Impulse vector

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.

[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[1] introduced the impulse vector (or Kang vector) to generalize Singhose's idea to undamped and underdamped systems with positive and negative impulses.

implies the time location of the impulse function, and

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

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

and a scaling factor for damping during time interval

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

represents the Dirac delta function with impulse time at

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

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

□ Consider an underdamped second-order system with the transfer function

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

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

on the impulse vector diagram to cancel the resultant

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

as shown in the red line of the bottom figure (b).

However, this canceling vector has a longer impulse time that can be as much as a half period compared to

Using impulse vectors, we can redesign[1] known input shapers [3] such as zero vibration (ZV), zero vibration and derivative (ZVD), and ZVDn shapers.

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

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

Now, consider equal shaping-time and magnitudes (ETM) shapers,[1] with the same magnitude of impulse vectors and with the same angle between impulse vectors.

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 The ETM5 shaper with five impulse vectors is obtained similarly as In the same way, the ETMn shaper with

In general, ETM shapers are less sensitive to modeling errors than ZVDn shapers in a large positive error range.

Consider a negative equal-magnitude (NMe) shaper,[1] in which the magnitudes of three impulse vectors are

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

, and impulse magnitudes are obtained easily by solving the simultaneous equations The resulting NMe shaper

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.

Impulse functions and corresponding impulse vectors (or Kang vectors) . (a) For a positive impulse function ( ), the initial point of the impulse vector is located at the origin. (b) For a negative impulse function ( ), the terminal point of the impulse vector is located at the origin.
Two impulse vectors and their corresponding impulse responses for a second-order system. (a) Two impulse vectors with the same magnitude and 180 deg 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 two impulse vectors are exactly same after the final impulse time
(a) Two representations of the resultant of two impulse vectors , and (b) the corresponding impulse responses.
Cancelling impulse vector causes no residual vibration after the final impulse time when the impulse sequence is applied to an undamped or underdamped second-order system because .
Impulse vector diagrams for (a) ZV, (b) ZVD, (c) ZVD 2 , and (d) ZVD 3 shapers.
Impulse vector diagram of (a) an ETM4 shaper, (b) ETM5 shaper, and (c) ETM6 shaper.
Impulse vector diagram for an NMe shaper with a negative impulse.
Residual vibration suppression by input shapers. (a) Block diagram of a typical input-shaping control system, and (b) step responses of a vibratory system with various input shapers when there is no modeling error.