withit chatlatanagulchai , supparat damyot , and pisit...
TRANSCRIPT
Abstract— Input shapers have been used to suppress residual
vibration of flexible systems. The shaped reference, which is the
output from the input shaper, avoids exciting the vibratory
modes of the flexible closed-loop systems, resulting in zero
residual vibration. Nevertheless, in practice, actuator
saturation normally exists. The saturation changes the shaped
control effort, intended for residual vibration suppression. As a
result, the performance of the input shaper is degraded. This
paper presents a technique to avoid violating the saturation by
choosing the right input shaper for every reference step size.
Using the rigid plant model, the control effort is computed for
the shaped reference. Hence, given a saturation level of the
control effort, the right input shaper that does not produce the
shaped reference that violates the saturation level can be
selected. The step size of the baseline reference can be varied,
leading to switching among input shapers. The ZVDk input
shaper was used. The proposed technique applied favorably
both in simulation and in experiment.
I. INTRODUCTION
Residual vibration takes place at the end of the move
when a flexible system is commanded to move rapidly from
point to point. Input shaping technique suppresses residual
vibration by using destructive interference of impulse
responses. It was originated in the name Posicast by [1].
Later, the Posicast was made more robust to mode parameter
uncertainties by [2] and was given the name Input Shaping.
The so-called zero vibration and its derivatives to the kth
order (ZVDk) input shaper is a robust input shaper. When k
increases, the shaper is more robust whereas producing
slower shaped reference with smaller step size.
Hard nonlinearities, which are saturation, rate limit, dead-
zone, and backlash, can reduce the effectiveness of input
shaping in suppressing residual vibration because they
change the shaped reference produced by the input shapers.
Ref. [3] quantified their detrimental effects and presented
mitigation strategies. A limited number of work has been
seen dealing with the hard nonlinearities problem.
Saturation is a type of hard nonlinearities that is seen
most often. Ref. [4] proposed an input shaping algorithm
called saturation-compensating shaper for closed-loop
system with saturation. The controller considered was a
simple proportional controller, and the plant is a simple mass.
W. Chatlatanagulchai was with Purdue University, USA. He is now an
associate professor of the Department of Mechanical Engineering, Kasetsart
University, Bangkok 10900 Thailand (phone: 662-942-8555; fax: 662-579-
4576; e-mail: [email protected]). S. Damyot is with the Department of Mechanical Engineering, Kasetsart
University, Bangkok 10900 Thailand.
P. Intarawirat is with the Department of Mechanical Engineering, Kasetsart University, Bangkok 10900 Thailand.
The saturation-compensating input shaper is obtained from
iteration process in which the design parameters of the input
shaper is iteratively adjusted until the simulation response is
satisfactory. Ref. [5] studied a closed-loop system with
saturation when the controller was a proportional and
derivative controller, and the plant is a simple mass. A
maximum allowable step reference that will not saturate the
actuator was computed for this closed-loop system, leading to
the design of the saturation-reducing input shaper.
This paper presents switching ZVDk input shaper. A new
ZVDk shaper is selected whenever the baseline reference is
changed so that the resulting control effort is not saturated.
The maximum allowable step reference for a ZVDk input
shaper is computed off-line and is stored in a look-up array.
When the step size of the baseline reference is varied, the
system automatically switches to a right ZVDk shaper that
will not saturate the control effort.
Advantages of the proposed technique over existing
techniques are
It is applicable to a more general plant than a simple
mass and to a more complicated controller. A two-
mass rigid-flexible plant is used in simulation and a
flexible-joint robot manipulator is used in
experiment.
At each time step, an input shaper is selected with the
shortest duration without violating the actuator
saturation. The control effort can be kept near its
saturation level at all times, leading to the quickest
movement.
This paper is organized in the following order. Section 2
contains the two-mass rigid-flexible plant, which represents a
large number of actual flexible systems. Section 3 presents
the ZVDk input shaper. Section 4 discusses closed-loop
system with saturation. Section 5 presents the switching input
shaper. Simulation result is given in Section 6. Section 7
contains the experimental result using a laboratory-scaled
flexible-joint robot manipulator. Conclusions are given in
Section 8.
II. TWO-MASS RIGID-FLEXIBLE PLANT
Consider a two-mass rigid-flexible system, as shown in
Fig. 1. This system represents general two-degree-of-freedom
flexible systems. , 1, 2,ix i are absolute position
coordinates of masses , 1, 2.im i 1m is the rigid-body mass
whose damping is 1c and control effort is .u 2m is the
flexible-body mass whose damping is 2c and spring constant
is 2.k
Switching ZVDk Input Shaper for Flexible Closed-Loop System with
Saturation
Withit Chatlatanagulchai, Supparat Damyot, and Pisit Intarawirat
2017 American Control ConferenceSheraton Seattle HotelMay 24–26, 2017, Seattle, USA
978-1-5090-5992-8/$31.00 ©2017 AACC 4492
The equations of motion of the plant are
1 1 2 2 1 2 2 1 1 1
2 2 1 2 2 1 2 2
,
.
u c x c x x k x x m x
c x x k x x m x
Fig. 1 Two-mass rigid-flexible plant.
Taking the Laplace transform of the equations above, the
transfer function from u to 1x is obtained as
2
1 2 2 21 4 3
1 2 1 2 1 2 2 2
2
1 2 1 2 2 2 1 2
,X m s c s k
G sU m m s m c c m c m s
m k c c k m s c k s
(1)
and from 1x to 2x is obtained as
2 22
2 21 2 2 2
.c s kX
G sX m s c s k
III. ZVDK INPUT SHAPER
The ZVDk input shaper, where 0,1, 2, ...k , has the total
of 2k impulses in the sequence and was shown to have
normalized impulse amplitudes and timings as
1
1 2
0
1
1, 1 ,
1 1
1, 2, ..., 2,
i
i ikj n
j
kK
iA t i
kK
j
i k
(2)
where
21.K e
!
! !
n n
r r n r
is the combinations of n things taken r at a time, n is the
natural frequency, and is the damping ratio.
IV. CLOSED-LOOP SYSTEM WITH SATURATION
Consider a closed-loop system, as shown in Fig. 2, where
IS is the input shaper, C is the controller, SAT is the actuator
saturation, 1G is the plant from the control effort to the rigid-
body position, and 2G is the plant from the rigid-body
position to the flexible-body position. The input shaper is
placed outside the loop. br is the baseline reference. sr is the
shaped reference. e is the tracking error. u and su are the
control effort in and out of the saturation. The control effort,
u, can be computed as
1
.1
s
Cu r
CG
(3)
Fig. 2 Closed-loop system with input saturation and with an input shaper
outside the loop.
Consider the case when C is a PI controller, that is,
/ .p iC K K s (4)
With 1sr t and using (1), (3), and (4), the control effort in
the s-domain is given by
6
0
7
0
,
i
i
i
j
j
j
a s
u s
b s
(5)
where
0 1 2 1 2
3 1 2 1 2 1 2 2 2
4 1 2 1 2 2 2 1 2 1 2 2 2
5 1 2 1 2 2 2 1 2 6 1 2
0 1 2 2 3 2 2
4 1 2 2 2 5 1 2 1 2 2 2 2
6 1 2 1
0, ,
,
,
, ,
0, , ,
, ,
i
p i
p i
p i p
i p i
p i p
a a a K c k
a K c k K m k c c k m
a K m k c c k m K m c c m c m
a K m c c m c m K m m a K m m
b b b K k b K k K c
b c k K c K m b m k c c k m K m
b m c c m
2 2 2 7 1 2, .c m b m m
The inverse Laplace transform of (5) gives the control effort
in the time domain as
7
1
,mt
m
m
u t B e
(6)
where
6
0
7
0
/
m
i
i
im
j
j
js
a s
B
d b s ds
and m are the roots of
7
0
0.j
j
j
b s
V. SWITCHING INPUT SHAPER
The input shaper is a sequence of impulses whose
amplitudes are iA and time locations are .it When the
baseline reference, ,br is a step signal, convolution with the
sequence of impulses produces a staircase signal, as shown in
Fig. 3. The step height of the staircase shaped reference, ,sr
equals .b ir A The final step reaches the final value of br due
to the normalization of the impulse amplitudes.
1c
2m
1x 2x
2c
2k1mu
G1CIS SATbr sr e u
su1x
G2
2x+
_
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Fig. 3 Shaped reference, sr , resulting from convolution between baseline
reference, br , and input shaper, IS .
For most controllers (especially those that do not have
strong influence from the integral term), the control effort is
the highest when the reference value is changed abruptly. For
the staircase shaped reference, this happens at the time
locations of the impulses, , 1, 2, ..., 2.it i k
From (6), the control effort at a time location, ,it is given
by
1
7
1 1
,m i l
it
i l m
l m
u t A B e
(7)
which is the accumulation of the control efforts due to the
present and previous staircase steps.
Let the actuator saturation be max .u u For each ZVDk
input shaper, the maximum step size of the baseline
reference, ,br that will not violate the saturation limit, max ,u
is then given by
max
, ,max min , 1, 2, ..., 2.b k
i
ur i k
u t
(8)
This maximum step size can be computed off-line.
During on-line implementation, for each step size br of
the baseline reference, an input shaper ZVDk can then be
selected so that
, ,maxb b kr r (9)
to ensure that the actuator saturation limit will not be
exceeded.
VI. SIMULATION
For simulation, let the plant parameters of the two-mass
rigid-flexible plant in Fig. 1 be 1 10 kg,m 2 1 kg,m
1 50 kg/s,c 2 0.1 kg/s,c and 2
2 10 kg/s .k Choose the PI
controller (4) as 50pK and 1.iK The actuator saturation
is set equal to max 20 N.u The closed-loop system in Fig. 2
is used.
Fig. 4 shows the result from using only the ZV input
shaper for all step sizes of the baseline reference. The ZV
input shaper is the ZVDk input shaper when 0.k Hence,
there are 2 2k impulses. The impulse amplitudes, ,iA
and time locations, ,it are given by (2). The baseline
reference, ,br consists of several step sizes, as given in Table
1. It is designed to test the switching algorithm. In Fig. 4(a),
the baseline reference, ,br is given in the dotted line, and the
shaped reference, ,sr is given in the solid line. Note that
there are two steps in the shaped reference, resulting from
convolution between the ZV input shaper and the baseline
reference.
The control effort, ,u is given by (3), which depends on
the controller, ,C the rigid-body plant, 1,G and the shaped
reference, .sr This control effort, ,u is intended for 1x to
track ,sr without residual vibration. However, because the
actuator saturation is set equal to 20 N,u the saturated
control effort, ,su is different from the intended effort, .u
Fig. 4(b) shows the intended control effort, ,u in the dotted
line, and the saturated control effort, ,su in the solid line.
As a result, the flexible-body position, 2 ,x vibrates as can
be seen from Fig. 4(c). High level of vibration can be seen
especially at the times 1, 20, and 80 seconds when the
actuator saturation is violated. This clearly shows that the
input shaper performance in suppressing the residual
vibration is degraded due to the violation of the actuator
saturation.
Fig. 4 Simulation result when the ZV shaper is used. (a) Baseline reference,
,br and shaped reference, .sr (b) Control effort before saturation, ,u and
after saturation, .su (c) Flexible-body mass position, 2 .x
Next, the proposed switching input shaper will be applied.
From (7) and (8), the maximum step size, , ,max ,b kr of the
baseline reference that will not violate the saturation limit for
each type of input shaper, ,k is shown in Fig. 5. The ZVDk
input shaper has 2k impulses, producing the shaped
reference, ,sr with 2k steps. Therefore, to reach the same
final position, the higher the k, the shorter the step size. Since
br
1A
1t
2A
2t
2kA
2kt
…
1t 2t 2kt
1br A
1 2br A A
br
Time(s)(a)
Reference (m)
sr
Time(s)(b)
Control effort (N)
Time(s)(c)
Flexible-body position (m)
br
u
su
2x
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short step size leads to less control effort; therefore, , ,maxb kr
increases or decreases with .k
Table 1 contains the step size, ,br of the baseline
reference, used in the simulation. From (9) and Fig. 5, the
input shaper type, k, can be selected as in Table 1. Fig. 6(d)
also contains k versus time.
TABLE I. BASELINE REFERENCE, ,br AND INPUT SHAPER TYPE,
.k
Time
0 ≤
t <
1
1 ≤
t <
20
20 ≤
t <
40
40 ≤
t <
60
60 ≤
t <
80
80 ≤
t ≤
100
Baseline
Reference
mbr 0 1 0.2 0.7 0 1.15
Step Size
mbr 1 0.8 0.5 0.7 1.15
Input
Shaper
Type k
7 3 0 2 10
Fig. 5 Maximum allowable step size of the baseline reference that will not
violate the saturation limit.
Fig. 6(a) shows the baseline reference, ,br in the dotted
line and the shaped reference, ,sr in the solid line. Note that
the number of steps in the shaped reference, ,sr is different
for each step size because different ZVDk input shaper is
used.
Fig. 6(b) shows that the control effort does not violate the
actuator saturation limit. As a result, the intended control
effort, u, is equal to the control effort given to the plant, .su
Note that the input shaper ZVDk is chosen as fast as possible
without violating the actuator saturation limit. Therefore, for
any step size, ,br the proposed switching input shaper
results in operation near the actuator saturation limit and
fastest possible shaped reference.
Because the switching input shaper keeps ,su u the
input shaping performance is not degraded by the actuator
saturation. Fig. 6(c) shows that flexible-body position, 2 ,x
where no residual vibration is seen.
VII. EXPERIMENT
To show the applicability of the proposed technique to
actual system, the switching input shaper was implemented
with a laboratory-scaled flexible-joint robot manipulator, as
shown in Fig. 7. Note that the manipulator is analogous to the
two-mass system in Fig. 1.
The manipulator has an optical encoder (Encoder 1) to
measure the absolute motor shaft’s angular position, 1.x
Another encoder (Encoder 2) is used to measure the relative
link’s angular position, 2 1.x x
Fig. 6 Simulation result when the proposed switching input shaper is used.
(a) Baseline reference, ,br and shaped reference, .sr (b) Control effort
before saturation, ,u and after saturation, .su (c) Flexible-body mass
position, 2 .x (d) Input shaper type, k.
The plant model of the manipulator was obtained from
system identification as
011 2 2
1 0
129.34,
31.98 26.88
nXG s
U s d s d s s
where u is the command voltage given to the motor driver
and 1x is the motor shaft’s angular position.
A PI controller in the form of (4) was used with 5pK
and 1.iK With a unit-step reference, 1,sr t the control
effort in the s-domain can be computed from (3) as
0.5
69
2
0.6
558
0.7
42
0
0.8
14
6
0.8
78
5
0.9
406
0.9
95
2
1.0
48
8
1.0
99
0
1.1
456
1.1
93
6
, ,max mb kr
k
Time(s)(a)
Reference (m)
Time(s)(b)
Control effort (N)
Time(s)(c)
Flexible-body position (m)
Time(s)(d)
Input shaper (ZVDk)
sr
br
, su u
2x
k
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3
0
4
0
,
i
i
i
j
j
j
a s
u s
b s
(10)
where
0 0 1 0 1 2 1 3
0 1 0 2 0 0 3 1 4
, , , ,
0, , , , 1.
i p i p i p
i p
a K d a K d K d a K d K a K
b b K n b d K n b d b
Fig. 7 Flexible-joint robot manipulator used in the experiment.
The inverse Laplace transform of (10) is given by
4
1
,mt
m
m
u t B e
where
3
0
4
0
/
m
i
i
im
j
j
js
a s
B
d b s ds
and m are the roots of
4
0
0.j
j
j
b s
The control effort at a time location, ,it is given by
1
4
1 1
,m i l
it
i l m
l m
u t A B e
where lA and it are amplitudes and time locations of the
impulses, as given by (2).
With an actuator saturation, max 1u volt, the maximum
step size for an input shaper ZVDk is given by (8), and for
each step size br of the baseline reference, an input shaper
ZVDk can then be selected from (9).
The closed-loop system in Fig. 2 was applied. Fig. 8
shows the experimental result, using only the ZV input
shaper for all step sizes of the baseline reference. Fig. 8(a)
shows the baseline reference, ,br in the dotted line and the
shaped reference, ,sr in the solid line. The shaped reference
has two steps from convolution between the baseline
reference and the ZV input shaper. Fig. 8(b) shows the
intended control effort, u, in the dotted line and the saturated
control effort, ,su in the solid line. Because of the saturation,
the intended control effort is changed, resulting in the
remaining residual vibration, as can be seen in Fig. 8(c),
which shows the absolute link’s angular position, 2.x
Note that, even with saturation, the input shaper still
performs better than without input shaper at all. Fig. 8(d)
shows the absolute link’s angular position, 2 ,x without the
input shaper.
Fig. 8 Experimental result when the ZV shaper is used. (a) Baseline
reference, ,br and shaped reference, .sr (b) Control effort before
saturation, ,u and after saturation, .su (c) Flexible-body angular position,
2 .x (d) Flexible-body angular position when the input shaper is not used.
Table 2 contains the baseline reference, rad ,br and the
corresponding step size, rad .br Fig. 9 shows the
maximum allowable step size of the baseline reference. For a
step size, ,br a ZVDk can be selected by comparing br to
, ,max ,b kr given in Fig. 9.
Fig. 10 contains the experimental result when the
proposed switching input shaper is used. Fig. 10(a) shows the
baseline reference, ,br in the dotted line and the shaped
reference, ,sr in the solid line. The number of steps in sr
varies according to the ZVDk used. Fig. 10(b) shows the
control effort before and after saturation as well as the
Encoder 1
Encoder 2
Motor
Link
Spring
5 10 15 20 25
0
0.5
1
5 10 15 20 25-2
-1
0
1
2
5 10 15 20 25
0
0.5
1
5 10 15 20 25
0
2
4
6
8
Time(s)(a)
Reference (rad)
sr
Time(s)(b)
Control effort (volt)
Time(s)(c)
Flexible-body position (rad)
br
u
su
2x
Time(s)(d)
Flexible-body position (rad)
2x
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saturation limit max 1u volt. The control effort is kept within
the saturation limit by the switching input shaper. Fig. 10(c)
shows the resulting link’s angular position. The residual
vibration is suppressed as is designed for. Note that a low
level of residual vibration at the time 15t second is still
seen because during that time the ZV input shaper is used.
The ZV shaper is the least robust to plant model uncertainty.
Fig. 10(d) shows the changing of the ZVDk input shaper.
TABLE II. BASELINE REFERENCE, ,br AND INPUT SHAPER TYPE,
.k
Time
0 ≤
t <
1
1 ≤
t <
5
5 ≤
t <
10
10 ≤
t <
15
15 ≤
t <
20
20 ≤
t ≤
25
Baseline
Reference
radbr 0 0.698 0 0.524 0.174 0.873
Step Size
radbr 0.698 0.698 0.524 0.350 0.699
Input
Shaper
Type k
8 8 4 0 8
Fig. 9 Maximum allowable step size of the baseline reference that will not
violate the saturation limit.
VIII. CONCLUSIONS
Violating the actuator saturation limit degrades the
performance of the input shaper in suppressing the residual
vibration. The proposed switching ZVDk input shaper
automatically selects the right ZVDk shaper that will not
violate the actuator saturation limit. The algorithm only
requires the rigid-plant model, which in most cases can be
obtained quite accurately, and the type of the feedback
controller used. The technique is applicable to general
flexible systems, especially those that can be put in the
transfer function form. Both simulation and experiment have
shown the effectiveness of the proposed technique.
Future work includes applying the switching input shaper
to human control of flexible system and to multi-mode
flexible systems.
REFERENCES
[1] O. J. M. Smith, “Posicast control of damped oscillatory systems,”
Proc. of the IRE, vol. 45, no. 9, pp. 1249–1255, Sep. 1957.
[2] N. C. Singer and W. C. Seering, “Preshaping command inputs to
reduce system vibration,” J. of Dynamic Systems, Measurement and
Control, vol. 112, no. 1, pp. 76–82, Mar. 1990.
[3] K. Sorensen and W. Singhose, “Oscillatory effects of common hard
nonlinearities on systems using two-impulse ZV input shaping,” in
Proc. ACC 2007, New York City, NY, pp. 5539–5544.
[4] R. Eloundou and W. Singhose, “Saturation compensating input
shapers for reducing vibration,” in Proc. MOVIC 2002, Saitama,
Japan, pp. 625–630.
[5] M. J. Robertson and R. S. Erwin, “Command shapers for systems with
actuator saturation,” in Proc. ACC 2007, New York City, NY, pp.
760–765.
[6] W. Chatlatanagulchai and P. H. Meckl, “Model-independent control
of a flexible-joint robot manipulator,” J. of Dynamic Systems, Measurement, and Control, vol. 131, no. 4, pp. 1–10, Apr. 2009.
Fig. 10 Experimental result when the proposed switching input shaper is
used. (a) Baseline reference, ,br and shaped reference, .sr (b) Control
effort before saturation, ,u and after saturation, .su (c) Flexible-body
angular position, 2 .x (d) Input shaper type, k.
0 1 2 3 4 5 6 7 8 9 100.3
0.4
0.5
0.6
0.7
0.8
0.9
0.3
503
0.3
801
0.4
542
0.5
077
0.5
449
0.6
13
8
0.6
267
0.6
952
0.7
02
0
0.7
481
0.7
726
, ,max radb kr
k
5 10 15 20 25
0
0.5
1
5 10 15 20 25-2
-1
0
1
2
5 10 15 20 25
0
0.5
1
5 10 15 20 25
0
2
4
6
8
Time(s)(a)
Reference (rad)
Time(s)(b)
Control effort (volt)
Time(s)(c)
Flexible-body position (rad)
Time(s)(d)
Input shaper (ZVDk)
sr
br
, su u
2x
k
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