Impulse vector explained

An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified manner. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically.^[1] A vector concept for an input shaper was first introduced by W. Singhose^[2] for undamped systems with positive impulses. Building on this idea, C.-G. Kang introduced the impulse vector (or Kang vector) to generalize Singhose's idea to undamped and underdamped systems with positive and negative impulses.

Definition

For a vibratory second-order system

	2
\omega
	n

/(s²+2\zeta\omega_n+

	2
\omega
	n

)

with undamped natural frequency

\omega_n

and damping ratio

\zeta

, the magnitude

I_i

and angle

\theta_i

of an impulse vector (or Kang vector)

I_i

corresponding to an impulse function

A_i\delta(t-t_i)

i=1,2,...,n

is defined in a 2-dimensional polar coordinate system as

I_i=A_i

	\zeta\omega_nt_i
e

\theta_i=\omega_dt_i

where

A_i

implies the magnitude of an impulse function,

t_i

implies the time location of the impulse function, and

\omega_d

implies damped natural frequency

\omega_n\sqrt{1-\zeta^2}

. For a positive impulse function with

A_i>0

, the initial point of the impulse vector is located at the origin of the polar coordinate system, while for a negative impulse function with

A_i<0

, the terminal point of the impulse vector is located at the origin. □

In this definition, the magnitude

I_i

is the product of

A_i

and a scaling factor for damping during time interval

t_i

, which represents the magnitude

A_i

before being damped; the angle

\theta_i

is the product of the impulse time and damped natural frequency.

\delta(t-t_i)

represents the Dirac delta function with impulse time at

t=t_i

. Note that an impulse function is a purely mathematical quantity, while the impulse vector includes a physical quantity (that is,

\omega_n

and

\zeta

of a second-order system) as well as a mathematical impulse function. 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

I₁

and

I₂

in the figure on the right-hand side, in which

I₁

is an impulse vector with magnitude

I₁(>0)

and angle

\theta₁

corresponding to a positive impulse with

A₁>0

, and

I₂

is an impulse vector with magnitude

I₂=-I₁

and angle

\theta₂=\pi+\theta₁

corresponding to a negative impulse with

A₂<0

. Since the two time-responses corresponding to

I₁

and

I₂

are exactly same after the final impulse time

t₂

as shown in the figure, the two impulse vectors

I₁

and

I₂

can be regarded as the same vector for vector addition and subtraction. Impulse vectors satisfy the commutative and associative laws, as well as the distributive law for scalar multiplication.

The magnitude of the impulse vector determines the magnitude of the impulse, and the angle of the impulse vector determines the time location of the impulse. One rotation,

2\pi

angle, on an impulse vector diagram corresponds to one (damped) period of the corresponding impulse response.

If it is an undamped system (

\zeta=0

), the magnitude and angle of the impulse vector become

I_i=A_i

and

\theta_i=\omega_nt_i

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 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 impulse vectors is zero, the time response of a second-order system for the input of the impulse sequence corresponding to the impulse vectors becomes zero also after the final impulse time regardless of whether the system is undamped or underdamped. □

4\pi²/(s²+0.4\pis+4\pi²)

. This system has

\omega_n=2\pi

and

\zeta=0.1

. For given impulse vectors

I₁

and

I₂

as shown in the figure, the resultant can be represented in two ways,

I_R1

and

I_R2

, in which

I_R1

corresponds to a negative impulse with

A_R1=I_R1/

	\zeta\omega_nt_R1
e

and

t_R1=\theta_R1/\omega_d

, and

I_R2

corresponds to a positive impulse with

A_R2=I_R2/

	\zeta\omega_nt_R2
e

and

t_R2=\theta_R2/\omega_d

The resultants

I_R1

I_R2

can be found as follows.

R_x=I₁+I₂\cos\theta₂, R_y=I₂\sin\theta₂

I_R1=-

	2
\sqrt{R
	x

	2},
R
	y

\theta_R1=\pi+\tan^-1(R_y/R_x)

I_R2=

	2
\sqrt{R
	x

	2},
R
	y

\theta_R2=\tan^-1(R_y/R_x)

Note that

-\pi/2<\tan^-1(a)<\pi/2

. The impulse responses

y_R1

and

y_R2

corresponding to

I_R1

and

I_R2

are exactly same with

y₁+y₂

after each impulse time location as shown in green lines of the figure (b).

Now, place an impulse vector

I₃

on the impulse vector diagram to cancel the resultant

I₁+I₂

as shown in the figure. The impulse vector

I₃

is given by

I₃=

	2
\sqrt{R
	x

	2},
R
	y

\theta₃=\pi+\tan^-1(R_y/R_x)

When the impulse sequence corresponding to three impulse vectors

I_1,I₂

and

I₃

is applied to a second-order system as an input, the resulting time response causes no residual vibration after the final impulse time

t₃

as shown in the red line of the bottom figure (b). Of course, another canceling vector

	'
I
	3

can exist, which is the impulse vector with the same magnitude as

I₃

but with an opposite arrow direction. However, this canceling vector has a longer impulse time that can be as much as a half period compared to

I₃

Applications: Design of input shapers using impulse vectors

ZVDⁿ shaper

Using impulse vectors, we can redesign known input shapers ^[3] such as zero vibration (ZV), zero vibration and derivative (ZVD), and ZVDⁿ shapers. The ZV shaper is composed of two impulse vectors, in which the first impulse vector is located at 0°, and the second impulse vector with the same magnitude is located at 180° for

I₁+I₂=0

. Then from the impulse vector diagram of the ZV shaper on the right-hand side,

\theta₁=0, \theta₂=\pi

I₁=I₂=I

Therefore,

t₁=0, t₂=\pi/\omega_d

Since

A₁+A₂=1

(normalization constraint) must be hold, and

A₁=I_1, A₂=I₂/

	\zeta\omega_nt₂
e

I₁+

I₂

	\zeta\omega_nt₂
e

=I+

	I
	K

=1,

	\zeta\pi/\sqrt{1-\zeta²
K=e

} .

Therefoere,

I=K/(K+1)

Thus, the ZV shaper

A₁\delta(t)+A₂\delta(t-t₂₎

is given by

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&\pi/\omega_d\\ K/(K+1),&1/(K+1) \end{bmatrix}

The ZVD shaper is composed of three impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at

\pi

rad, and the third vector at

2\pi

rad, and the magnitude ratio is

I₁:I₂:I₃=1:2:1

. Then

I₁+I₂+I₃=0

. From the impulse vector diagram,

\theta₁=0, \theta₂=\pi, \theta₃=2\pi

Therefore,

t₁=0, t₂=\pi/\omega_d, t₃=2\pi/\omega_d

Also from the impulse vector diagram,

I₁=I₃=I, I₂=2I

Since

A₁+A₂+A₃=1

must be hold,

I₁+

I₂

	\zeta\omega_nt₃
e

I₃

	\zeta\omega_nt₃
e

=I+

	2I
	K

	I
	K²

=1,

	\zeta\pi/\sqrt{1-\zeta²
K=e

} .

Therefore,

I=K²/(K+1)²

Thus, the ZVD shaper

A₁\delta(t)+A₂\delta(t-t₂₎+A₃\delta(t-t₃)

is given by

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&\pi/\omega_d,&2\pi/

	2/(K+1)
\omega
	d\\ K

^2,&2K/(K+1)^2,&1/(K+1)^{2
\end{bmatrix}}

The ZVD² shaper is composed of four impulse vectors, in which the first impulse vector is located at 0 rad, the second vector at

\pi

rad, the third vector at

2\pi

rad, and the fourth vector at

3\pi

rad, and the magnitude ratio is

I₁:I₂:I₃:I₄=1:3:3:1

. Then

I₁+I₂+I₃+I₄=0

. From the impulse vector diagram,

\theta₁=0, \theta₂=\pi, \theta₃=2\pi, \theta₄=3\pi

Therefore,

t₁=0, t₂=\pi/\omega_d, t₃=2\pi/\omega_d, t₄=3\pi/\omega_d

Also, from the impulse vector diagram,

I₁=I₄=I, I₂=I₃=3I

Since

A₁+A₂+A₃+A₄=1

must be hold,

I₁+

I₂

	\zeta\omega_nt₂
e

I₃

	\zeta\omega_nt₃
e

I₄

	\zeta\omega_nt₄
e

=I+

	3I
	K

	3I
	K²

	I
	K³

=1,

	\zeta\pi/\sqrt{1-\zeta²
K=e

} .

Therefore,

I=K³/(K+1)³

Thus, the ZVD² shaper

A₁\delta(t)+A₂\delta(t-t₂₎+A₃\delta(t-t₃₎+A₄\delta(t-t₄₎

is given by

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&\pi/\omega_d,&2\pi/\omega_d,&3\pi/\omega_d\\ K^3/(K+1)^3,&3K^2/(K+1)^3,&3K/(K+1)^3,&1/(K+1)^{3
\end{bmatrix}}

Similarly, the ZVD³ shaper with five impulse vectors can be obtained, in which the first vector is located at 0 rad, the second vector at

\pi

rad, third vector at

2\pi

rad, the fourth vector at

3\pi

rad, and the fifth vector at

4\pi

rad, and the magnitude ratio is

I₁:I₂:I₃:I₄:I₅=1:4:6:4:1

. In general, for the ZVDⁿ shaper, i-th impulse vector is located at

(i-1)\pi

rad, and the magnitude ratio is

I₁:I₂:I₃: … :I_n+2=\tbinom{n+1}{0}:\tbinom{n+1}{1}:\tbinom{n+1}{2}: … :\tbinom{n+1}{n+1}

where

\tbinom{m}{k}

implies a mathematical combination.

ETM shaper

Now, consider equal shaping-time and magnitudes (ETM) shapers, with the same magnitude of impulse vectors and with the same angle between impulse vectors. The ETMn shaper satisfies the conditions

\theta₁=0, \theta₂=

	2\pi
	n-1

, … ,\theta_n-1=

	(n-2)2\pi
	n-1

I₂=I₃= … =I_n-1=I₁+I_n, I_n=mI₁ (m>0)

	n
\sum
	i=1

A_i=1

Thus, the resultant of the impulse vectors of the ETMn shaper becomes always zero for all

n\ge2

. One merit of the ETMn shaper is that, unlike the ZVDⁿ or extra insensitive (EI) shapers,^[4] the shaping time is always one (damped) period of the time response even if n increases. The ETM4 shaper with four impulse vectors is obtained from the above conditions together with impulse vector definitions as

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&(2\pi/3)/\omega_d,&(4\pi/3)/\omega_d,&2\pi/\omega_d\\ I/(1+m),&I/K^2/3,&I/K^4/3,&mI/[(1+m)K^{2]
\end{bmatrix}}

I=	(1+m)K²
	K²+(1+m)(K^4/3+K^2/3)+m

	\zeta\pi/\sqrt{1-\zeta²
K=e

The ETM5 shaper with five impulse vectors is obtained similarly as

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&0.5\pi/\omega_d,&\pi/\omega_d,&1.5\pi/\omega_d,&2\pi/\omega_d\\
I/(1+m),&I/K^1/2,&I/K,&I/K^3/2,&mI/[(1+m)K^{2]
\end{bmatrix}}

I=	(1+m)K²
	K²+(1+m)(K^3/2+K+K^1/2)+m

	\zeta\pi/\sqrt{1-\zeta²
K=e

In the same way, the ETMn shaper with

n\ge6

can be obtained easily. In general, ETM shapers are less sensitive to modeling errors than ZVDⁿ shapers in a large positive error range. Note that the ZVD shaper is an ETM3 shaper with

m=1

NMe shaper

Moreover, impulse vectors can be applied to design input shapers with negative impulses. Consider a negative equal-magnitude (NMe) shaper, in which the magnitudes of three impulse vectors are

I₁=I(>0), I₂=-I, I₃=I

, and the angles are

\theta₁=0, \theta₂=\pi/3, \theta₃=2\pi/3

. Then the resultant of three impulse vectors becomes zero, and thus the residual vibration is suppressed. Impulse time

t_2,t₃

of the NMe shaper are obtained as

t₂=(\pi/3)/\omega_d, t₃=(2\pi/3)/\omega_d

, and impulse magnitudes are obtained easily by solving the simultaneous equations

A₁=I,A₂=

	\zeta\omega_nt₂
-I/e

, A₃=

	\zeta\omega_nt₃
I/e

A₁+A₂+A₃=1

The resulting NMe shaper

A₁\delta(t)+A₂\delta(t-t₂₎+A₃\delta(t-t₃₎

\begin{bmatrix} t_i\\ A_i\end{bmatrix} =\begin{bmatrix} 0,&(\pi/3)/\omega_d,&(2\pi/3)/\omega_d\\ I,&-I/K^1/3,&I/K^2/3\end{bmatrix}

I=K/(K-K^2/3+K^1/3),

	\zeta\pi/\sqrt{1-\zeta²
K=e

The NMe shaper has faster rise time than the ZVD shaper, but it is more sensitive to modeling error than the ZVD shaper. Note that the NMe shaper is the same with the UM shaper^[5] if the system is undamped (

\zeta=0

Figure (a) in the right side shows a typical block diagram of an input-shaping control system, and figure (b) shows residual vibration suppressions in unit-step responses by ZV, ZVD, ETM4 and NMe shapers.

Refer to the reference for sensitivity curves of the above input shapers, which represent the robustness to modeling errors in

\omega_n

and

\zeta

Notes and References

Kang . Chul-Goo . August 2019 . Impulse vectors for input shaping control: A mathematical tool to design and analyze input shapers . IEEE Control Systems Magazine . 39 . 4 . 40–55 . 10.1109/MCS.2019.2913610 . 198145461.
Singhose . W. . Seering . W. . Singer . N. . Residual vibration reduction using vector diagrams to generate shaped inputs . Journal of Mechanical Design . 1994 . 116 . 2 . 654–659 . 10.1115/1.2919428 .
Singhose . W. . Command shaping for flexible systems: A review of the first 50 years . International Journal of Precision Engineering and Manufacturing . 2009 . 10 . 4 . 153–168 . 10.1007/s12541-009-0084-2. 111341954 .
Singhose . W. . Derezinski . S. . Singer . N. . Extra-insensitive input shapers for controlling flexible spacecraft . Journal of Guidance, Control and Dynamics . 1996 . 19 . 2 . 385–391 . 10.2514/3.21630 . 1996JGCD...19..385S .
Singhose . W. E. . Seering . W. P. . Singer . N. C. . Time-optimal negative input shapers . Journal of Dynamic Systems, Measurement, and Control . 1997 . 119 . 2 . 198–205 . 10.1115/1.2801233 .

Impulse vector explained

Definition

Properties

Property 1: Resultant of two impulse vectors.

Property 2: Zero resultant of impulse vectors.

Applications: Design of input shapers using impulse vectors

ZVDn shaper

ETM shaper

NMe shaper

Notes and References

ZVDⁿ shaper