8/7/2019 10.1.1.46.1587

1/3

AN ANALYSIS AND COMPARISON OF

FREQUENCY-DOMAIN AND TIME-DOMAIN INPUT SHAPING*

Lucy Y. Pao and Craig F. Cutforth

Electrical and Computer Engineering Department

University of ColoradoBoulder, CO 80309-0425

pao@colorado.edu and craig.cutforth@colorado.edu

ABSTRACT

The technique of input shaping has been successfully

applied to the problem of maneuvering flexible structures without

excessive residual vibration. With input shaping, non-negative

impulse shapers are often desired because they can be used with

any arbitrary (unshaped) commands and not cause actuator

saturation (if these original unshaped commands do not cause

actuator saturation). We outline conditions when non-negative

amplitude shapers will result when using frequency-domain

methods of input shaping, and we draw comparisons with time-domain input shaping in terms of shaper length (speed) and

number of impulses (ease of implementation).

1.0 INTRODUCTION

Input shaping is a feedforward technique used to reduce

residual vibration in flexible structures. The system parameters

(frequency and damping) are used to design an input shaper,

which is an impulse sequence that is convolved with the input.

This shaped input is the new input to the system. If the

parameters used to design the shaper are accurately known, all or

most of the residual vibration would disappear after the time of

the last impulse. This time, of the last impulse, is referred to as

the length or duration of a shaper.

Favorable characteristics result if the following twoconstraints are satisfied: (1) All impulse amplitudes are non-

negative. (2) The sum of the impulse amplitudes is one. If the

amplitudes sum to one, the final set point resulting from the

shaped input will be the same as that resulting from the original

unshaped input. In addition, all non-negative amplitudes prevent

undue actuator saturation.

A number of researchers have investigated command

shaping methods in the frequency domain [1,5,6]. The purpose of

this paper is to establish some constraints that guarantee the

existence of a non-negative impulse shaper solution when using a

frequency-domain (FD) zero-placement method [6] and to

compare the performance of this frequency-domain method with

conventional time-domain (TD) input shapers. Section 2 reviews

TD and FD input shaping. Section 3 establishes conditions

which guarantee that FD input shaping will yield non-negative

impulse shapers. Section 4 determines the shortest length

shapers that can be achieved for two-mode, zero damping,

flexible systems using FD zero-placement input shaping. Finally,

concluding remarks are given in section 5.

2.0 INPUT SHAPING TECHNIQUES

Singer and Seering [4] developed several TD input-shaping

methods that only require knowledge of the natural frequency and

damping of each flexible mode of a system. The simplest type of

shaper is one that only guarantees zero residual vibration (a ZV

shaper) at the modeling frequencies and damping ratios. For a

multi-mode system, a one-mode shaper is computed for each

mode, then the multi-mode shaper is obtained by convolving all

single-mode shapers [4]. The length of a multi-mode ZV shaper

is 1 /(2 fi )i where fi is the frequency of the ith mode.

Tuttle and Seering [6] established a FD method of placing

zeros to cancel system vibration where the unwanted system

poles are calculated in the z-plane. A discrete shaper is then

formed by placing a zero on each pole (or more to add

robustness). In order to keep the shaper causal, for each zero

placed, an additional pole is placed (at the origin so that no new

vibration is added to the system). Once the discrete transfer

function is established, it is transformed from thez-plane to the s-

plane by the mapping z = exp(sT) , where T is the impulse

spacing. Taking the inverse Laplace transform yields a sequence

of impulses. Sequences of impulses are solved for a range ofTs,

and the smallest T that yields all non-negative amplitudes is

chosen as the desired impulse spacing since it leads to the

shortest shaper with all non-negative amplitudes.

Using this zero-placement method, the general solution to a

single mode shaper is: C(A0z (t) + A1z(tT) + A2z (t 2T))

where: A0 = 1, A1 = )exp()1cos(22

TT z wzw , and

A2 = exp(zwT) . A0 , A1 , and A2 are the impulse amplitudes,

w and z are the natural frequency and damping of the flexible

mode of the system, Tis the impulse spacing, and Cis a scaling

constant. Since the first impulse is at t= 0 , the length of this

shaper is 2T. A FD designed shaper for n modes is just the

convolution ofn single mode shapers, and the n-mode shaper is

of length 2nT. Placing two zeros on one mode for added

robustness is just a special case of a two-mode shaper where

wa = wb and za = zb .The number of impulses for TD ZV multi-mode shapers,

consisting of the convolution ofn single-mode shapers is 2n

; as

the number of modes increases, this quickly leads to a large

number of impulses. The FD ZV zero-placement method results

in 2n +1 impulses. Because the conventional TD method yields

spacing between the impulses that is not constant, one might

solve for all the impulse amplitudes simultaneously to decrease

the number of impulses to 2n +1 , where the impulses satisfy 2n

nonlinear plus one linear constraint equations [2,3]. Numerical

routines are generally required to obtain shaper designs, and

convergence to a solution is not always guaranteed. The FD zero-

*This work was supported in part under a National Science

Foundation Early Faculty CAREER Development Award (Grant

CMS-9625086), a University of Colorado Junior Faculty

Development Award, and an Office of Naval Research Young

Investigator Award (Grant N00147-97-1-0642).

0-7803-4530-4/98 $10.00 1998 AACC

8/7/2019 10.1.1.46.1587

2/3

placement method provides a good alternative for multi-mode

systems and can often be easier to implement. The number of

impulses increases linearly with the number of modes, not

exponentially; the impulse spacing is constant; and solving for

the amplitudes does not involve complex numerical optimization

packages.

3.0 EXISTENCE OF A POSITIVE SOLUTIONIt is not clear that a non-negative shaper solution always

exists when using the zero-placement method. A sufficiency

condition to ensure that the final multi-mode shaper has all non-

negative amplitudes is to ensure that each single mode shaper has

all non-negative amplitudes. A0 (which equals one) is always

positive. Knowing w > 0 , z 0 , and T 0 , A2 = exp(2zwT)

will always be positive. The factor 2 exp(2zwT) in A1 is

always negative, leaving the remaining factor, cos(wdT) , as the

only one that can change sign, where zww2

1 =d . If one can

show that cos(wdT) 0 , then all the amplitudes will be non-

negative. For a multi-mode zero-placement shaper, if one can

show that there exists a T such that the shaper amplitudes for

each mode are non-negative, then there indeed exists a Tfor theresultant convolved shaper such that all its amplitudes are non-

negative.

Is it possible to demonstrate that for a number of flexible

modes, wdi , that there exists a T such that cos(w di T) 0

for all i ? A simple example can be used to show that there does

not always exist a T such that these constraints are met for any

number of frequencies. Pick wda ; wdb = 2wda ; wdc = 3wda ;

wdd = 4w da . For any four frequencies meeting these conditions;

there is no Tat which all four signals cos(wdiT) , i = a, b,c,d, are

non-positive.

It can be shown, however, that a Talways exists if there are

two frequencies.

THEOREM

Given A(t) = cos(2pfat+ f a ) and B(t) = cos(2pfa t+ f a) where

f a = f b = 0 then t* s.t.A(t*) 0, B(t*) 0

fa

andfb, where fb fa .

Proof:

Let ta and tb be the period of A(t) and B(t) , respectively, that

is ta =1/fa and tb =1/fb .

If fa fb 3fa then tb/4 ta/ 4 3tb / 4 . We know that

B(t) 0 t s.t. tb/ 4 t 3tb/ 4 and that A(ta/4) = 0 .

a t* s.t.A(t*) 0,B( t*) 0 when fa fb 3fa .

If fb >3fa , let T be the interval: [ ta/ 4,3ta/ 4] ,A(t) 0 t T . For fb > 3fa , tb < ta/3 and the

length(T) = 3ta / 4 ta / 4 = ta/ 2 , and tb < ta / 3 < ta / 2 =

length(T). Therefore B(t) must complete at least one cycle

within T and a t* Ts.t.B(t*) 0 .

a t* s.t.A(t*) 0, B(t*) 0 when fb > 3fa .

a t* s.t.A(t*) 0, B(t*) 0 when fb fa .

Q.E.D.

Work is currently being done to establish an analogous theorem

for the case of three frequencies.

4.0 FINDING THE SHORTESTSOLUTION

In the previous section we proved the existence of an

impulse spacing Twhere all shaper amplitudes are non-negative

for a two-mode FD designed zero-placement shaper. Let us now

find the minimum T where all the shaper amplitudes are non-

negative, thus leading to the zero-placement shaper design that

yields the fastest maneuvers.

Consider the sufficiency constraints from the previoussection: cos(wdiT) 0 for all i. A cosine wave of frequencyf*,

will only be non-positive in the range [t* / 4 + nt*,3t* / 4 + nt*]

where t* = 1/f* , n =1,2,3,... Thus, for the two-mode system,

the earliest A(t) can be non-positive is at ta / 4 . Since we will

only be varying fb , which is greater than fa , our minimum

guaranteed Tcan not come before ta/ 4 . If we consider the range

fa fb 3fa then the minimum Tfor this range of fb is ta/ 4

because B(ta/ 4) 0 . If we consider the range 3fa < fb 5fa ,

the minimum Tfor this range of fb is 5tb/ 4 . These results can

be extended to all possible ranges: Tm = ta/4 where

nfa < fb (n + 2) fa for n =1,5,9,13,... and Tm = (n + 2)tb/4

where nfa < fb (n + 2) fa for n = 3,7,11,15,... This isrepresented in Figure 4.1. There probably is a T< Tm where all

the amplitudes are non-negative; Tm is the smallest T

guaranteed by the above theorem to yield non-negative

amplitudes.

If the minimum impulse spacing possible for a two-mode FD

zero placement shaper is Tm = ta / 4 , then the duration of the

shaper is 4Tm = ta . The minimum time for a two-mode TD

convolved ZV shaper is ta/2 + tb/ 2 . Hence, because fb fa ,

the best the FD zero-placement method can be guaranteed to do

is match the length of the TD convolved ZV shaper, which occurs

when fb = fa . A parametric plot of the two cosine waves shows

that the sufficient constraints we have established, cos(wdaT) 0

and cos(wdbT) 0 , consist of the third quadrant.

Because the constraints leading to the Theorem are

sufficient and not necessary, the question arises: Can we establish

other constraints that yield a smaller impulse spacing? To make

the situation a little less complex, let us add one more constraint:

za = zb = 0 . With zero damping the amplitudes for a FD two-

mode shaper are: A0 = 1, A1 = 2(cos(wdaT) + cos(wdbT)) ,A2=4 cos(wdaT)cos(wdbT) +2 , A3 = A1 , A4 = 1. Again, the need

is for all the amplitudes to be non-negative. A1 0 requires

cos(wdbT) cos(wdaT) ; A2 0 requires

cos(wdaT)cos(wdb T) 1/ 2 . These constraints are represented

on a parametric plot as the shaded region in Figure 4.2; this

region is certainly larger than the earlier sufficiency conditions

which yielded the third quadrant. It should be noted that for

za = zb = 0 , these are necessary andsufficient constraints. Any

impulse spacing T that gives a (cos(2pfaT),cos(2pfbT) )

coordinate in the shaded region leads to all positive impulses in

the shaper.

The rightmost point in this new constraint region is

)2/2,2/2( . The earliest T at which cos(wda T)= 22 is

T= ta/ 8 . Thus it is possible for all the amplitudes of a zero

damping, two-mode system to be non-negative at an impulse

spacing, T= ta/ 8 . As one might guess this would require wdb

8/7/2019 10.1.1.46.1587

3/3

to be a certain frequency so that cos(wdb T)= 22 at

T= ta /8. Ifwdb would vary from this particular value,

entrance into the region would occur elsewhere at a different T

greater than ta / 8. Examples with various mode ratios

(r= fb/fa ) are shown in Figure 4.2.

Figure 4.3 shows the minimum T for a two-mode zero

damping system as the mode ratio varies for both the FD zero-placement method and the conventional TD convolved ZV

method. For a two mode, zero damping system the zero-

placement shaper will be shorter than the traditional ZV method

when the mode ratio is less than five. For mode ratios greater

than five, the shaper method that leads to faster maneuvers

varies. These results could be used as a guide for choosing the

shaper design method when determining shapers for two mode

systems with small damping factors.

5.0 CONCLUSION

We have proven that a non-negative shaper solution always

exists when using a frequency-domain zero-vibration zero-

placement method for a two-mode system. We have also

analyzed the shortest shaper lengths possible when applying thezero-placement method for undamped systems. For small mode

ratios of undamped systems, the zero-placement method yields

shorter length shapers than convolving traditional time-domain

zero-vibration input shapers; for larger mode ratios, the method

that produces a shorter shaper varies. Our analysis has allowed

us to gain further insight in the advantages and disadvantages of

frequency-domain and convolved time-domain input shaping

methods for multi-mode flexible systems.

REFERENCES

[1] S. P. Bhat and D. K. Miu. Solutions to Point-to-Point

Control Problems Using Laplace Transform Technique,

ASME J. Dynamic Systems, Measurement, and Control,

113(3): 425-431, Sept. 1991.

[2] J. M. Hyde and W. P. Seering. Using Input Command Pre-

Shaping to Suppress Multiple Mode Vibration , Proc. IEEE

Robotics and Automation Conf., Sacramento, CA, pp. 2604-

2609, April 1991.

[3] B. W. Rappole, N. C. Singer, and W. P. Seering. Multiple-

Mode Input Shaping Sequences for Reducing Residual

Vibrations,ASME Mechanisms Conf., 1994.

[4] N. Singer and W. Seering. Preshaping Command Inputs to

Reduce System Vibration, ASME J. Dynamic Systems,

Measurement, and Control, 112(1): 76-82, March 1990.

[5] T. Singh and S. R. Vadali. Robust Time-Optimal Control: A

Frequency Domain Approach, Proc. AIAA Guidance,

Navigation, and Control Conf., Scottsdale, AZ, pp. 241-251,

Aug. 1994.

[6] T. D. Tuttle and W. P. Seering. A Zero-Placement

Technique for Designing Shaped Inputs to Suppress

Multiple-Mode Vibration, Proc. American Control Conf.,

Baltimore, MD, pp. 2533-2537, June 1994.

Figure 4.2: Parametric Plots of various mode ratios and the

constraint region. The solid line representsr= 5; the dashed line

represents r= 3; the dotted line represents r= 3.5.

Figure 4.3: Length Comparison of two-mode shapers,

Zero-Placement vs. Traditional Convolved ZV

Figure 4.1: The minimum impulse spacingTproven by the

Theorem, as a fraction ofta, versus the mode ratio r =fb/fa