) as 
(7) 
Output due to signal 
Output due to noise 
R =  
and is a measure of the sensitivity of the system 
to perturbation signals (in this case the Coulomb 
friction). The comparison of the ratios of the 
schemes of figures 1 and 4 is done here for the 
same levels of input 0; and perturbation p (which 
is added to the current of the motor). 
The output of the nested double loop scheme 
in terms of the reference input 6; and the noise p 
can be written from figure 1 as 
The signal-to-noise ratio for this scheme is given 
by 
Similarly the output and the signal-to-noise ra- 
tio for the system shown in figure 4 can be written 
as 
Comparing both results, expression (9) can be 
made large by designing the controller Rz(s) with 
arbitrarily high gain because the inner loop is 
minimum phase. However, ICl in expression (ll), 
and all the parameters of R(s) in general, are lim- 
ited by the stability margin since the system is 
non-minimum phase. The gains of R1 (s) in equa- 
tion (9) are also limited for the same reason. 
From all this it follows that, in general, (9) 
may be made larger than (11) by properly choos- 
ing the gains of the controller of the inner loop. 
It must be mentioned here that this is a theoret- 
ical analysis. In practice, the gains of the inner 
loop will be limited by the saturation of the am- 
plifier, instability because of unmodelled high fre- 
quency dynamics, or even instability because of 
the discretization of the signals when using dig- 
ital controllers. But in any case these limits are 
much larger than the ones imposed by the non- 
minimum phase characteristic. 
3.2 Sensitivity to the Dynamic Friction 
Dynamic friction is the second manifestation of 
friction. This is normally assumed to be linear 
and is included in the model of the plant. In 
many cases, however, this assumption of linearity 
is not correct. Often, the dynamic friction coeffi- 
cient changes noticeably depending on the sense 
of rotation of the motor [9] or on the position of 
the rotor of the motor relative to the stator (con- 
Coefficient 
43 1 
fronted poles), etc. It will be shown here (by per- 
forming the sensitivity analysis of both systems to 
changes in the dynamic friction coefficient) that 
the robustness of the system due to changes in the 
dynamic friction coefficient may be significantly 
improved by using the nested double loop scheme. 
The ratio given by equation (16) is significantly 
smaller than 1 since Rz(s)(l + R1(s )GB(s ) )  >> 
R ( s ) G B ( s ) .  Notice also that the gains of R ( s )  
and Rl( s )  are bounded by a stability margin and 
will thus have the same order of magnitude; but 
the gain of Rz(s) may be increased arbitrarily. 
Consequently, scheme of figure 1 is more robust 
in general to changes in the dynamic friction than 
the scheme of figure 4, by a factor of approxi- 
mately R ~ ( s ) .  
The sensitivity to changes in the parameter v 
of a system whose closed-loop transfer function is 
M(s),  is defined (Kuo [8] ) by 
(12) 3.3 Comparison of the Characteristic d M ( s )  /W )  dv/v  su,u = 
Equations 
The previous analysis gives a quantitative justi- 
fication of how the robustness of the system is 
increased using the two nested loops scheme. A 
qualitative understanding of how the nested loops 
modify the stability of the system allowing higher 
gains may be obtained by looking at the charac- 
teristic equations. of the two systems. 
In order to do this analysis, let us express G M ( s )  
in the form 
(13) 
Kln 
G M ( s )  = 592 + U S  + T ( s )  
which is the typical transfer function of a D.C. 
motor, except the term T(s) which represents the 
coupling torque between the beam and the mo- 
tor. This allows us to characterize the influence 
of the dynamic friction coefficient, v ,  in the gen- 
In the traditional control scheme, the robust- 
ness depends on R(s). The characteristic equa- 
tion of the system shown in figure 4 is given by 
eral transfer function. 1 + R(s )GM(s )GB(s )  = 0 (17) 
Performing some calculations, we find the sen- 
sitivity to v for the closed-loop system of figure 1 
is given by 
Substituting equation (6) in (17), the charac- 
teristic equation can be expressed in terms of the 
feedback gains as 
The sensitivity for the system of figure 4 is 
Notice that in equation (18) the right half- 
plane zeroes of G B ( s )  limit the gain, K, and con- 
sequently the gains of R(s). - su 
s',uz = (1 + R(s)GM(s)GB(s))(JsZ + U S + T ( s ) )  
(15) In the proposed scheme, we note from equation 
(9) that the signal-to-noise robustness depends 
on the product RI (s)R2( s), while from equation 
(14), that the sensitivity depends both on this 
product and also on the &(s) term. The charac- 
teristic equation of the system can now be written 
The ratio of the two sensitivities given by equa- 
tions (14) and (15) can be written as 
432 
REFERENCES 
If the factor l+GB(s)Rl(s)  had all its zeroes in 
the left half-plane, then the gains of R~(s) could 
be made arbitrarily large. Because GB(s) is sta- 
ble, there always exists an Rl(s) of moderately 
low gains that makes the above factor have all 
its zeroes in the left half-plane ( this may be jus- 
tified from the root locus plot of G B ( s )  R l ( s )  ) 
and, consequently, the product R1 ( s ) R ~ ( s )  may 
be arbitrarily large, improving the robustness of 
the system. 
4 CONCLUSIONS 
In this paper, a new control scheme to control 
flexible arms with friction in the joints has been 
presented. The proposed scheme is based on a 
nested double loop structure. It has been shown 
to be more robust to changes in the dynamic fric- 
tion coefficient as well as less sensitive to the pres- 
ence of Coulomb friction as compared to existing 
methods. 
The non-minimum phase feature that appears 
in the most of the flexible arms precludes the 
use of very high gains in the controller, which 
is the classical way of increasing the robustness. 
The proposed scheme divides the control problem 
into two: the first consists of the design of an in- 
ner loop controller for a minimum phase system, 
and the second consists of the design of an outer 
loop to control the tip position where the non- 
minimum phase feature must be considered. 
Analytical comparative studies of the sensitiv- 
ity to both kinds of perturbation have shown that 
the proposed scheme is more robust to these per- 
turbations than traditional schemes by a ratio of 
roughly the gain of the controller of the inner 
loop. This gain may be made arbitrarily high 
(the only limitation will be the saturation of the 
amplifier of the motor and the presence of high 
frequency unmodelled modes). 
1. Cannon, Robert H and Schmitz, E., “Precise 
Control of Flexible Manipulators”, Robotics 
Research, 1985. 
2. Harahima, F. and Ueshiba, T., “Adaptive 
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3. Siciliano, B., Yuan, B.S. and Book, W.J., 
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on Decision and Control, Athens, December 
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4. Ower, J. C. and Van de Vegte, J., “Classi- 
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5. Rovner, D.M. and Cannon, R.H., “Experi- 
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tor”, International Journal of Robotics Re- 
search, Vol. 6, No. 4, Winter 1987. 
6. Matsuno, F., Fhkushima, S. and Coworkers, 
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with a Parallel Drive Mechanism”, Interna- 
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7. Craig, J.J., “Introduction to Robotics, Me- 
chanics & Control”, Addison- Wesley Pub- 
lishing Company, 1986. 
Prentice-Hall, 1982. 
8. Kuo, B.C., “Automatic Control Systems”, 
9. Feliu, V., Rattan, K.S., and Brown, H.B., 
“Model Identification of a Single-Link Flexi- 
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ence, Pittsburgh, May 1988. 
433 
I I 
Figure 1. Proposed General Control Scheme. 
I I x1 : 
I I I 
Figure 2. Existing Control Scheme for Flexible Arms. 
434 
Figure 3. Existing Control Scheme Modified for Ouput Feedback. 
I 
Figure 4. Existing Control Scheme Transformed into Equivalent Transfer Function Form. 
435 


A robust control scheme for flexible arms with friction in the joints

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