C.-G. Kang, “Impulse vectors for input-shaping control: A mathematical tool to design and analyze input shapers,” IEEE Control

Systems Magazine, vol. 39, no. 4, pp. 40-55, Aug. 2019, doi: 10.1109/MCS.2019.2913610 (accepted version) © 2019 IEEE

Impulse Vectors for Input-Shaping Control

A MATHEMATICAL TOOL TO DESIGN AND ANALYZE INPUT SHAPERS

CHUL-GOO KANG

Input shaping is an open-loop control technique for reducing

residual vibrations in computer-controlled machines that is

implemented by convolving a sequence of impulses (an

input shaper) with any desired command [1]. The timing

instants and amplitudes of the impulses are determined from

the natural frequency and damping ratio of a vibratory

system by solving a set of constraint equations [2]. Because

the input shaper is generally located outside the feedback

loop (Figure 1), input-shaping control does not affect the

stability of the overall control system, which is a benefit

compared to conventional feedback control.

Input shaper Controller Plant

Command input r(t) Output

+

-

Disturbance

+ +

rIS(t)

FIGURE 1 A typical block diagram of an input-shaping control system. Input shaping is an open-loop control technique for reducing residual vibrations in computer-controlled machines; it is implemented by convolving a sequence of impulses (an input shaper) with any desired command.

The earliest works on input shaping were performed in

the late 1950s by O.J.M. Smith at the University of

California, Berkeley, who studied posicast control [3]

(which was motivated by a simple wave cancellation concept

for eliminating the oscillatory motion of an underdamped

system), and Calvert and Gimpel, who studied signal

component control [4]. However, the drawback was that

these techniques were sensitive to modeling errors of natural

frequencies and damping ratios of the investigated system.

In the late 1980s, Seering, Singer, and Singhose at the

Massachusetts Institute of Technology significantly

improved the robustness to modeling errors by adding

additional constraints on the derivatives of residual-

vibration magnitudes [5]-[9]. Following this, Lim et al. [10]

investigated input-shaping design for multi-input systems

using quasi-convex optimization, and Singh et al. [11], [12]

achieved robustness by cascading multiple time-delay filters,

which cancel the poles of the system.

Given its robustness to modeling errors, input-shaping

control has been successfully implemented using widespread

microprocessor technology in a variety of industrial systems

(including cranes [13], [14], disk drives [15], flexible

spacecrafts [16]-[18], coordinate-measuring machines [19],

[20], and industrial robots [21], [22]) to suppress unwanted

residual vibrations in point-to-point maneuvers. One

advantage of input-shaping control is that if the input shaper

causes no residual vibration when applied to a vibratory

system, then the command generated by convolving it with

any function will also cause no residual vibration. Moreover,

input shaping can be easily implemented without any

additional sensors. (Input Shaping and Input Shaper are

trademarks of Convolve, Inc. [23].)

To generate a sequence of impulses that could reduce

residual vibration, Singhose introduced the concept of a

vector diagram [24]. That definition of a vector diagram

could be effectively applied to undamped systems

effectively [25]-[27], but it was inconvenient for application

to underdamped systems because the effect of damping must

be considered separately. Additionally, its application to

negative shapers was ambiguous because the vector

corresponding to a negative impulse was not defined. For an

underdamped system, Singh et al. [12] introduced a

transformation to shift the Laplace variable s with s p = − ,

which permits converting the transfer function of an

underdamped system to an undamped system for the purpose

of conveniently designing input shapers.

As discussed in “Summary,” this article defines an

impulse vector that can be applied to the design and analysis

of input shapers for both undamped and underdamped

systems (and for both positive and negative impulses) in a

unified way without ambiguity. Because the impulse time of

an input shaper is one of the key parameters to its design,

and is, in general, not known in advance, it is necessary to

solve a highly nonlinear set of equations to obtain the

impulse time and magnitude of an input shaper for an

underdamped system. The proposed impulse vector, a

graphical tool that can handle negative shapers and

underdamped systems in a straightforward manner, makes it

easier to graphically obtain impulse time and magnitude.

Note that in Singh et al.’s approach [12], an underdamped

2

system is converted algebraically to an undamped system by

variable transposition. The magnitude of the impulse vector

in a polar coordinate system represents the magnitude of an

impulse function, and its angle represents the time location

of the impulse function.

To demonstrate the usefulness of impulse vectors, in this

article, they are used to design new classes of input shapers,

such as negative equal-magnitude (NMe), negative

magnitude (NM), generalized zero-vibration and derivative

(gZVD), and equal shaping-time and magnitude (ETM)

shapers. The impulse vector concept introduced in this

article provides a fundamental mathematical tool for

designing and analyzing input shapers more clearly and

easily. Moreover, it provides insights into the design of

better input shapers, and deeper understanding of input-

shaping control.

DEFINITION OF AN IMPULSE VECTOR

An impulse vector iI is first defined by using the magnitude

and angle of the vector in a polar coordinate system, in which

the subscript i implies 1, 2, 3, ….

Definition: Impulse Vector

For a second-order system with undamped natural frequency

n and damping ratio , the magnitude iI and angle

i

of an impulse vector iI corresponding to an impulse

function ( )i iA t t − is defined in a 2D polar coordinate

system as

,n it

i i i d iI A e t = = , (1)

where iA implies the magnitude of an impulse function,

it

the time location of the impulse function, and d damped

natural frequency 21n − . For a positive impulse

( 0iA ), the initial point of the impulse vector is located at

the origin of the polar coordinate system, while for a

negative impulse ( 0iA ), the terminal point of the impulse

vector is located at the origin. ￭

In this definition, the magnitude iI of the impulse vector

is the product of iA and a scaling factor for damping during

time interval it , which represents the magnitude

iA of the

impulse function before being damped. The angle i of the

impulse vector is the product of the impulse time and

damped natural frequency. Note that the absolute value of

iA is always smaller than the absolute value of iI . ( )it t −

represents the Dirac delta function with impulse time at

it t= . In this article, the term impulse function is sometimes

shortened to impulse. Figure 2(a) and (b) illustrates the

definition of the impulse vector iI with a corresponding

positive and negative impulse. For a positive impulse with

0iA , the initial point of the impulse vector is located at

the origin of the polar coordinate system [Figure 2(a)]; for a

negative impulse with 0iA , the terminal point of the

impulse vector is located at the origin, as shown in Figure

2(b). This impulse vector definition allows for a negative

vector magnitude, which can be illustrated using an arrow

pointing to the origin, unlike the usual position vectors.

Impulse vectors with negative magnitudes are useful for

designing and analyzing input shapers that have negative

impulses. The impulse vectors defined here satisfy the

commutative and associative laws, as well as the distributive

law for scalar multiplication.

As shown in Figure 2, the impulse vector definition is

one-directional because only one impulse vector is defined

for a given impulse function and a second-order system with

fixed n and . However, many impulse functions and

second-order systems can exist for a given impulse vector. If

a second-order system is given with fixed n and , then

an impulse vector and an impulse function have a one-to-one

correspondence relationship. Note that an impulse function

is a purely mathematical quantity, while the impulse vector

includes a physical quantity (in other words, n and of

a second-order system) as well as a mathematical impulse

function. Note that position vectors are defined in an n-

dimensional Euclidean space, while impulse vectors are only

defined in a 2D polar coordinate system. Representing more

than two impulse vectors in the same polar coordinate

system makes an impulse vector diagram. The impulse

vector diagram is a graphical representation of an impulse

sequence.

Consider two impulse vectors (as shown in Figure 3) first

to clarify the key concepts. In Figure 3(a), 1I is the impulse

Summary An impulse vector is introduced as a mathematical tool

for the design and analysis of input shapers and can be

applied conveniently to both undamped and

underdamped systems for both positive and negative

impulses in a unified way. The impulse vector approach

can enrich the comprehension of input-shaping theory by

providing fast and clear intuitions graphically in a

practical way. Additionally, it can provide insight to

improving input shapers or designing new input shapers.

The usefulness of the impulse vector is demonstrated by

designing a new class of input shapers, for example,

negative equal-magnitude, negative magnitude, and

equal shaping-time and magnitude (ETM) shapers. ETM

shapers are less sensitive to modeling errors than well-

known zero-vibration and derivative (ZVDn) shapers in a

large modeling error of higher natural frequency range

than the nominal value. Furthermore, they have a fixed

shaping time of one (damped) period for all n, which is

usually much smaller than the shaping time of ZVDn

shapers.

3

(a)

(b)

FIGURE 2 Impulse functions and their corresponding impulse

vectors. (a) For a positive impulse ( 0iA ), the initial point of the

impulse vector is located at the origin of the polar coordinate

system. (b) For a negative impulse ( 0iA ), the terminal point of

the impulse vector is located at the origin. An impulse function

( )i iA t t − is a purely mathematical quantity, while the impulse

vector iI is defined using a mathematical impulse function with

physical quantities ,d of a second-order system.

vector with magnitude 1I and angle

1 corresponding to a

positive impulse (1 0A ), and

2I is the impulse vector with

magnitude 2 1I I= − and angle

2 1 = + corresponding to a

negative impulse (2 0A ). Figure 3(b) shows impulse

responses corresponding to each impulse vector, in which

arrows representing impulse functions are overlapped. In

Figure 3(b), two time-responses corresponding to 1I and

2I are exactly the same after the final impulse time 2t .

Figure 3 shows that angle difference between impulse

vectors implies impulse-time difference of a half (damped)

period, and the arrow direction of the impulse vector

pointing to the origin implies a negative impulse with

2 0A . The two impulse vectors 1I and

2I can be

regarded as the same vector in that corresponding impulse

responses are identical after the final impulse time. The two

impulse vectors 1I and

2I in Figure 3 are regarded as the

same vector for vector addition and subtraction.

The magnitude of the impulse vector determines the

magnitude of the impulse function, and the angle of the

impulse vector determines the time location of the impulse

function. One rotation (for example, a 2 angle) on an

impulse vector diagram means one (damped) period of the

(a)

(b)

FIGURE 3 Two impulse vectors and their corresponding impulse

responses for a second-order system with 2 ,n

= 0.2, =

11,A = and 1

0.2s.t = (a) Two impulse vectors 1I and 2I with the

same magnitude and angle difference (with one pointing to the origin and the other pointing to the outside) are regarded as the same vector for vector addition and subtraction. (b) Two impulse

responses corresponding to 1I and 2I are exactly the same

after the final impulse time 2.t The angle difference of the

impulse vector implies a half-period impulse-time difference, and the arrow direction pointing to the origin implies a negative

impulse with 20.A

corresponding impulse response. If it is an undamped system

( 0= ), the magnitude and angle of the impulse vector

become i iI A= and ,i n it = respectively, from (1). The

definition of (1) is derived from the amplitude and phase of

the unit-impulse response (2) of a second-order system with

natural frequency n and damping ratio

( )2

( ) sin1

ntn

dy t e t

−=

−. (2)

In (2), the amplitude at it t= is decreased by n ite

times, compared to the amplitude at 0t = . This damping

effect can be plotted as a damping spiral [9] for the

magnitude iA of the impulse function in terms of the angle

in the polar coordinate system, as shown in Figure 4. In the

figure, the solid line circle represents impulse vectors with

constant magnitudes iI , and the dotted spiral represents

decreasing impulse magnitudes iA due to the damping

effect.

4

FIGURE 4 The damping spiral in a polar coordinate system. As the angle of the impulse vector with constant magnitude increases, the magnitude of corresponding impulse function

decreases, in other words, / n it

i iA I e

= and i d it = .

Impulse vector diagrams are beneficial because they

enable the design and analysis of input shapers for both

undamped and underdamped systems in a unified way.

Moreover, impulse vector diagrams may provide ideas for

designing a new class of input shapers. The impulse vectors’

usefulness is mainly based on the following two properties.

Property 1: Resultant of Two Impulse Vectors

The impulse response of a second-order system

corresponding to the resultant of two impulse vectors is the

same as the time response of the system with a two-impulse

input corresponding to two impulse vectors after the final

impulse time, regardless of whether the system is undamped

or underdamped.

Property 2: Zero Resultant of Impulse Vectors

If the resultant of the impulse vectors is zero, then the time

response of a second-order system for the input of the

impulse sequence corresponding to the impulse vectors also

becomes zero after the final impulse time for both undamped

and underdamped systems.

To demonstrate these two properties, consider an

underdamped second-order system with the transfer function 2 2 24 / ( 0.4 4 )s s + + . This system has an undamped

natural frequency of 1 Hz, and a damping ratio of 0.1. The

first impulse with magnitude one is applied at time 0.1 s as

an input, and the second impulse with magnitude two is

applied at time 0.2 s. What will be the time response after

0.2 s for this impulse sequence input? In this case, the two

impulse vectors 1I and

2I are obtained using (1)

1

1 1 1 11.065, 0.625 rad,nt

dI Ae t = = = =

2

2 2 2 22.268, 1.250 rad.nt

dI A e t = = = =

The resultant of the two vectors can then be found by

calculating two components ,x yR R in x and y directions

1 1 2 2

1 1 2 2

cos cos 1.578,

sin sin 2.776.

x

y

R I I

R I I

= + =

= + =

(a)

(b)

FIGURE 5 The resultant of two impulse vectors and the corresponding impulse responses. (a) The resultant

RI of two

impulse vectors represents the total effect of two impulse

responses by two impulse inputs 1 1( )A t t − and

2 2( )A t t − .

(b) The impulse response corresponding to the resultant RI is

identical to the red solid line 1 2y y+ after the final impulse time

2 0.2t = s.

2 2 13.193, tan ( / ) 1.054 rad.R x y R y xI R R R R − = + = = =

Thus, the resultant RI can be represented in a polar

coordinate system, as in Figure 5(a). Figure 5(b) shows the

impulse responses corresponding to three impulse vectors,

in which the red solid line implies the total effect 1 2y y+ of

two impulse vectors, 1I and

2I . As shown in Figure 5(b),

the impulse response (thick dotted line) for the input

( )R RA t t − by the resultant RI is the same as

1 2y y+

after the final impulse time 0.2 s, in which RA and

Rt are obtained as

/ 0.169 s, / 2.872.n Rt

R R d R Rt A I e = = = =

Moreover, an impulse vector 3I can be placed on an

impulse vector diagram to cancel the resultant 1 2+I I , as

shown in Figure 6(a). The impulse vector 3I is

2 2 1

3 33.193, tan ( / ) 4.196 radx y y xI R R R R −= + = = + = .

0 0.1 0.2 0.3 0.4 0.5 0.6

5

The sum 1 2 3+ +I I I of these three impulse vectors then

becomes zero. When these three impulse vectors

1 2 3, and I I I are converted into an impulse sequence

1 1 2 2 3 3( ) ( ) ( )A t t A t t A t t − + − + − and this impulse sequence

is applied to the second-order system as an input, the residual

vibration is completely removed after the final impulse time

3t , as shown in Figure 6(b). 3A and

3t are

3

3 3 3 3/ 0.671 s, / 2.094.nt

dt A I e = = = =

Note that if the system is undamped, then iA is always the

same as iI , and

d is the same as n . Consider again the

underdamped second-order system with the transfer function 2 2 24 / ( 0.4 4 )s s + + to demonstrate the use of negative

impulses. Let the first impulse with magnitude two be

applied at time zero as an input and the second impulse with

magnitude –2.5 at time 0.2 s. In this case, two impulse

vectors 1I and

2I are obtained using (1)

2

1 1 1

2 2 2 2

2, 0,

2.835, 1.250 rad.nt

d

I A

I A e t

= = =

= = − = =

(a)

(b)

FIGURE 6 The cancelling impulse vector. (a) Impulse vector 3I

cancels the resultant 1 2+I I of the two impulse vectors. (b) The

input of the impulse sequence 1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + − causes

no residual vibration after the final impulse time 3t when applied

to a second-order system ( 2 , 0.1n = = ).

(a)

(b)

FIGURE 7 Two representations of the resultant of two impulse vectors and their corresponding impulse responses. (a) The

resultant of two impulse vectors can be represented as 1RI or

2RI , in which 1RI corresponds to a negative impulse with

magnitude 1

1 1 / n Rt

R RA I e

= and time 1 1 /R R dt = .

2RI corresponds

to a positive impulse with magnitude 2

2 2 / n Rt

R RA I e

= and time

2 2 /R R dt = . (b) Each impulse response corresponding to each

resultant 1RI and

2RI is identical to the thick red line 1 2y y+ after

each impulse time location.

The resultant RI of the two vectors can then be found by

calculating two components , x yR R in x and y directions

1 2 2 2 2cos 1.107, sin 2.690.x yR I I R I = + = = = −

2 2 2.909,R x yI R R = + =

1tan ( / ) 1.180 rad.R y xR R −= = −

Thus, the resultant RI can be represented in two ways (

1RI

and 2RI ) in a polar coordinate system [Figure 7(a)], in which

1RI corresponds to a negative impulse with magnitude

1

1 1 / n Rt

R RA I e

= and time 1 1 /R R dt = , and 2RI , which

corresponds to a positive impulse with magnitude 2

2 2 / n Rt

R RA I e

= and time 2 2 /R R dt = . Figure 7(b)

shows impulse responses corresponding to four impulse

vectors, in which the thick red line implies the total effect

1 2y y+ of two impulse vectors, 1I and

2I . As shown in

Figure 7(b), each impulse response by each resultant 1RI and

0 0.5 1 1.5

6

2RI is exactly the same as 1 2y y+ after each impulse time

location.

It is also possible to place an impulse vector 3I on an

impulse vector diagram to cancel the resultant 1 2+I I , as

shown in Figure 8(a). The impulse vector 3I is

2 2 1

3 32.676, tan ( / ) 2.047 radx y y xI R R R R −= + = = + = .

When the three impulse vectors 1 2 3, and I I I are converted

into an impulse sequence and this impulse sequence is

applied to the second-order system as an input, the resulting

time response causes no residual vibration after the final

impulse time 3t , as shown in Figure 8(b). However, another

canceling vector can exist, such as the impulse vector with

the same magnitude and angle as 1 2+I I , but with an

opposite arrow direction. This canceling vector has a longer

impulse time, which can be as much as a half period

compared to the previously mentioned 3I . Therefore, the

canceling vector 3I is preferred.

IMPULSE VECTOR DIAGRAMS FOR KNOWN INPUT SHAPERS

Using impulse vector diagrams, known input shapers [28]-

[36] such as ZV, ZVD, ZVDn, unity-magnitude (UM),

partial sum (PS), and specified-negative-amplitude (SNA)

shapers can be redesigned. Brief descriptions for the

principles of ZV and ZVD shapers are provided in “The

Principle of Input-Shaping Control.”

Let the first impulse vector be located at 0° on an impulse

vector diagram because the first impulse can be given at

0t = without losing generality. To design an input shaper

with two impulse vectors, the second impulse vector must be

located at 180° with the same magnitude, as shown in Figure

9(a). Together with a normalization constraint 1 2 1A A+ = ,

the impulse vector diagram in Figure 9(a) results in the ZV

shaper, regardless of an undamped or underdamped system.

Using the impulse vector diagram together with 1 2 1A A+ = ,

the ZV shaper is easily obtained to be 1 20, / dt t = = ,

1 / ( 1)A K K= + ,2 1/ ( 1)A K= + , and

2/ 1K e

−= by solving

simultaneous equations.

If the goal is to design an input shaper with three impulse

vectors, there can be infinitely many cases, as expected from

the impulse vector diagram. As a specific example, for the

first impulse vector at 0°, locate the second vector at 180°

and the third vector at 360° with the magnitude ratio

1 2 3: : 1: 2 :1I I I = , as shown in Figure 9(b). The sum of these

three vectors then becomes zero. The magnitude ratio 1: 2:1

can be obtained by solving the simultaneous equations of the

following residual-vibration constraint (3) and derivative

constraint (4) with all 3N =

(a)

(b)

FIGURE 8 The cancelling impulse vector. (a) Impulse vector 3I

cancels the resultant 1 2+I I of the two impulse vectors. (b) The

input of the impulse sequence 1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + −

causes no residual vibration after the final impulse time 3t when

applied to a second-order system ( 2 , 0.1n = = ).

1 1

cos 0, sin 0.N N

i i i i

i i

I I = =

= = (3)

1 1

cos 0, sin 0N N

i i i i i i

i i

I I = =

= = . (4)

Note that the constraints (3) and (4) are alternative

expressions of (S3) and (S5), respectively. The impulse

vector diagram in Figure 9(b) (together with a normalization

constraint 1 2 3 1A A A+ + = ) results in the ZVD shaper with

1 20, / dt t = = , 3 2 / dt = , 2 2

1 / ( 1)A K K= + , 2

2 2 / ( 1)A K K= + ,

and 2

3 1/ ( 1)A K= + , in which K is the same as above.

By extending the aforementioned idea, input shapers can

be designed with four and five impulse vectors, as shown in

Figure 9(c) and (d). The magnitude ratio of the impulse

vector diagram in Figure 9(c) is 1 2 3 4: : : 1: 3 : 3 :1I I I I = ,

which can be obtained by simultaneously solving (3) and (4),

and the double-derivative constraints

2 2

1 1

cos 0, sin 0N N

i i i i i i

i i

I I = =

= = (5)

with 4N = .

7

The Principle of Input-Shaping Control

If an impulse input 1 ( )A t is applied to a second-order

system with a transfer function 2 2 2/ ( 2 ),n n ns s + +

then a residual vibration will occur as shown in Figure S1.

If a second impulse 2 2( )A t t − with an appropriate

magnitude 2A is applied exactly at the half-period time

point 2 / dt = , then the residual vibration can be

suppressed completely, as shown in Figure S1. This shaper

1 2 2( ) ( )A t A t t + − is known as the zero-vibration (ZV)

shaper.

FIGURE S1 The principle of zero-vibration (ZV) shaping. The

residual vibration generated by the first impulse 1 ( )A t can be

suppressed completely by the second impulse with appropriate

magnitude applied exactly at the half-period time 2 / dt = .

For a second-order system, the unit-impulse response

for an input ( )t is

2

( ) sin , 01

n tndy t e t t

−=

−. (S1)

where n is the undamped natural frequency, is the

damping ratio, and 21d n = − is the damped natural

frequency. When an input 1 2 2( ) ( )A t A t t + − is applied to

the system, the response is obtained as

2( )

21

2

2 1

2 22

2

( ) sin ( )1

1

(sin )(cos ) (cos )(sin )

( , ) ( , ) sin( ),1

n i

n n i

n

t ti nd i

i

t tni

i

d d i d d i

tnn n d

Ay t e t t

e A e

t t t t

e C S t t t

− −

=

−

=

−

= −−

=−

−

= + − −

2

1

2

11

( , ) cos ,

( , ) sin , 0

n i

n i

tn i d i

i

tn i d i

i

C A e t

S A e t t

=

=

=

= = . (S2)

Therefore, the response (S2) will be zero at 2t t without

residual vibration if the following residual vibration constraints are satisfied

2

1

( , ) cos 0,n itn i d i

i

C A e t

=

= =

2

1

( , ) sin 0.n itn i d i

i

S A e t

=

= = (S3)

If an input shaper is composed of N impulses, then the upper bound 2 of summation must be replaced with N in (S3). By solving (S3) simultaneously with a normalization

constraint 1 2 1A A+ = and the first impulse time 1 0t = , the ZV

shaper 1 2 2( ) ( )A t A t t + − is obtained as

210,, .

(1 ), 1 (1 )

i d

i

tK e

A K K K

− = =

+ + (S4)

The normalization constraint ensures that the total move by the shaped command is the same as the move by the unshaped command. To increase the robustness to modeling error of the ZV shaper, Singer et al. [5]-[7] added the following derivative constraints to be the zero-vibration and derivative (ZVD) shaper with three impulses

3 3

1 1

cos 0, sin 0.n i n it ti i d i i i d i

i i

A t e t A t e t

= =

= = (S5)

The derivative constraints (S5) are obtained from / 0ndC d =

and / 0ndS d = [S1]. Simultaneously solving the equations

( ) ( )

( ) ( )

( ) ( )

( ) ( )

2 3

2 3

2 3

2 3

1 2 2 3 3

2 2 3 3

2 2 2 3 3 3

2 2 2 3 3 3

1 2 3

cos cos 0,

sin sin 0,

cos cos 0,

sin sin 0,

1

n n

n n

n n

n n

t t

d d

t t

d d

t t

d d

t td d

A A e t A e t

A e t A e t

A t e t A t e t

A t e t A t e t

A A A

+ + =

+ =

+ =

+ =

+ + = .

then yields the ZVD shaper 1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + − as

2

2 2 2 2

1

0, , 2,

(1 ) , 2 (1 ) , 1 (1 )

.

i d d

i

t

A K K K K K

K e

−

=

+ + +

=

(S6)

The ZVD shaper is more robust to modeling errors than the ZV shaper; however, the shaping time of the ZVD shaper is longer than that of the ZV shaper. Note that

0C S= = and / / 0n ndC d dS d = = result in / / 0dC d dS d = = ,

and thus, the robustness of the ZVD shaper to modeling

errors in n guarantees the robustness to modeling error in

[5]. If the impulse sequence of the input shaper can

remove residual vibrations of the system, then the shaped command generated by convolution of an arbitrary command and the impulse sequence also can remove residual vibration since the superposition principle holds in a linear system.

REFERENCE

[S1] C.-G. Kang, “On the derivative constraints of input shaping control,” J. Mech. Sci. Technol., vol. 25, no. 2, pp. 549-554, 2011.

8

The impulse vector diagram in Figure 9(c) (together with

1 2 3 4 1A A A A+ + + = ) results in a ZVD2 shaper with 1 0,t =

2 / dt = , 3 2 / dt = , 4 3 / dt = , 3 3

1 / ( 1)A K K= + , 2 3

2 3 / ( 1)A K K= + , 3

3 3 / ( 1)A K K= + , and 3

4 1/ ( 1)A K= + , in which K

is also the same as above. The magnitude ratio of the impulse

vector diagram in Figure 9(d) is 1 2 3 4 5: : : : 1: 4 : 6 : 4 :1I I I I I = ,

which can be obtained by simultaneously solving (3)-(5),

and the triple-derivative constraints

3 3

1 1

cos 0, sin 0,N N

i i i i i i

i i

I I = =

= = (6)

with 5N = . The impulse vector diagram in Figure 9(d),

together with 1 2 3 4 5 1A A A A A+ + + + = , results in a ZVD3 shaper

with 1 20, / dt t = = ,

3 2 / dt = , 4 3 / dt = ,

5 4 / dt = , 4 4

1 / ( 1)A K K= + , 3 4

2 4 / ( 1)A K K= + , 2 4

3 6 / ( 1)A K K= + , 4

4 4 / ( 1)A K K= + , and 4

5 1/ ( 1)A K= + ,

in which K is also the same as above.

Moreover, impulse vector diagrams can be applied to

design input shapers with negative impulses. For example,

consider an input shaper with three impulses shown in Figure

10(a), in which the magnitudes of three impulse vectors are

1 2 3, , ( 0)I I I I I I I= = − = , and the angles are 1 0, =

2 60 , = and 3 120 = , respectively. The resultant of three

vectors then becomes zero, and thus, the residual vibration is

suppressed. Impulse instants 2 3 and t t of this input shaper

1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + − are obtained from the angles

of the impulse vectors as2 3( / 3) / , (2 / 3) /d dt t = = .

(a) (b)

(c) (d)

FIGURE 9 Impulse vector diagrams for the well-known zero-vibration (ZV), zero-vibration and derivative (ZVD), ZVD2, ZVD3 shapers. (a) The ZV shaper is composed of two impulse vectors in which the first impulse vector is located at 0°, the second impulse vector is uniquely determined at 180° for the resultant to be zero. (b) The ZVD shaper is composed of three impulse vectors in which the first impulse vector is located at 0°, the second at 180°, and the third at 360°, and the magnitude ratio is

1 2 3: : 1: 2 :1I I I = . (c) The ZVD2 shaper is composed of four impulse

vectors in which the first impulse vector is located at 0°, the second at 180°, the third at 360°, and the fourth at 540°, and the

magnitude ratio is 1 2 3 4: : : 1: 3 : 3 :1I I I I = . (d) The ZVD3 shaper is

composed of five impulse vectors in which the first impulse vector is located at 0°, the second at 180°, the third at 360°, the fourth at 540°, and the fifth at 720°, and the magnitude ratio is

1 2 3 4 5: : : : 1: 4 : 6 : 4 :1I I I I I = .

(a) (b)

FIGURE 10 Impulse vector diagrams for input shapers with negative impulses. (a) The negative equal-magnitude shaper corresponding to this impulse vector diagram has a negative impulse due to the arrow direction of

2I and thus has a shorter

shaping time, which is similar to the well-known unity-magnitude shaper in sensitivity curves. (b) If the angle of

3I is further

decreased, then the shaping time is further decreased and the magnitude of

2I is increased, which results in the negative

magnitude shaper being similar to the partial sum shaper with a big negative impulse.

Impulse magnitudes iA of this input shaper are easily

obtained by solving equations

2

1

2

,

/ ,nt

A I

A I e

=

= −

3

3

1 2 3

/ ,

1.

ntA I e

A A A

=

+ + =

The resulting input shaper 1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + − is

2

1/3 2/3

2/3 1/3

/ 1-

0, ( / 3) / , (2 / 3) /,

, / , /

/ ( - )

.

i d d

i

t

A I I K I K

I K K K K

K e

=

−

= +

=

(7)

The input shaper (7) is called an NMe shaper, which is

the same as the UM shaper [34], [35] if the system is

undamped; in other words, 0 = . The NMe shaper in (7)

has an advantage in that it has a shorter shaping time than

that of the ZV shaper, but it has the following disadvantages:

It is more sensitive to modeling errors than the ZV shaper,

and it could excite high-frequency unmodeled dynamics.

This input shaper is very similar to the UM shaper in

sensitivity curves [7], [34], [37], [38] (see “Sensitivity

Curves” for further details), as shown in Figure 11, in which

1 2 31, 1, 1A A A= = − = instead of 1 2 3, ,I I I I I I= = − = .

However, the NMe shaper is easier to compute and handle

than the UM shaper because there is a closed-form solution

(7) for nonzero damping ratios, whereas the UM shaper only

has numerical solutions for nonzero damping ratios [34].

If the angle of 3I is decreased further [Figure 10(b)],

then the shaping time is decreased, and the magnitude of 2I

(the diagonal of a rhombus) is increased. For a given angle

3 of the impulse vector 3I , the magnitude

2I and angle 2

9

Sensitivity Curves The robustness to modeling errors of an input shaper has been conventionally evaluated by using the sensitivity

function ( , )nV , which roughly correlates the ratio of

the vibration amplitude with input shaping to that without input shaping [5], [7]. When an impulse sequence

1 2 2( ) ( ) ( )N NA t A t t A t t + − + + − is applied to

a second-order system with natural frequency n and

damping ratio , the response is

( )

21

2 2

2

( ) sin ( )1

( , ) ( , ) sin( ),1

n i

n

Nt ti n

d ii

tnn n d N

Ay t e t t

e C S t t t

− −

=

−

= −−

= + − −

1

1

( , ) cos ,

( , ) sin

n i

n i

Nt

n i d ii

Nt

n i d ii

C A e t

S A e t

=

=

=

= . (S7)

The sensitivity function ( , )nV is defined by the ratio of

the amplitude of (S7) at Nt t= to the amplitude of (S1)

at 0t = [34], [38], [S2],

2 2 2

2

( / 1 ) ( , ) ( , )( , )

/ 1

n Ntn n n

n

n

e C SV

−− +

=−

2 2( , ) ( , ) .n Ntn ne C S

−= + (S8)

To plot a sensitivity curve, the modeled natural

frequency ˆn and damping ratio ̂ are fixed and then

used to compute the impulse magnitude iA and impulse

time it . The V values in (S8) are then plotted for varying

actual n and using fixed iA and it . In this article,

notation means a measured value or an estimated value for a system parameter. Figure S2 shows the sensitivity curves of the zero-vibration (ZV) and zero-vibration and derivative (ZVD) shaper. As shown, the ZVD shaper is more robust to modeling errors than the ZV shaper because the sensitivity curve of the ZVD shaper is much smoother than that of the ZV shaper in

the neighborhood of ˆ/ 1n n = . In general, the sensitivity

to modeling errors in the damping ratio is less than the sensitivity to modeling errors in the natural frequency.

FIGURE S2 Sensitivity curves for modeling errors in natural frequency for the zero-vibration (ZV) and zero-vibration and derivative (ZVD) shapers (no modeling error in damping ratio

0.1 = ). The ZVD shaper is more robust to modeling errors

than the ZV shaper because the sensitivity curve of the ZVD shaper is much smoother than that of the ZV shaper in the

neighborhood of ˆ/ 1n n = .

REFERENCE

[S2] W. Singhose, E. Crain, and W. Seering, “Convolved and simultaneous two-mode input shapers,” IEE Control Theory Appl., vol. 144, no. 6, pp. 515-520, 1997.

10

FIGURE 11 Sensitivity curves for modeling errors in natural frequency. The sensitivity curves of the negative equal-magnitude (NMe) and negative magnitude (NM) shaper are similar to those of the unity-magnitude (UM) and partial sum (PS) shaper, respectively. The NMe and NM shapers have closed-form solutions for nonzero damping ratios, while the UM and PS shapers have only numerical solutions for nonzero damping ratios.

of the impulse vector 2I can be computed in terms of

1I

and 3I as

1 3

1 3 3 3

3 3 3

2 2

2 3

1 1 3 3

2

3

,

cos (1 cos ),

sin sin ,

2 2cos ,

sintan tan .

1 cos 2

x

y

x y

y

x

I I I

R I I I

R I I

I R R I

R

R

− −

= =

= + = +

= =

= − + = − +

= = =

+

Impulse instants 2 3 and t t of this input shaper 1 ( )A t +

2 2 3 3( ) ( )A t t A t t − + − are then obtained from the angles of

the impulse vectors as 2 2 3 3/ , /d dt t = = . The impulse

magnitudes iA of this input shaper are obtained by solving

2 2

3 3

1 1

/

2 2 3

/

3 3

1 2 3

,

/ 2 2cos / ,

/ / ,

1.

n

n

t

t

A I I

A I e I K

A I e I K

A A A

= =

= = − +

= =

+ + =

For a given 3 ( ) , the resulting input shaper

1 2 2 3 3( ) ( ) ( )A t A t t A t t + − + − is obtained as

( )

32

2 3

//

3

1

2 3 3 3

0, / , /,

, 2 2cos / , /

tan sin / (1 cos ) / 2,

d di

i

t

A I I K I K

−

=

− +

= + =

(8)

2 3 2 3 3 2

2

( )/ ( )/ / /

3

/ 1-

/ ( - 2 2cos ),

.

I K K K K

K e

+ += + +

=

The input shaper (8) is called an NM shaper, which is similar

to the PS shaper [34] and has a larger negative impulse than

that of the NMe shaper. Note that the NM shaper becomes a

ZV shaper if 3 = . The NM shaper has sensitivity curves

similar to the PS shaper (see Figure 11), in which

1 2, 2 ,A P A P= = − and3 1, 1A P P= + , instead of

1 ,I I=

2 32 2cos ,I I = − + and 3I I= . However, the NM shaper

can be a closed-form solution for a nonzero damping ratio,

while the PS shaper only has a numerical solution for

nonzero damping ratios [34], [36]. If 3 is smaller in the NM

shaper, then faster rise time can be achieved; however, the

absolute value of 2I becomes larger, and thus, actuator

saturation can occur eventually. The NMe or NM shaper

with a negative impulse has faster rise times than those of

the ZVDn shapers; however, it is more sensitive to modeling

errors than the ZVDn shapers. Therefore, the usage of NMe

and NM shapers is limited to the case where the modeling

errors of n and are very small.

Using the impulse vector diagram, it can be shown why

the ZVD or ZVDn shaper has better robustness to modeling

errors than the ZV shaper. An explanation was first offered

by Singer for undamped systems [24]; however, this article

extends this explanation to underdamped and undamped

systems using impulse vectors. If there is a modeling error

in the natural frequency, then the angle of 2I of the ZV

shaper is no longer and has an angle offset [see Figure

12(a)] as a result of an inaccurate calculation of 2t in the

2 2d t = computation. Note that the impulse time 2t is

computed using the modeled natural frequency ˆd even if it

differs from the actual natural frequency d . Therefore, the

resultant of 1I and 2I is a nonzero vector errI , owing to an

offset for a small offset angle and

1 2

2 2

cos( ) 0,

sin( ) 0.

x

y

R I I

R I I

= + −

= −

However, for the ZVD shaper shown in Figure 12(b), the

angle offset of 3I becomes 2 because 3 3d t = and

3 22t t= . Thus, 0err I for the ZVD shaper due to the small

offset angle and

1 2 3

2 3

cos( ) cos(2 2 ) 2 0,

sin( ) sin(2 2 ) (2 ) (2 ) 0.

x

y

R I I I I I I

R I I I I

= + − + − − + =

= − + − − =

Furthermore, for the ZVD2 shaper shown in Figure 11(c),

0err I because

11

1 2 3 4

2 3 4

cos( ) cos(2 2 ) cos(3 3 )

3 3 0

sin( ) sin(2 2 ) sin(3 3 )

(3 ) (3 )(2 ) (3 ) 0 .

x

y

R I I I I

I I I I

R I I I

I I I

= + − + − + −

− + − =

= − + − + −

− + =

The resultant 0err I in the ZVD and ZVD2 shapers results

in small residual vibrations, while the nonzero errI of the ZV

shaper for a modeling error results in a big residual vibration.

Discussions of the sensitivity or robustness to modeling

errors in this article consider only modeling errors in natural

frequency (no modeling errors in damping ratio) because it

is known that input shapers are less sensitive to modeling

errors in the damping ratio than modeling errors in the

natural frequency [5], [6].

(a) (b) (c)

FIGURE 12 Robustness to modeling errors. When there is a modeling error in the natural frequency, (a) the zero-vibration shaper has a big residual vibration, as 0err I . However, (b) the

zero-vibration and derivative (ZVD) and (c) ZVD2 shapers have

small residual vibrations, as 0err I .

DESIGNING NEW GENERALIZED ZVD AND EQUAL SHAPING-TIME AND MAGNITUDE SHAPERS USING IMPULSE VECTORS

From the concept of impulse vectors, there exist infinitely

many input shapers that can suppress residual vibrations of

a second-order system [28] because there are infinitely many

impulse vector diagrams that make the resultant zero. A

gZVD shaper with three impulses, which is a generalization

of ZV and ZVD shapers, is introduced. The magnitudes of

impulse vectors of the ZVD shaper satisfy 3 1 ,I I=

1 3 2I I I+ = , while the impulse vectors of the gZVD shaper

satisfy

3 1 1 3 2, ( 0)I mI I I I m= + = . (9)

Figure 13 shows the impulse vector diagram and the

sensitivity curve of the gZVD shaper, which becomes a ZV

shaper if 0m = and a ZVD shaper if 1m= . From the

normalization constraint 1 2 3 1A A A+ + = with 1 1,A I=

22 2 3 3/ and /A I K A I K= = [and (9)], the gZVD shaper is

obtained as

2

0, / , 2 /

/ (1 ), / , / {(1 ) }

i d d

i

t

A I m I K mI m K

=

+ + , (10)

22

/ 1

2

(1 ),

(1 )

m KI K e

K m K m

−+= =

+ + + .

Figure 13(b) shows the sensitivity curves of the ZV, ZVD,

and gZVD shapers when the actual natural frequency n

varies from 0.6 times to 1.4 times the fixed modeled

frequency ˆn . From the sensitivity curve of the gZVD

shaper shown in Figure 13(b), the robustness to modeling

errors of the gZVD shaper lies between that of the ZV and

ZVD shaper, according to the value of m.

The following designs a new class of input shapers, ETM

shapers, with the same magnitude of impulse vectors and the

same angle between impulse vectors, as shown in Figure 14.

The ETMn shaper satisfies conditions

1 2 1

2 ( 2)20, , , , 2

1 1n n

n

n n

−

−= = = =

− −, (11)

2 3 1 1 1, ( 0)n n nI I I I I I m I m−= = = = + = . (12)

The resultant of impulse vectors of an ETMn shaper is then

zero for all integers 3 . Note that when 3n = , the ETM3

shaper is a gZVD shaper. One advantage of the ETMn shaper

is that, in contrast to the ZVDn or extra insensitive shapers

[17], [39], [40], the shaping time is always one (damped)

period of the time response, even if n increases.

(a)

(b)

FIGURE 13 An impulse vector diagram and sensitivity curve of a generalized zero-vibration and derivative (gZVD) shaper. (a) The

gZVD shaper with 3 1 ( 0)I mI m= becomes a ZV shaper if

0m = and a ZVD shaper if 1m= . (b) The sensitivity curve of

the gZVD shaper with 0 1m exists between that of a ZV

shaper and a ZVD shaper.

0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

Resid

ual vib

ration V

ZV shaper ( = 0.03)

ZVD ( = 0.03)

gZVD ( = 0.03), x = 0.5

ˆ/n n

m

Resid

ual V

ibra

tio

n V

12

(a) (b) (c)

FIGURE 14 The impulse vector diagrams of the equal shaping-time and magnitude (ETM) shapers. (a) An ETM4 shaper, (b) an ETM5 shaper, and (c) an ETM6 shaper.

The ETM4 shaper has four impulse vectors, as shown in

Figure 14(a). The impulse instants 1 2 3, ,t t t and

4t of the

ETM4 shaper are obtained by the angles of the impulse

vectors in (11), in other words, 1 0,t = 2 (2 / 3) / ,dt =

3 (4 / 3) / ,dt = and 4 2 / dt = . The impulse magnitudes

1 2 3 4, , and A A A A of the ETM4 shaper are obtained by solving

2 3 4

2 3 1 4 4 1

1 1 2 2 3 3 4 4

1 2 3 4

, ( 0),

, , , ,

1.

n n nt t t

I I I I I mI m

I A I A e I A e I A e

A A A A

= = + =

= = = =

+ + + =

The resulting ETM4 shaper is

2/3 4/3 2

0, (2 / 3) / , (4 / 3) / , 2 /,

/ (1 ), / , / , / {(1 ) }

i d d d

i

t

A I m I K I K mI m K

=

+ +

22

/ 1

2 4/3 2/3

(1 ), .

(1 )( )

m KI K e

K m K K m

−+= =

+ + + + (13)

When the actual natural frequency of the system is much

larger (for example, more than 20%) than the modeled

natural frequency, the ETM4 shaper shows a robustness

larger than the ZVD shaper, as shown in Figure 15.

Specifically, when the actual natural frequency is 1.5 times

larger than the modeled frequency (that is, ˆ1.5n n = ), the

sensitivity V of the ETM4 shaper becomes almost zero, due

to changes in the angles of the impulse vectors, as shown in

Figure 16. Thus, the residual vibration of the system is nearly

suppressed when there is a 50% modeling error in the natural

frequency (in other words, ˆ1.5n n = ).

To obtain the optimal value of m, define a performance

index J (see Figure 17) that is integral to the sensitivity curve

from, for example, ˆ/ 0.2n n = to 1.8

1

( , ) , , , 0.2, 1.8.ˆ

u

l

rn n

l ur

n

IJ V r m dr r m r r

I

= = = = = (14)

Finding a value m that minimizes J gives the optimal value

optm . Table 1 represents optimal values optm for various

ETM shapers. For different damping ratios of the system,

FIGURE 15 The sensitivity curves of equal shaping-time and

magnitude (ETM) shapers. When the actual natural frequency of

the system is much larger than the modeled natural frequency,

the ETM shapers show a stronger robustness than the zero-

vibration and derivative (ZVD) shaper.

(a) (b)

FIGURE 16 Impulse vector diagrams of the equal shaping-time and magnitude 4 shaper (m = 1). (a) The resultant of four impulse vectors is zero when there is no modeling error in the natural frequency. (b) The resultant of four impulse vectors is also zero, owing to changes in angles of the vectors when there is a 50% modeling error in the natural frequency, in other words,

ˆ1.5 .n n =

FIGURE 17 The performance index J, which obtains the optimal

value of m. J is the integral of the sensitivity curve from ˆ/n n

= 0.2 to 1.8.

13

TABLE 1 The optimal values optm for various equal shaping-time and magnitude (ETM) shapers. For different damping ratios

of the system, each ETM shaper has a different optimal value optm , which minimizes the performance index J in (14) with

[ ,l ur r ] = [0.2, 1.8].

0 0.01 0.03 0.05 0.07 0.1 0.2 0.3 0.4 0.5

mopt, ETM4 1 0.99 0.97 0.95 0.93 0.90 0.79 0.69 0.56 0.40

mopt, ETM5 1 0.98 0.95 0.92 0.89 0.85 0.69 0.53 0.34 0.08

mopt, ETM7 1 0.97 0.91 0.85 0.79 0.71 0.46 0.22 0 0

mopt, ETM10 1 0.95 0.85 0.76 0.68 0.56 0.23 0 0 0

mopt, ETM20 1 0.86 0.62 0.43 0.28 0.09 0 0 0 0

each ETM shaper has a different optimal value optm that

minimizes the performance index J. As the damping ratio is

increased from zero, optm is decreased from 1; however, as

the damping ratio is increased, the change of J (and thus the

change in the sensitivity) becomes small. Of course, optm

depends on the upper and lower limits of the integral J. For

example, optm becomes 0.89 instead of 0.90 for 0.1 = if

[ ,l ur r ] = [0, 2]. Note that if the first and the last impulse

vectors are replaced with one impulse vector with the same

magnitude as others at 0 angle, then the resultant of the

whole impulse vectors is also zero. However, this input

shaper has a larger sensitivity to modeling errors than that of

the ETM shaper, so input shapers of this type will not be

considered further in this article.

The ETM4 shaper can be extended to ETM5 and ETM6

shaper, as shown in Figure 14(b) and (c). The ETM5 shaper

has five impulse vectors at instants 1 2, ,t t 3 4, ,t t 5and t that

are obtained by the angles of the impulse vectors in (11), in

other words, 1 20, ( / 2) / ,dt t = = 3 / ,dt = 4 (3 / 2) / ,dt =

and 5 2 / dt = . The impulse magnitudes 1 2 3 4, , , ,A A A A and

5A of the ETM5 shaper are obtained by solving

2 3 4 5

2 3 4 1 5 5 1

1 1 2 2 3 3 4 4 5 5

1 2 3 4 5

, ( 0),

, , , , ,

1

n n n nt t t t

I I I I I I mI m

I A I A e I A e I A e I A e

A A A A A

= = = + =

= = = = =

+ + + + = .

The resulting ETM5 shaper is

1/2 3/2 2

0, ( / 2) / , / , (3 / 2) / , 2 /,

/ (1 ), / , / , / , / {(1 ) }

i d d d d

i

t

A I m I K I K I K mI m K

=

+ +

22

/ 1

2 3/2 1/2

(1 ), .

(1 )( )

m KI K e

K m K K K m

−+= =

+ + + + + (15)

As shown in Figure 15, the sensitivity of the ETM5 shaper is

nearly zero when the actual natural frequency is 100% larger

than the modeled frequency, in other words, ˆ2n n = .

The ETM6 shaper with six impulse vectors can be

obtained similarly as

2/5 4/5 6/5

8/5 2

0, (2 / 5) / , (4 / 5) / , (6 / 5) / ,

/ (1 ), / , / , / ,

(8 / 5) / , 2 /,

/ , / {(1 ) }

i d d d

i

d d

t

A I m I K I K I K

I K mI m K

=

+

+

22

/ 1

2

(1 ), ,

(1 ) s

m KI K e

K m K m

−+= =

+ + +

8/5 6/5 4/5 2/5.sK K K K K= + + + (16)

An Algorithm for an Equal Shaping-Time and Magnitude (ETM) Shaper Design

(1) Choose the number of impulses n. (2) Draw an impulse vector diagram composed of n

impulse vectors with equal angle intervals.

(3) Obtain the angles of impulse vectors 1 20, = =

12 / ( 1), , ( 2)2 / ( 1), 2 .n nn n n −− = − − =

(4) Let the magnitudes of impulse vectors 2 3I I= =

1 1 1, ( 0).n n nI I I I m I m−= = + =

(5) Compute impulse time 1 2 20, / , ,dt t = =

1 1 / , 2 / .n n d n dt t − −= =

(6) Compute impulse magnitude iA from n it

i iI Ae

=

and 1

1n

ii

A=

= ..

(7) Obtain optimal m from Table 1 and interpolation.

Resulting ETM shaper is 1 2 1( ) ( )A t A t t + − +

( )n nA t t+ − .

14

In the same way, the ETMn shaper with 7n can easily

be obtained. In general, ETM shapers are less sensitive to

modeling errors than ZVDn shapers in a large error range of

ˆn n . Moreover, they have a fixed shaping time of one

(damped) period for all n, which is usually much smaller than

the shaping time of ZVDn shapers. The design procedure of

an ETMn shaper is summarized in “An Algorithm for an

Equal Shaping-Time and Magnitude (ETM) Shaper Design.”

To implement an input shaper in a control system,

convolution with the input shaper and an arbitrary command

r(t) is used to create a shaped command rIS(t), as shown in

Figure 1. For example, to implement an ETM4 shaper for a

system with a unit-step command, the shaped command is

generated as

1 2 2 3 3 4 4

1 2 2 3 3 4 4

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ).

IS s

s s s s

r t u t A t A t t A t t A t t

Au t A u t t A u t t A u t t

= + − + − + −

= + − + − + −

where * implies convolution operation and ( )su t implies

unit-step function. Mathematical operations of convolution

and Dirac delta functions are summarized and partially

proved in “The Operations of Convolution and Impulse

functions.”

SIMULATION AND EXPERIMENTAL STUDIES

To show the validity of the input shapers designed by

impulse vector diagrams, simulations on step responses for

a second-order system were conducted with a damping ratio

of 0.1 and a modeled natural frequency ˆ 2n = rad/s.

Figure 18 shows the simulation results using the ETM4 and

ZVD2 shaper with four impulses when the actual natural

frequency is exact, or 40% larger than the modeled

frequency. When the actual natural frequency is much larger

FIGURE 18 A simulation comparison of the ZVD2 and ETM4 shaping control for the unit-step responses of a second-order

system ( ˆ0.1; 2 ; 2 or 2.8n n = = = ). When the actual

natural frequency is 40% larger than the modeled frequency, the ETM4 shaper generates less residual vibration than the ZVD2 shaper.

than the modeled frequency (for example, more than 30%),

the ETM4 shaper generates less residual vibration than the

ZVD2 shaper, as shown in Figure 18.

To demonstrate the validity of the ETM4 shaper

experimentally, a motion-control testbed that can produce

vertical up-down or horizon motions by easily reassembling

components was developed (Figure 19). This motion-control

device is composed of an ac servo motor, a ball-screw, two

The Operations of Convolution and Impulse Functions

For any two function f and g that satisfy f(t) = g(t) = 0 for all t < 0, the convolution of f and g is defined as

0

0

( ) ( ) ( ) ( )

( ) ( )

t

t

f t g t f t g d

f g t d

= −

= −

The impulse function is defined as

, 0( ) , ( ) 1

0, 0

tt t dt

t

−

== =

The properties are

(S9) ( ) ( ) ( ) ( )sf t t f t u t =

(S10) 1 1 1( ) ( ) ( ) ( )sf t t t f t t u t t − = − −

(S11) 1 2 1( ) { ( ) ( )}f t A t A t t + −

1 2 1 1( ) ( ) ( ) ( )s sA f t u t A f t t u t t= + − −

(S12) ( ) ( ) ( )t t t =

(S13) 1 1( ) ( ) ( )t t t t t − = −

(S14) 1 1 2 2 1 2 1 2{ ( )} { ( )} ( )A t t A t t A A t t t − − = − −

(S15) ( ) ( )t

sd u t −

=

(S16) 0

1 ( ) ( )t

f t f d =

(S17) ( ) ( ) (0)f t t dt f

−=

(S18) 1 1( ) ( ) ( )f t t t dt f t

−− =

where ( )su t is unit-step function, and all 1 20, 0t t .

For example, (S10) and (S18) can be proved as

1 10

1 10

1 1 1 10

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

t

t

t

s

f t t t f t t d

f t t t d

f t t t d f t t u t t

− = − −

= − −

= − − = − −

and

1 1 1

1 1 1

( ) ( ) ( ) ( )

( ) ( ) ( )

f t t t dt f t t t dt

f t t t dt f t

− −

−

− = −

= − =

The proofs of other properties are similar.

ZVD2

ZVD2

15

flexible cantilever beams, and two laser sensors. The

vibration of each beam is detected by each laser sensor

below it. Input shaping and feedback control logics are

implemented in 1 ms sampling time using the free Linux

operating system Ubuntu [41] and the free real-time kernel

RealTime Application Interface [42]. The feedback control

logic is a proportional-and-derivative control ( ) ( )pu t K e t= +

( )dK e t with 0.135, 0.01p dK K= = , respectively. For simulation

purposes, motor dynamics including a timing belt is

modeled as a first-order system with a time constant of 5 ms.

The natural frequency of the flexible beam can be

changed by adding masses at the end of the cantilever beam.

In Figure 19, the left beam with three masses is modeled as ˆ ˆ0.001, 25.25n = = rad/s, and the dynamics of the right beam

is assumed to be unknown but with a much higher natural

frequency (actually is 40.53 rad/s) than the left beam. When

this occurs, the ETM4 shaper shows better vibration-

suppression performance than the ZVD2 shaper, as shown in

Figure 20. Figure 20(a) shows the experimental step

responses of the device when the actual natural frequency is

nearly the same as the modeled frequency, while Figure 20(b)

shows the step responses when it is 40% larger than the

modeled one. For modeled dynamics, the ETM4 shaper has

a vibration-suppression performance similar to the ZVD2

shaper [Figure 20(a)]. However, for unmodeled dynamics

with higher natural frequency, the ETM4 shaper performs

better than the ZVD2 shaper during residual vibrations.

Moreover, the shaping time of the ETM4 shaper is much

smaller than the ZVD2 shaper, as shown in Figure 20(b). For

comparison, the simulation results are plotted together in

Figure 20.

In Figure 21, experimental sensitivities to modeling

errors are compared with the simulated sensitivity curve for

an ETM4 shaper. In Figure 21, an ETM4 shaper was

FIGURE 19 The experimental testbed for input-shaping control. This motion-control device can be driven vertically or horizontally by reassembling its components. The natural frequency and damping ratio of the flexible beam can be changed by adding masses at the end of the beam. The beam vibrations are detected by laser sensors.

designed for a flexible cantilever beam with three (different)

masses at the end ( ˆ ˆ0.001, 24.35 rad/sn = = ). Then several

experiments were conducted to obtain experimental

sensitivity ( points) by changing masses at the end of the

beam. Figure 21 shows that experimental sensitivities to

modeling errors have the same trend with simulated ones

with very little discrepancy, as a result of measurement

errors and other uncertainties.

The results suggest that the newly designed ETM4

shaper works well for a system in which the natural

frequency varies significantly with a known lower limit

during operation or in which there are several modes of

vibrations and the smallest natural frequency is known.

(a)

(b)

FIGURE 20 The experimental step responses of the motion- control device with two flexible beams. (a) The input-shaping results for the modeled natural frequency and damping ratio of the left beam with three masses. The shaping time of the ETM4 shaper is smaller than that of the ZVD2 shaper. (b) The input- shaping results for the unmodeled right beam with a high natural frequency (here, 40.53 rad/s). The ETM4 shaper designed from the beam with three masses can suppress residual vibration of the unmodeled beam better than the ZVD2 shaper.

16

FIGURE 21 A comparison of experimental sensitivities with a simulated sensitivity curve for an ETM4 shaper. For a flexible

cantilever beam with three masses at the end ( ˆ ˆ0.001, n = =

24.35 rad/s ), an ETM4 shaper was designed, and several

experiments were conducted to obtain experimental sensitivity (* points) by changing masses at the end of the beam.

CONCLUSION

In this article, an impulse vector was introduced as a

mathematical tool for the design and analysis of input shapers,

which can be applied conveniently to both undamped and

underdamped systems and for both positive and negative

impulses in a unified way. The impulse vector approach can

enrich the comprehension of input-shaping theory by providing

fast and clear intuitions graphically in a practical way. The

impulse vector’s usefulness was demonstrated by designing a

new class of input shapers (for example, NMe, NM, and ETM

shapers). If the resultant of the impulse vectors is zero, then the

time response for the impulse sequence corresponding to the

impulse vectors becomes zero after the final impulse time,

regardless of whether the system is undamped or underdamped.

Moreover, impulse vectors can provide insight on improving

input shapers or designing a new class of input shapers.

NM and NMe shapers with negative impulses were

introduced. These shapers are more convenient than the well-

known UM and PS shapers because they have closed-form

solutions (even for nonzero damping ratios). The ZVD shaper

was generalized to the gZVD shaper, and a new class of input

shapers (ETM), designed using impulse vectors, were

introduced. The ETM shaper can be well applied to a system

in which the natural frequency varies significantly with a

known lower limit during operation or in which there are

several modes of vibrations and the smallest natural frequency

is known. The performance of the ETM shaper was verified

through simulation and experimental studies by comparing the

results of ZVD2 shaping control using an up-down motion-

control device with flexible beams.

ACKNOWLEDGMENTS

This work was supported by Konkuk University in 2017 under

Grant 2017A0190617. The author thanks Mr. Manh-Tuan Ha

for helping with the experiments.

AUTHOR INFORMATION

Chul-Goo Kang (cgkang@konkuk.ac.kr) is a professor in the

Mechanical Engineering Department at Konkuk University,

Seoul, South Korea. He received the B.S. and M.S. degrees in

mechanical design and production engineering from Seoul

National University, South Korea, in 1981 and 1985,

respectively, and the Ph.D. degree in mechanical engineering

from the University of California, Berkeley, in 1989. In 1990,

he joined the faculty of mechanical engineering at Konkuk

University, where he is the director of Intelligent Control and

Robotics Laboratory and Railway Vehicle Laboratory. He was

general chair of the International Conference on Ubiquitous

Robots and Ambient Intelligence in 2011, and organizing chair

of the International Conference on Control, Automation and

Systems in 2012. In 2015, he was president of the Korea

Robotics Society and editor-in-chief of the Journal of the

Korean Society for Urban Railway. In 2017, he received the

Presidential Commendation of Korea. He is a member of the

National Academy of Engineering of Korea.

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