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: fengwtc@ku.ac.th). 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+

_

4493

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

4494

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

4495

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

4496

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

4497