Tutorial on Input Shaping/Time Delay Control

of Maneuvering Flexible Structures

Tarunraj SinghDept. of Mech. & Aero. Eng.,

University of Buffalo,Buffalo, NY 14260

tsingh@eng.buffalo.eduhttp://code.eng.buffalo.edu/tdf/

William SinghoseDept. of Mech. Eng.,

Georgia Institute of TechnologyAtlanta, Georgia, 30332

Bill.Singhose@me.gatech.eduhttp://www.me.gatech.edu/inputshaping/

Abstract

Precise position control and rapid rest-to-rest mo-tion is the desired objective in a variety of applica-tions. The desire for reducing the maneuver timerequires reducing the inertia of the structure whichsubsequently results in low frequency dynamics.The requirement of precise position control im-plies that the residual vibration of the structureshould be zero or near zero. This paper presentstechniques to shape the input to the system soas to minimize the residual vibration. A secondclass of problems which includes design of inputprofiles for systems with rigid-body modes drivenby actuators with finite control authority, is alsopresented. The common thread which connects allthe techniques presented in this paper is related tothe design of controllers which are robust to mod-eling uncertainties. The proposed techniques areillustrated by simulations and experiments.

1 Introduction

Smith [1], Calvert and Gimple [2] proposed a sim-ple technique to generate non-oscillatory responsefrom an lightly-damped system subject to a stepinput. This is achieved by exciting two transientoscillations so as to result in constructive cancel-lation of the oscillations. Smith termed the tech-nique Posicastmotivated by what he states: “ Thisis what happens when a fisherman drops his flyin the water at the maximum-position and zerovelocity instant” [1]. Tallman and Smith [3] illus-trated the Posicast technique using an analog com-puter and noted the sensitivity of the controller to

variation in the location of the poles of the sys-tem caused by nonlinear components in the sys-tem or variation of the parameters of the systemas a function of temperature.

Between the late 50’s and the publication of theInput Shaping paper by Singer and Seering [8],there was some work on the shaping of inputprofiles for control of residual vibration [4], [5].Swigert [5] proposed techniques for the determi-nation of torque profiles which considered the sen-sitivity of the terminal states to variations in themodel parameters. Publication of the Input Shap-ing paper renewed interest in prefiltering refer-ence inputs for robust vibration control, whichhas resulted in dozens of papers with applicationto spacecrafts, robots, cranes, chemical processes,etc. A chronological listing of papers relevant torobust vibration control of slewing structures ispresented in the bibliography.

This paper will consider two classes of problems:The first, involves real-time shaping or time-delayfiltering of the reference command to stable sys-tems with the objective of minimizing the resid-ual vibration [1] [8], [16], [23]. This will be dealtin detail in Section 2. The second class of prob-lems considered is the design of controllers for sys-tems with rigid body modes with constraints onthe control input [7], [9], [14], [29], [49]. This canbe further classified into two categories which willbe expounded in Section 3. Sections 2 and 3 willdescribe in detail, technique for reducing the sen-sitivity of the control profile to errors in modelparameters such as damping ratio, and natural fre-quency. Section 4 will briefly address the designof robust controllers for nonlinear systems [41].

Finally, Section 5 will describe some applicationswhere the proposed techniques have been success-fully implemented.

2 Real Time Command Shaping

θ

Trolley

Cable

Payload

g

x

Figure 1: Overhead Gantry Crane

There are all types of possible solutions to theproblem of flexible dynamics including feedbackcontrol, feedforward control, command shaping,and even redesigning the physical hardware. Asimple example of this challenging area is pre-sented by an overhead gantry crane like the oneshown schematically in Figure 1 . The payload ishoisted up by a cable. The upper end of the cableis attached to a trolley which travels along a railto position the payload.

Cranes are controlled by a human operator whomoves levers or presses buttons to cause the trol-ley to move. If the operator simply presses thecontrol button for a finite time period, then thetrolley will move a finite distance and come to rest.The payload on the other hand, will usually oscil-late about the new trolley position. The payloadposition for a typical trolley movement is shownin Figure 2a.

An experienced crane operator can sometimes pro-duce the desired payload motion with a smallamount of residual vibration by pressing the but-ton multiple times at the proper instances. Thepayload position for such a situation is shown inFigure 2b.

2.1 Simple Zero Vibration CommandAs a first step to understanding how to generatecommands that move systems without vibration,

0

1

2

3

4

5

6

7

8

0 5 10 15

Trolley

Payload

Pos

itio

n

Time

Button On

(a)

0

1

2

3

4

5

6

7

8

0 5 10 15

Trolley

Payload

Pos

itio

nTime

Button On

(b)

Figure 2: Crane Response: a) Unshaped Commandb) Shaped Command

it is helpful to start with the simplest such com-mand. We know that giving the system an impulsewill cause it to vibrate; however, if we apply a sec-ond impulse to the system, we can cancel the vi-bration induced by the first impulse. This conceptis shown in Figure 3.

At this point, it is useful to derive the amplitudesand time locations of the two-impulse commandshown in Figure 3. If we have a reasonable es-timate of the system’s natural frequency, ω, anddamping ratio, ζ, then the residual vibration thatresults from a sequence of impulses can be de-scribed by:

V (ω, ζ) = e−ζωtn√C(ω, ζ)2 + S(ω, ζ)2 (1)

where,

C(ω, ζ) =n∑

i=1

Aieζωticos(ωdti),

S(ω, ζ) =n∑

i=1

Aieζωtisin(ωdti) (2)

-0.4

-0.2

0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3

A1 ResponseA2 ResponseTotal Response

Pos

itio

n

Time

A1

A2

Figure 3: Two Impulse Response

Ai and ti are the amplitudes and time locationsof the impulses, n is the number of impulses inthe impulse sequence, and ωd = ω

√1− ζ2. Equa-

tion 1 is actually the percentage residual vibration.It tells us how much vibration a sequence of im-pulses will cause, relative to the vibration causedby a single, unity-magnitude impulse. By setting(1) equal to zero, we can solve for the impulseamplitudes and time locations that would lead tozero residual vibration. However, we must placea few more restrictions on the impulses, or thesolution will converge to zero-valued or infinitely-valued impulses. To avoid the trivial solution of allzero-valued impulses and to obtain a normalizedresult, we require the impulses to sum to one:

∑Ai = 1. (3)

Impulses could satisfy (3) by taking on very largepositive and negative values. One way to obtaina bounded solution is to limit the impulse ampli-tudes to finite values or to positive values:

Ai > 0, i = 1,2,....,n (4)

The problem we want to solve can now be statedexplicitly: find a sequence of impulses that makes(1) equal to zero, while also satisfying (3) and (4).For a two-impulse sequence, the problem has fourunknowns - the two impulse amplitudes (A1, A2)and the two impulse time locations (t1, t2). With-out loss of generality, we can set the time locationof the first impulse equal to zero, t1 = 0. The prob-lem is now reduced to finding three unknowns (A1,A2, t2). In order for (1) to equal zero, the expres-sions in (2) must both equal zero independently

because they are squared in (1). Therefore, theimpulses must satisfy:

0 = A1 +A2eζωt2cos(ωdt2) (5)

0 = A2eζωt2sin(ωdt2) (6)

Equation (6) can be satisfied in a nontrivial man-ner, when the sine term equals zero. This occurswhen:

ωdt2 = nπ,⇒ t2 =nπ

ωd=

nTd

2n=1,2,.. (7)

where Td is the damped period of vibration. Thisresult tells us that there is an infinite number ofpossible values for the location of the second im-pulse - they occur at multiples of the half period ofvibration. To cancel the vibration in the shortestamount of time, choose the smallest value for t2:

t2 =Td

2(8)

For this simple case, the amplitude constraintgiven in (3) reduces to:

A1 +A2 = 1 (9)

Using the expression for the damped natural fre-quency and substituting (8) and (9) into (5) gives:

0 = A1 − (1−A1)exp(ζπ√1− ζ2

) (10)

Rearranging (10) and solving for A1 gives:

A1 =exp( ζπ√

1−ζ2)

1 + exp( ζπ√1−ζ2

)(11)

Defining K = exp( −ζπ√1−ζ2

), the sequence of two

impulses that leads to zero residual vibration cannow be summarized as:[

Ai

ti

]=

[1

1+KK

1+K

0 0.5Td

](12)

2.2 Using Zero-Vibration Impulse Se-quences to Generate Zero-VibrationCommandsReal systems cannot be moved around with im-pulses, so we need to convert the properties of theimpulse sequence given in (12) into a usable com-mand. This can be done in a very simple way.

0 2 0*

0 2 ∆ 2+∆Shaped Command

Initial Command Input Shaper∆

Figure 4: Multi Pulse Shaped Input

The impulse sequence is convolved with any de-sired command signal. The convolution productis then used as the command to the system. Ifthe impulse sequence causes no vibration, then theconvolution product will also cause no vibration.

This command generation process, called inputshaping, is demonstrated in Figure 4 for an ini-tial command that is a pulse function and a two-impulse input shaper. Note that the convolutionproduct in this case is the two-pulse command sim-ilar to that shown in Figure 2b. But in most casesthe impulse sequence will be much shorter thanthe command profile. When this occurs, the com-ponents of the shaped command that arise fromthe individual impulses run together as shown inFigure 5.

0 0 ∆*

Initial Command Input Shaper

0 ∆Shaped Command

D

D+∆

Figure 5: Continuous Shaped Input

2.3 Robustness to Modeling ErrorsThe amplitudes and time locations of the impulsesdepend on the system parameters (ω and ζ). If

there are errors in these values (and there alwaysare), then the impulse sequence will not result inzero vibration. In fact, for the two-impulse se-quence discussed above, there can be a lot of vi-bration for a small modeling error. This lack of ro-bustness was a major stumbling block for the orig-inal formulation that was developed in the 1950’s.This problem can be visualized by plotting a sen-sitivity curve that shows the amplitude of residualvibration as a function of the system parameters.One such sensitivity curve for the zero-vibration(ZV) shaper is shown in Figure 6 with the normal-ized frequency on the horizontal axis and the per-centage vibration on the vertical axis. Note thatas the actual frequency deviates from the model-ing frequency, the amount of vibration increasesrapidly. The robustness can be measured quan-titatively by measuring the width of the curve atsome low level of vibration. This non-dimensionalrobustness measure is called the shaper’s insensi-tivity. The 5% insensitivity has been labeled inFigure 6.

0

5

10

15

20

25

0.5 0.75 1 1.25 1.5

ZV ShaperRobust (ZVD) ShaperVery Robust Shaper

Per

cent

age

Vib

rati

on

Normalized Frequency (ωωωω/ωωωωm)

Vtol

Figure 6: Sensitivity Curves

In order to increase the robustness of the inputshaping process, the shaper must satisfy addi-tional constraints. One such constraint sets thederivative of (1), with respect to frequency, equalto zero [8]. That is:

0 =d

dωV (ω, ζ) (13)

When this additional constraint is satisfied with V= 0, the result is a Zero Vibration and Derivative(ZVD) shaper [8]. By comparing the 5% insen-sitivities shown in Figure 6, it can be concluded

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.6 0.81

1.21.4

0

0.1

0.20

0.05

0.1

0.15

ResidualVibration

DampingRatio, ζ

Frequency (Hz)

Figure 7: Three Dimensional Curve

that the ZVD shaper is significantly more robustto modeling errors than the ZV shaper.

Since the development of the ZVD shaper, severalother robust shapers have been developed. In fact,shapers can now be designed to have any amountof robustness to modeling errors [38]. The sen-sitivity curve for a very robust shaper is shownin Figure 6. Robustness is not restricted to er-rors in the frequency. Figure 7 shows a three-dimensional sensitivity curve for a shaper thatwas designed to suppress vibration between 0.7Hz and 1.3 Hz and also over the range of damp-ing ratios between 0 and 0.2. The shapers corre-sponding to these curves were designed using theSpecified-Insensitivity (SI) approach. The moststraightforward method for generating a shaperwith specified insensitivity to modeling errors isthe technique of frequency sampling [19], [38].This method requires repeated use of the vibra-tion amplitude equation, (1). In each case, V(ω,ζ)is set less than or equal to a tolerable level of vi-bration, Vtol:

Vtol > e−ζωstn√C(ωs, ζ)2 + S(ωs, ζ)2, s = 1,..., m

(14)where ωs represents the m unique frequencies atwhich the vibration is limited.

For example, if the shaper needs to suppress vi-bration for frequency errors of 20%, then the con-straint equations limit the vibration to below Vtol

at specific frequencies between 0.8ωn and 1.2ωn.This procedure is illustrated in Figure 8 for Vtol

0

5

10

15

20

25

30

35

0.6 0.8 1 1.2 1.4

Per

cent

age

Vib

rati

on

Normalized Frequency (ωωωωa/ωωωωm)

Limit Vibration at Specific Frequencies

Vtol

Figure 8: Specified-Insensitivity Shaper

= 5%. Another technique related to SI shapingminimizes the expected level of residual vibrationover a specified frequency range [52]. This tech-nique has the advantage of taking into account anyknown distribution of the frequencies in the rangebeing suppressed.

Any shaped command will have its rise time in-creased by the duration of the shaper as is shownin Figure 9a. Because the duration of the ZVDshaper is twice that of the ZV shaper, the ZVDshaper increases the rise time more than the ZVshaper. This increased rise time is the price thatis paid for the increased robustness to modelingerrors. With the SI shapers, increasing robustnessincreases rise time in a nonlinear manner. Thisleads to certain operating points that are advan-tageous [3].

For example, the ZVD shaper has a duration ofonly 1 period of the natural frequency. This timepenalty is a small price to pay for the excellent ro-bustness to modeling errors. To demonstrate thistradeoff, Figure 9 shows the response of a spring-mass system to step commands shaped with thethree shapers shown in Figure 6. Figure 9a showsthe response when the model is perfect and Fig-ure 9b shows the case when there is a 30% error inthe frequency estimate. The increase in rise timecaused by the shapers is apparent in Figure 9a,while Figure 9b shows the vast improvement in vi-bration reduction that the robust shapers providein the presence of modeling errors.

The techniques mentioned above produce robust-ness built into the design of the input shaper.There are other approaches to achieve robust in-

0

0.5

1

1.5

0 1 2 3 4 5

ZVZVDVery Robust

Pos

itio

n

Time

(a)

0

0.5

1

1.5

0 1 2 3 4 5

ZVZVDVery Robust

Pos

itio

n

Time

(b)

Figure 9: System Response: a) Perfect Knowledge,b) 30% Error

put shaping. Several researchers have used adap-tive input shaper modification techniques to ob-tain robustness. Sensor feedback is used to tunethe input shaper such that the residual vibrationis decreased. Some of these approaches includethat of Tzes and Yurkovich [26] and Khorrami, etal. [25], who used on-line adaptive schemes to up-date the input shaper parameters. Bodson used arecursive least-squares technique to tune the inputshaper parameters [36]. Magee and Book modifiedthe input shaper as a function of the system con-figuration [18].

2.4 Concurrent Design of Command Shap-ing and Feedback ControlGiven that command shaping can greatly reducethe vibration of the system, it reduces the burdenon the feedback controller. Therefore, the feed-back control design becomes easier; it does nothave to be concerned with reducing vibration in-duced by the reference command. The design ofthe feedback controller can be primarily based on

disturbance rejection and stability, which are itsnatural strengths. Given this realization, the ques-tion arises as to how to optimize the combined de-sign of the feedback and command shaping com-ponents.

One method assumes a PD feedback controller andthen concurrently chooses the PD gains and the in-put shaper impulses while satisfying performancespecifications [56], [63]. The design method takesinto account limits on allowable overshoot, resid-ual vibration, and actuator effort. Furthermore,the structure of the method allows a wide rangeof performance requirements, such as disturbancerejection, to be integrated into the design. The re-sults indicate that PD feedback control enhancedwith input shaping provides better performancethan PD control alone. This effect is demonstratedin Figure 10 where the settling time is shown as afunction of the damping ratio of the system. WithPD control alone, the minimum settling time oc-curs near ζ=0.7 - the classic solution. However,when input shaping is used, the settling time canbe greatly reduced and this occurs when the con-troller is tuned to have a small damping ratio.

0

0.5

1

0 0.2 0.4 0.6 0.8 1

PD

PD + Shaping

5% S

ettl

ing

Tim

e, s

Damping Ratio (ζζζζ))))

0.39

0.15

Figure 10: Settling Time vs. Damping Ratio

2.5 Shaper Design in the S-Plane and Z-PlaneThe effect of command shaping is to place zeros ator near the flexible poles of the system. This ideawas well documented by Bhat and Mui [10]. Thisrealization leads to straightforward design proce-dures in the z-plane that were first discussed byMurphy and Watanabe [16]. Singh and Vadali [23]illustrated that a time-delay filter designed to can-cel the poles of the system results in the same solu-

tion as the posicast control developed by Smith [1].They also illustrated that cascading multiple ver-sions of the time-delay filter resulted in the robustshaper that was proposed by Singer and Singer.Singh and Vadali also proposed a simple techniqueto design time-delay filters using the specified timespacing of the sampling period [32].

Seth used z-plane analysis to design a digitalshaper for reducing vibration in a coordinate mea-suring machine [27]. Tuttle developed a simplestep-by-step method to design multiple-mode in-put shapers in the discrete time domain by bring-ing together previous methods [30]. Additionally,Tuttle directly addressed the issue of time opti-mality for digital shapers by presenting a methodfor finding a positive impulse shaper that had theshortest time duration. Like Seth, Jones used z-plane analysis to design a digital shaper for re-ducing vibration in a coordinate measuring ma-chine [54]. Additionally, Jones indicated the re-quirements on shaper duration to obtain an inputshaper with only positive impulses.

Magee applied a digital shaping filter to a systemwith varying parameters by modifying the inputshaper duration to account for system parametervariations [18]. This work verified the difficultyof changing shaper duration that was predictedby Murphy and Watanabe. More recently, Parket al. extended the z-plane based design of digi-tal input shapers to more robust shapers [59]. Inparticular, Park devised a discrete time sensitiv-ity expression. This expression was used to designvery robust multiple hump input shapers directlyin the z-plane [47].

3 Saturating Controllers

The problem of design of optimal control with lim-its on control authority has been of interest fordecades. When cost functions such as maneuvertime, or fuel or a weighted combination of fuel andtime are considered, the resulting optimal controlprofiles are bang-bang or bang-off-bang implyingthat the controller is turned on to the extremevalues or is turned off. The problem of designof time-optimal control profiles for flexible struc-tures has been of increasing interest over the pasttwo decades. There have been numerous computa-

tional approaches presented to deal with the effectof flexibility. Most of these deal with single in-put rest-to-rest problems under two classes: near-minimum time control and exact minimum-timecontrol.

The first category is based on smooth approxima-tions to the time-optimal control for an equiva-lent rigid body. This is applicable where the ap-plied input can be smoothly varied and are notrestricted to an on-off set. Junkins et al. [9] pa-rameterize a single switch bang-bang profile usingcubic polynomials in time and illustrate that theresidual vibration of a flexible structure can be sig-nificantly reduced for a small penalty in maneuvertime. Vadali et al. [33] used the arctan to ap-proximate the signum function and used a param-eterized smooth control profiles to determine near-time optimal control profiles for three dimensionalattitude control of ASTREX, a flexible spacecrafttestbed.

The second category studies the exact time-optimal control problem. The determination oftime-optimal control profiles for flexible slewingstructures with limited control authority has beenaddressed by Singh et al. [7]. They illustrate thatthe time-optimal control profile for un-dampedsystems is antisymmetric about the mid-maneuvertime. Hablani [11] studied the same problem,but with damped modes. Ben-Asher et al. [14],present an elegant technique to prove the time-optimality of the control profiles. It is well knownthat the time-optimal control profile is highly sen-sitive to errors in system parameters. Liu and Wie[15] present a technique to “robustify” the time-optimal control by including additional switchesto the control profile. Singh and Vadali [29] pro-pose a frequency domain approach for the designof time-optimal controllers for flexible structures.The motivation behind their work is the fact thata bang-bang input can be viewed as a summationof time-delayed step commands. They pose theproblem as the design of a time-delay filter de-signed to cancel all the poles of the system andsatisfy the rigid body boundary conditions. Theyuse the knowledge that locating multiple zeros ofthe time-delay filter at the estimated location ofthe poles of the system, results in robustness tomodeling uncertainties. They illustrate their tech-nique by designing time-optimal and robust time-

optimal control profiles for rest-to-rest and spin-up maneuvers for the benchmark floating oscillatorproblem.

This section will discuss the design of robust timeand weighted fuel-time optimal controllers. Thebenchmark floating oscillator problem [21], will beused to illustrate the design technique. The designproblem is posed as the design of a time-delay fil-ter which generate the bang-bang or bang-off-bangcontrol profile when it is driven by a step input.The knowledge that locating multiple zeros of thetime-delay filter at the estimated location of thepoles of the system will result in robustness to un-certainty in the location of the pole of the systemwill be used in the design process. For a functionf(s) = 0, to have a minimum of two roots at s =s0 requires that

f(s0) = 0 anddf(s)ds

∣∣∣∣s0

= 0. (15)

This fact is utilized to develop constraint equa-tions to design time-delay filters with multiple ze-ros at specified locations.

m2m1 ��❅❅❅�

��❅❅❅��

k✲y1

✲y2

✲u

Figure 11: Floating Oscillator

The equations of motion of the benchmark floatingoscillator problem illustrated in Figure 11 are[m1 00 m2

] [y1

y2

]+

[k −k−k k

] [y1

y2

]=

[10

]u.

(16)The state constraint for all the optimization prob-lems considered in this section are

y1(0) = y2(0) = 0, and y1(0) = y2(0) = 0y1(tf ) = y2(tf ) = 1, and y1(tf ) = y2(tf ) = 0 (17)

and the normalized control is subject to the con-straint

−1 ≤ u ≤ 1. (18)

The nominal values of the parameters of the sys-tem are

m1 = m2 = k = 1 (19)

The equations of motion can be decoupled by thesimilarity transformation

θ =12(y1 + y2) and q =

12(y2 − y1) (20)

resulting in the equations of motion

θ =12u

q + 2q = −12u, (21)

and the corresponding boundary conditions are

θ(0) = q(0) = 0 and θ(0) = q(0) = 0θ(tf ) = 1, q(tf ) = 0 and θ(tf) = q(tf ) = 0. (22)

The decoupled equations are used to derive theconstraint equations for the optimization prob-lem since this approach can be generalized for anynumber of modes.

✲ ∑Ni=0 Ai e

−sTi ✲ GP (s) ✲U∗(s) U(s) Y (s)

Step Time Delay Filter Plant

Figure 12: Time Delay Filter Structure

In this paper, the design of control profiles subjectto the aforementioned control constraint is posedas the design of a time-delay filter. The outputof this time-delay filter subject to a step inputis the optimal switching control profile, as shownin Figure 12. For instance to generate a singleswitch bang-bang control profile which is the time-optimal control profile for a rigid body system, thetransfer function of the time-delay filter is

G(s) = 1− 2exp(−sT ) + exp(−2sT ), (23)

and the optimal control profile is given by theequation

u(s) =1s(1− 2exp(−sT ) + exp(−2sT )). (24)

The generic transfer function of a time-delay filteris represented as

G(s) =N∑

i=0

Aiexp(−sTi), where T0 = 0, (25)

and where Ai belongs to the set

Ai =[ −2 −1 1 2

](26)

to guarantee that the output of the time-delay fil-ter is either bang-bang or bang-off-bang. For rest-to-rest maneuver of flexible structures, the con-straint which guarantees zero residual vibration isderived by requiring a set of zeros of the time-delayfilter to cancel the under-damped poles of the sys-tem. For a system with a set of under-dampedpoles at

s = σ ± jω, (27)

the constraint equations are

N∑i=0

Aiexp(−σTi)cos(ωTi) = 0, (28)

andN∑

i=0

Aiexp(−σTi)sin(ωTi) = 0, (29)

which are derived by forcing the real and imag-inary parts of the transfer function of the time-delay filter to zero at s = σ±jω. Note, that this isequivalent to the conditions given in Equations 2.

To satisfy the boundary conditions for the rigidbody for the rest-to-rest maneuver, the transferfunction of the time-delay filter should have twozeros at the origin of the complex plane to cancelthe rigid body poles, resulting in the constraintequation

N∑i=0

Ai = 0 (30)

andN∑

i=0

AiTi = 0 (31)

The constraint to satisfy the total rigid body dis-placement is

θ(tf = TN ) =12

N∑i=0

Ai(TN − Ti)2

2. (32)

Finally, to desensitize the control profile to uncer-tainties in the location of the under-damped polesof the system, constraints are derived which placemultiple zeros of the time-delay filter at the esti-mated location of the poles of the system. These

constraints are

N∑i=0

AiTiexp(−σTi)cos(ωTi) = 0, (33)

andN∑

i=0

AiTiexp(−σTi)sin(ωTi) = 0 (34)

which are equivalent to the zero derivative con-straint given in Equation 13.

Since the constraints are nonlinear, there are po-tentially numerous parameter sets which satisfy allof the constraints. The sufficient conditions for theoptimality of the control profile are dependent onthe cost function to be optimized for and a gen-eral approach to verify the optimality is not avail-able. For un-damped system, it has been shownthat the control profile is anti-symmetric about themid-maneuver time. This fact can be exploited toreduce the number of parameters to be optimizedfor.

3.1 Time-Optimal ControlThe time-optimal control profile for the un-damped benchmark problem can be determinedby solving the following parameter optimizationproblem which is derived by exploiting the anti-symmetric properties of the control profile (Fig-ure 13):

min J = T 22 (35)

−2cos(ω(T2 − T1)) + 1 + cos(ωT2) = 0 (36)12(2T 2

2 − (2T2 − T1)2 + T 22 − T 2

1 ) = 1, (37)

where

T0 = 0, T1 = T1, T2 = T2, T3 = 2T2−T1, T4 = 2T2

A0 = 1, A1 = −2, A2 = 2, A3 = −2, A4 = 1. (38)

To desensitize the controller to the frequency ofthe flexible mode, two switches are added to thecontrol profile and the problem is

min J = T 23 (39)

−2cos(ωT31) + 2cos(ωT32) + 1 + cos(ωT3) = 0 (40)

−2T31sin(ωT31) + 2T32sin(ωT32) + T3sin(ωT3) = 0 (41)

2T 23 − (2T3 − T1)

2 + (2T3 − T2)2 − T 2

3 + T 22 − T 2

1 = 2. (42)

where the double subscript represent Pij = Pi−Pj .Having solved for the switch times, their optimal-ity have to be verified. This is achieved by solving

✲

✻

t

u

1

−1

T1 T2 2T2 − T1

2T2

Figure 13: Antisymmetric Control Profile

for the costates and the resulting switching curveis used to corroborate the optimality of the solu-tion. Details of these techniques have been pre-sented in [7], [15], [14], [29] etc.

Figure 14 illustrates the time-optimal control pro-file and the robust control profile and the corre-sponding energy sensitivity curves. It is clear thatlocal to the nominal spring stiffness there is a sig-nificant improvement of the insensitivity of the ro-bust control profile.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

k1

Res

idua

l Ene

rgy

Time−Optimal Profile Robust Control Profile

0 1 2 3 4 5 6

−1

−0.5

0

0.5

1

Time

Con

trol

Figure 14: Control and Residual Energy Variation

The variation of the structure of time-optimal con-trol profiles as a function of damping has beenillustrated by Pao [37] and Singh [35]. For a two-mass system connected by a spring and a damper,the control profile changes from a three switch forun-damped systems to a five switch and back toa three switch control profile, as the damping isincreased. Time-optimal control profile for mult-mode systems have been derived in [45]. Tuttleand Seering, developed a Matlab toolbox to gen-

erate time-optimal commands for a wide range offlexible systems [55].

3.2 Fuel/Time Optimal ControlThe Fuel/Time optimal control profile for thebenchmark problem can be represented as shownin Figure 15 where the anti-symmetric propertyfor undamped systems is exploited.

✲

✻

t

u

1

−1

T1

T2 T3

2T4 − T3

2T4 − T2

2T4 − T1 2T4

Figure 15: Fuel-Time Optimal Control Profile

Minimizing the cost function

J =∫ tf

0(1 + α|u|) dt (43)

where the cost function J is a weighted (α > 0)combination of the maneuver time and fuel con-sumed results in the parameter optimization prob-lem:

min J = 2T4 + 2α [T43 + T21] (44)

cos(ωT43)− cos(ωT42)− cos(ωT41) + cos(ωT4) = 0 (45)

1

2(−T 2

1 − T 22 + T 2

3 + 2T4(T1 + T2 − T3)) = 1. (46)

Solving for the optimal control profile as a func-tion of α, the control profile changes from a threeswitch bang-bang profile for α = 0, to a six switchbang-off-bang control profile which simplifies to atwo switch bang-bang profile for α > 0.6824, forthe benchmark problem, as shown in Figure 16.

Hartmann and Singh [50] present a general devel-opment of the necessary and sufficient conditionsfor optimality of the fuel/time optimal control pro-files for system of order higher than the benchmarkproblem.

3.3 Minimax ControlThe techniques to desensitize the controller tomodeling errors which have been presented to thispoint, only require information about the nomi-nal values of the model parameters. The result-ing controllers are robust in the vicinity of thenominal parameters of the system. However, in

0 2 4 6

−1

−0.5

0

0.5

1

0 1 2 3 4

−1

−0.5

0

0.5

1

α0.0 0.6284

0 2 4 6 8 10

−1

−0.5

0

0.5

1

Figure 16: Spectrum of Fuel/Time Optimal Control

numerous applications, the range of uncertainty isknow and often the distribution of the uncertaintyis know. There is thus a desire to formulate opti-mization problems which includes the informationabout the range of uncertainty, in the design ofcontrollers. The technique expounded in this sec-tion is germane to the design of the input shapers,as well as for the design of saturating controllers.

For a mechanical system undergoing rest-to-restmaneuvers, the model can be represented as

My + C(p)y +K(p)y = Dr (47)

whereM is a positive definite mass matrix, and Kand C, the stiffness and damping matrices. K ispositive semi-definite when the model of the sys-tem includes rigid body modes and is positive def-inite otherwise. p is a vector of uncertain param-eters whose elements satisfy the constraints:

plbi ≤ pi ≤ pub

i (48)

where plbi and pub

i represent the lower and upperbounds on the parameters respectively. The goalhere is to design a saturating controller with theobjective of minimizing the maximum value of theresidual energy

minx

maxp

F

F =12yTMy+

12(y−yf )TK(y−yf )+

12(yr−yrf )2

(50)where x is a vector of parameters which definethe saturating controller and yf corresponds to the

final displacement states of the system. The aboveequation will be referred to as the pseudo-energyfunction since it is associated with a hypotheticalspring whose potential energy is zero when y =yf . The pseudo-energy function is evaluated at thefinal time, i.e., the end of the maneuver. The lastterm is added to guarantee that the cost functionis positive definite.

Minimax bang-bang controllers are designed forthe floating oscillator benchmark problem to illus-trate the proposed technique. The time-optimalcontrol of the benchmark problem is a 3-switchanti-symmetric bang-bang control profile. Thethin solid line in Figure 17 illustrates the vari-ation of the residual energy of the time-optimalcontrol to variations in the spring stiffness. A 3-switch minimax controllers is designed without theconstraint that the residual energy be zero at thenominal value of the spring stiffness. The thindashed line illustrates that the maximum mag-nitude of the residual energy over the uncertainrange (0.7 < k < 1.3) has been minimized, but atthe cost of non-zero residual energy for the nomi-nal model. Next, the 5-switch robust time-optimalcontrol profile is designed to force the slope of thesensitivity curve for the nominal system to go tozero. The think solid line illustrate the signifi-cant reduction of the residual energy over the en-tire uncertain range. However, this is achievedat a cost of increased maneuver time. Finally, a5-switch minimax controller is designed and thethick dashed line illustrates the improvement overthe robust 5-switch time-optimal controller. Theresulting sensitivity curve is similar to the curvefor extra-insensitivity input shapers [40].

3.4 Finite Jerk Time-Optimal ControlRecently, Muenchhof and Singh [58] and Lim etal. [53] have proposed a optimal control formula-tion which includes limits on the rate of change ofcontrol (Jerk). The control rate profiles are bang-bang or bang-off-bang as a function of the per-mitted jerk and the maneuver distance, for a rest-to-rest maneuver. The problem in [58], is posedas the design of time-delay filter which is parame-terized to generate a bang-bang or bang-off-bangprofile whose integral is the control input to theplant. Figure 18 illustrates the variation of theswitches and change of the structure of the con-trol profile as a function of permitted jerk for the

0.7 0.8 0.9 1 1.1 1.2 1.30

0.05

0.1

0.15

0.2

0.25

0.3

Spring Stiffness k

Squ

are

root

of R

esid

ual E

nerg

y

Time−Optimal Control Minimax 3−Switch Control Robust Time−Optimal ControlMinimax 5−Switch Control

Figure 17: Residual Energy Distribution

benchmark floating oscillator problem.

0.02 0.05 0.1 0.5 1 5 10 50 100 5000.1

1

10

50

Jerk J

Sw

itchi

ng T

imes

Ti

Figure 18: Switch Time Trajectories

It can be seen that the structure of the control pro-file changes significantly. The thick vertical linesindicate the value of jerk where switches collapseor are spawned. It is interesting to note that themaneuver time only increases marginally. as thepermitted jerk is decreased from ∞ to 2.

4 Nonlinear Systems

The technique presented for the design of con-troller which are robust to modeling uncertaintiesincluded location of multiple zeros of the time-delay filter at the estimated location of the polesof the system. This approach obviously cannot be

used for the design of robust control profiles fornonlinear systems. Liu and Singh [41] proposeda technique where the sensitivity state equationsare included in the problem formulation with theconstraint that the sensitivity states be forced tozero at the final time. For the nonlinear system

x = f(x, u, p), (51)

where p is the vector of uncertain parameters, thesensitivity state equations are

dx

dpi=

∂f

∂pi+

n∑j=1

∂f

∂xj

dxj

dpi. (52)

The control profile should in addition to satisfyingthe boundary conditions of the system states x,must also force

dx

dpi

∣∣∣∣tf

= 0, ∀pi. (53)

To illustrate this approach, consider the bench-mark problem with a nonlinear spring whosemodel is

m1y1 + k1(y1 − y2) + k2(y1 − y2)3 = u (54)m2y2 − k1(y1 − y2)− k2(y1 − y2)3 = 0. (55)

The sensitivity state equations are

m1dy1

dk1+ y12 + k1

dy12

dk1+ 3k2y

212

dy12

dk1= 0 (56)

m2dy2

dk1− y12 − k1

dy12

dk1− 3k2y

212

dy12

dk1= 0 (57)

m1dy1

dk2+ k1

dy12

dk2+ y3

12 + 3k2y212

dy12

dk2= 0 (58)

m2dy2

dk2− k1

dy12

dk2− y3

12 − 3k2y212

dy12

dk2= 0 (59)

The time-optimal control profile is designed to sat-isfy the boundary conditions

y1 = y2 = y1 = y2 =dy1

dk1=

dy1

dk1=

dy1

dk2=

dy1

dk2= 0

∣∣∣∣t=0

y1 = y2 = 1, y1 = y2 =dy1

dk1=

dy1

dk1=

dy1

dk2=

dy1

dk2= 0

∣∣∣∣t=tf

.

(60)Figure 19 illustrates the reduction in residual vi-bration of the desensitized time-optimal control,compared to the time-optimal control in the vicin-ity of the nominal values of the spring coefficients.The maneuver time of the two time-optimal con-trol profile is tf = 4.1514, and for the desensitizedcontrol profile is tf = 6.2439. It is clear the insen-sitivity to model parameters is achieved at a costof increased maneuver time.

0.6

0.8

1

1.2

1.4

0.6

0.8

1

1.2

1.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

k2

k1

Res

idua

l Ene

rgy

Time Optimal Control

Robust Time Optimal Control

Figure 19: Sensitivity Plot for the 3/9 switch Control

5 Applications

Given the simple approach and ease of implemen-tation of basic input shaping techniques, they havebeen used in a variety of applications. The widespread use is also attributable to the robustnessthat can be added to many of the techniques.One major area of success has been on cranes andcrane-like structures. Starr wrote an early pa-per that implemented a ZV-like shaping schemeon a crane [6]. Groups at Sandia and Oak RidgeNational labs have been especially active in thisarea [28], [20], [43]. Their approach has been sim-ilar to the input shaping described in Section 2;however, the shaping filter utilized has often beenan IIR filter instead of a FIR filter. Their tech-nique has also been utilized to control sloshingfluids [43]. Another approach, which has been im-plemented on some large gantry cranes, designedinput shapers to suppress vibration over the ex-pected operating ranges of the cranes [44]. Fig-ure 20 shows the reduction in residual vibrationas a function of the hoist cable length.

0

20

40

60

80

100

5 10 15 20 25 30

TheoreticalMeasured

Per

cent

age

Vib

rati

on

Cable length (ft)

Figure 20: Vibration Reduction vs. Cable Length

High tech manufacturing is perhaps the area withthe highest number of input shaping applications.Shaping was an important component of a controlsystem developed for a wafer stepper [48]. Multi-ple modes of a silicon-handling robot were elimi-nated with input shaping [31]. Accuracy of coordi-nate measuring machines has been improved withcommand shaping [54], [27], [39]. The throughputof a hard-disk-drive-head testing machine was sig-nificantly improved with shaping [46]. Figure 21shows the position response of the reading headsduring testing both before and after input shap-ing was implemented. The greatly reduced set-tling time allowed for much higher throughput.Command shaping has been combined with visionsensing and learning control on x-y-z gantry-typeautomation machinery [62]. Figure 22 shows thedecrease in tip vibration of the machine as thelearning controller adapts the input shaper to thesystem vibration during repeated motion.

Input ShapedUnshaped

-50

0

50

100

150

200

250

-100

-50

0

50

100

150

200

0 0.01 0.02 0.03 0.04 0.05 0.06

Uns

hape

d P

osit

ion,

µµµµin Shaped P

osition, µµ µµin

Time (sec)Figure 21: Head Response During Testing

Applications involving shaping the commands toon-off actuators or saturating actuators are fewerthan with real-time shaping, but there have stillbeen a number of successes. In fact, the cranecontrol scheme whose results are shown in Fig-ure 20 uses an on-off actuator switching algorithmto accomplish the vibration reduction. Recently,a technique for shaping the momentum dumpingof spacecraft was adopted as a baseline design forthe next generation space telescope [60].

References

[1] Smith, O. J. M., “Posicast Control of Damped Oscil-latory Systems”, Proc. of the IRE, 1957, pp 1249-1255.

[2] Calvert, J. F. and Gimpel, D. J., “Method and Ap-paratus for Control of System Output Response to SystemInput”, U.S. Patent #2,801,351, 1957.

-8-6-4-202468

0 2000 4000 6000 8000

Tip

Acc

eler

atio

n

Time (msec)

Figure 22: Adaptive Command Shaping

[3] Tallman, G. H., Smith, O. J. M., 1958, “AnalogStudy of Posicast Control”, IRE Transactions on AutomaticControl, Vol. 3, pp 14-21.

[4] Farrenkopf, R. L., 1979, “Optimal Open-Loop Ma-neuver Profiles for Flexible Spacecraft”, J. of Guidance, andControl, Vol. 2, No. 6, pp 491-498

[5] Swigert, C. J., “Shaped Torques Techniques”, J. ofGuidance and Control, Vol. 3, 1980, pp 460-467.

[6] Starr, G. P., “Swing-Free Transport of SuspendedObjects With a Path-Controlled Robot Manipulator”, J.of Dynamic Systems, Measurement and Control, vol. 107,pp. 97-100, 1985.

[7] Singh, G., Kabamba, P. T., McClamroch, N. H.,1989, “Planar, Time-Optimal, Rest-to-Rest Slewing Ma-neuvers of Flexible Spacecraft”, J. of Guidance, Control,and Dynamics, Vol. 12, No. 1, pp 71-81.

[8] Singer, N. C., and Seering, W. P., “Preshaping Com-mand Inputs to Reduce System Vibrations”, ASME J.of Dynamic Systems, Measurement and Control, Vol. 112,1990, pp 76-82.

[9] Junkins, J., L., Rahman, Z., Bang, H., 1990, “Near-Minimum Time Maneuvers of Flexible Vehicles: A Lia-punov Control Law Design Method”, Mechanics and Con-trol of Large Flexible Structures, Published by the AmericanInstitute of Aeronautics and Astronautics, Inc., Washing-ton.

[10] Bhat, S. P. and Miu, D. K., 1990, “Precise Point-to-Point Positioning Control of Flexible Structures”, J. ofDynamic Sys., Meas., and Control, Vol. 112(4), pp. 667-674

[11] Hablani, B. H., 1990, “Zero-Residual-Energy, Single-Axis Slew of Flexible Spacecraft with Damping, UsingThrusters: A Dynamic Approach”. Proc. of the 1990 AIAAGuidance, Navigation, and Control Conference,

[12] Singhose, William E., Singer, Neil C., Seering, War-ren P.,1990, “Shaping Inputs to Reduce Vibration: A Vec-tor Diagram Approach” Proc. of the 1990 IEEE Inter-national Conference of Robotics and Automation, Vol. 2,Cincinnati, Ohio, pp. 922-927.

[13] Hyde, J. M., Seering, W. P., 1991, “Multiple ModeVibration Suppression in Controlled Flexible Systems”MITSpace Engineering Research Center report, SERC #3-91

[14] Ben-Asher, J., Burns, J. A. and Cliff, E. M., 1992,“Time-Optimal Slewing of Flexible Spacecraft”, J. of Guid-ance, Control, and Dynamics, Vol. 15(2), pp. 360-367.

[15] Liu, Q. and Wie, B., 1992, “Robust Time-OptimalControl of Uncertain Flexible Spacecraft”, J. of Guidance,Control, and Dynamics, Vol. 15(3), pp. 597-604

[16] Murphy, B. R. and Watanabe, I., 1992 “Digital Shap-ing Filters for Reducing Machine Vibration”, IEEE Trans-actions on Robotics and Automation, Vol. 8(April), pp. 285-289.

[17] Singer, N. C. and Seering, W. P., “An Extension ofCommand Shaping Methods for Controlling Residual Vi-bration Using Frequency Sampling”, IEEE InternationalConference on Robotics and Automation, Nice, France,1992.

[18] Magee, D. and Book, W., “The Application ofInput Shaping to a System with Varying Parameters”,Japan/USA Symposium on Flexible Automation, 1992.

[19] Singer, N. C. and Seering, W. P., “An Extension ofCommand Shaping Methods for Controlling Residual Vi-bration Using Frequency Sampling”, IEEE InternationalConference on Robotics and Automation, Nice, France,1992.

[20] Noakes, M. W. and Jansen, J. F., “Generalized In-puts for Damped-Vibration Control of Suspended Pay-loads”, Robotics and Autonomous Systems, vol. 10, pp. 199-205, 1992.

[21] Wie, B. and Bernstein, D., “Benchmark Problemsfor Robust Control Design”, J. of Guidance, Control andDynamics, Vol. 15, No. 5, 1992, pp 1057-1058.

[22] Singh, T. and Vadali, S. R., 1993, “Input-ShapedControl of Three-Dimensional Maneuvers of Flexible Space-craft”, J. of Guidance, Control, and Dynamics, Vol. 16(6),pp. 1061-1068.

[23] Singh, T. and Vadali, S. R., 1993, “Robust Time-Delay Control”, ASME J. of Dynamic Systems, Measure-ment, and Control, Vol. 115, pp. 303-306.

[24] Singh, T., Heppler, G. R., 1993, “Shaped Input forMultimode System” ASME J. of Dynamic Systems, Mea-surement and Control, Vol. 115, 1993, pp 341-347.

[25] Khorrami, F., Jain, S., and Tzes, A., “Adaptive Non-linear Control and Input Preshaping for Flexible-Link Ma-nipulators”, American Control Conf., San Francisco, CA,1993.

[26] Tzes, A., and Yurkovich, S., “An Adaptive InputShaping Control Scheme for Vibration Suppression in Slew-ing Flexible Structures”, IEEE Transactions on ControlSystems Technology, Vol. 1, pp. 114-121, 1993.

[27] Seth, N., Rattan, K. and Brandstetter, R., “Vibra-tion Control of a Coordinate Measuring Machine”, IEEEConf. on Control Apps., Dayton, OH, 1993.

[28] Feddema, J. T., “Digital Filter Control of RemotelyOperated Flexible Robotic Structures”, American ControlConf., San Francisco, CA, 1993.

[29] Singh, T. and Vadali, S. R., “Robust Time-OptimalControl: Frequency Domain Approach”, AIAA J. of Guid-ance, Control and Dynamics, Vol. 17, No. 2, 1994, pp 346-353.

[30] Tuttle, T. D. and Seering, W. P., “A Zero-placement Technique for Designing Shaped Inputs to Sup-press Multiple-mode Vibration”, American Control Conf.,Baltimore, MD, 1994.

[31] Rappole, B. W., Singer, N. C., and Seering, W. P.,“Multiple-Mode Impulse Shaping Sequences for ReducingResidual Vibrations”, 23rd Biennial Mechanisms Confer-ence, Minneapolis, MN, 1994.

[32] Singh, T., Vadali, S. R., 1995, “Robust Time-DelayControl of Multimode Systems”, International J. of Control,Vol. 62, No. 6, pp 1319-1339.

[33] Vadali, S. R., Carter, M. T., Singh, T., and Ab-hyankar, N. S., 1995,“Near-Minimum-Time Maneuvers ofLarge Structures: Theory and Experiment”, J. of Guid-ance, Control and Dynamics, Vol. 18, No. 6, pp 1380-1385.

[34] Singh, T., 1995, “Fuel/Time Optimal Control of theBenchmark Problem”, J. of Guidance, Control, and Dy-namics, Vol. 18(6), pp. 1225-31.

[35] Singh, T., 1996, “Effect of Damping On the Structureof Time-Optimal Controllers”, J. of Guidance, Control andDynamics, Vol.19, No. 5, pp 1182-1184.

[36] Bodson, M., “Experimental Comparison of Two In-put Shaping Methods for the Control of Resonant Systems”,IFAC World Congress, San Francisco, CA, 1996.

[37] Pao. L. Y., “Minimum-Time Control Characteristicsof Flexible Structures”, J. Guidance, Control, and Dynam-ics, 19(1): 123-129, Jan.-Feb. 1996.

[38] Singhose, W. E., Seering, W. P. and Singer, N. C.,“Input Shaping for Vibration Reduction with Specified In-sensitivity to Modeling Errors”, Japan-USA Sym. on Flex-ible Automation, Boston, MA, 1996.

[39] Singhose, W., Singer, N., and Seering, W., “Improv-ing Repeatability of Coordinate Measuring Machines withShaped Command Signals”, Precision Engineering, Vol. 18,pp. 138-146, 1996.

[40] Singhose, W., Derezinski, S. and Singer, N., “Extra-Insensitive Input Shapers for Controlling Flexible Space-craft”, AIAA J. of Guidance, Control, and Dynamics, vol.19, pp. 385-91, 1996.

[41] Liu, S-W., and Singh, T., 1997, “Robust Time-Optimal Control of Nonlinear Structures with ParameterUncertainties”, ASME J. of Dynamic Systems, Measure-ment and Control, Vol. 119, No. 4, 1997, pp 743-748.

[42] Singhose, W., Banerjee, A. and Seering, W., 1997,“Slewing Flexible Spacecraft with Deflection-Limiting In-put Shaping”, J. of Guidance, Control, and Dynamics, Vol.20(2), pp. 291-298.

[43] Feddema, J., Dohrmann, C. , Parker, G., Robinett,R., Romero, V. and Schmitt, D., “Control for Slosh-FreeMotion of an Open Container”, IEEE Control Systems, vol.17, pp. 29-36, 1997.

[44] Singer, N., Singhose, W., and Kriikku, E., “An In-put Shaping Controller Enabling Cranes to Move WithoutSway”, ANS 7th Topical Meeting on Robotics and RemoteSystems, Augusta, GA, 1997.

[45] Singhose, W., Pao, L. Y., and Seering, W. P., “Slew-ing Multi-Mode Flexible Spacecraft Using Zero DerivativeRobustness Constraints”, J. of Guidance, Control and Dy-namics, Vol. 20, 1997, pp 204-206.

[46] Singhose, W., Singer, N., and Seering, W., “Time-Optimal Negative Input Shapers”, J. of Dynamic Systems,Measurement, and Control, vol. 119, pp. 198-205, 1997.

[47] W. E. Singhose, L. J. Porter, T. D. Tuttle, and N.C. Singer, “Vibration Reduction Using Multi-Hump InputShapers”, J. of Dynamic Systems, Measurement, and Con-trol, vol. 119, pp. 320-326, 1997.

[48] deRoover, D., Sperling, F. B. and Bosgra, O. H.,“Point-to-Point Control of a MIMO Servomechanism”,American Control Conference, Philadelphia, PA, 1998.

[49] Pao, L. Y. and Singhose, W. E., “Robust MinimumTime Control of Flexible Structures”, Automatica, 34(2):229-236, Feb. 1998.

[50] Hartmann, R., and Singh, T., “Fuel/Time OptimalControl of Flexible Structures: A Frequency Domain Ap-proach’, Journal of Vibration and Control, Sept., 1999, Vol.5, No. 5, pp 795-817.

[51] Singhose, W., Singh, T., Seering W., 1999, “On-OffControl with Specified Fuel Usage”, ASME J. of DynamicSystems, Measurement and Control, Vol. 121(2), pp 206-212.

[52] Pao, L. and Lau, M. A., “The Expected Residual Vi-bration of Traditional and Hybrid Input Shaping Designs”,J. Guid., Contr., & Dyn., vol. 22, pp. 162-165, 1999.

[53] Sungyung Lim, Homer D. Stevens, and Jonathan P.How “Input Shaping Design for Multi-Input Flexible Sys-tems”, ASME J. of Dynamic Systems, Measurement andControl, Vol. 121(3), pp 443-447.

[54] Jones, S. and Ulsoy, A. G., “An Approach to Con-trol Input Shaping with Application to Coordinate Measur-ing Machines”, J. of Dynamics, Measurement, and Control,Vol. 121, pp. 242-247, 1999.

[55] Tuttle, and Seering, W. P. “Creating Time-OptimalCommands with Practical Constraints”, J. Guid., Contr.,& Dyn., Vol. 22, No. 2, 1999, pp. 241-250.

[56] Kenison, M. and Singhose, W. ,“Concurrent Designof Input Shaping and Feedback Control for Insensitivity toParameter Variations”, Sixth Int. Workshop on AdvancedMotion Control, Nagoya, Japan, 2000.

[57] Hindle, T., Singh, T., 2001, “Robust MinimumPower/Jerk control of Maneuvering Structures”, J. of Guid-ance, Control and Dynamics, Vol. 24, No. 4, pp 816-826.

[58] Muenchhof, M., Singh, T., 2001, “Jerk Limited TimeOptimal Control of Structures”, Proc. of the 2001 ASMEInternational Mechanical Engineering Congress and Expo-sition

[59] Park, U. H., Lee, J. W., Lin, B. D. and Sung, Y.G., “Design and Sensitivity Analysis of an Input ShapingFilter in the z-Plane”, J. of Sound and Vibration, vol. 243,pp. 157-171, 2001.

[60] Banerjee, A., Pedreiro, N., and Singhose, W., “Vibra-tion Reduction for Flexible Spacecraft Following Momen-tum Dumping with/without Slewing”, AIAA J. of Guid-ance, Control, and Dynamics, Vol. 24, 2001.

[61] Lau, M. A. and Pao, L. Y., “Comparison of InputShaping and Time-Optimal Control of Flexible Structures”,Proc. American Control Conf., Arlington, VA, pp. 1485-1490, June 2001.

[62] Rhim, S. and W. J. Book, “Noise Effect on Time-domain Adaptive Command Shaping Methods for FlexibleManipulator Control”, IEEE Transactions of Control Sys-tems Technology, Vol. 9, No. 1, 2001, pp. 84 - 92.

[63] Kenison, M. and Singhose, W., “Concurrent Designof Input Shaping and Proportional Plus Derivative Feed-back Control”, Accepted to the ASME J. of Dynamic Sys-tems, Measurement, and Control.