Command-induced vibration analysis using input shaping principles

Command-induced vibration analysis using input shaping principles

Automatica 44 (2008) 2392–2397 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief pape...

2MB Sizes 0 Downloads 111 Views

Automatica 44 (2008) 2392–2397

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Command-induced vibration analysis using input shaping principlesI Khalid L. Sorensen, William E. Singhose ∗ Georgia Institute of Technology, Department of Mechanical Engineering, Atlanta, GA 30332, USA

article

info

Article history: Received 5 February 2007 Received in revised form 15 August 2007 Accepted 24 January 2008 Available online 20 May 2008 Keywords: Command shaping Input shaping Deconvolution Phase plane Vector diagram Vibration analysis

a b s t r a c t Input shaping is a well-established technique used for reducing the vibratory response of dynamic systems. Analytical tools are available for systems utilizing input shaping. These tools aid in performance analysis by providing intuitive and computationally simple methods for determining key system attributes, such as the residual vibration in response to a command. This paper describes methods whereby arbitrary reference commands may be interpreted as input-shaped commands. This capability allows input shaping analysis tools to be used on systems without input shapers. Experimental results obtained from an industrial 10-ton bridge crane validate the theoretical developments. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Reference commands used to drive flexible systems have a tremendous influence on performance. For example, consider the velocity command shown with the solid line in Fig. 1(a). This command was generated by a human operator while driving a 10ton industrial bridge crane. The dashed line in Fig. 1(a) shows that the command induced a large amount of transient and residual swing in the crane payload. Now consider the velocity command in Fig. 1(b). This command is similar in form to the command of Fig. 1(a), but has more complicated amplitude variations. However, in spite of its more complicated appearance, the command induces much less oscillation, as shown by the dashed line in Fig. 1(b). Since reference commands can produce such widely varying responses, it is important to understand the qualities of a command that influence induced vibration. This paper investigates the vibration-inducing qualities of reference commands by analyzing them from the perspective of a control technique called input shaping. Input shaping is a

(a) Command inducing noticeable residual oscillation.

(b) Command inducing minimal residual oscillation. Fig. 1. Payload swing response (dashed) to different velocity commands (solid).

I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Pedro Albertos under the direction of Editor Mituhiko Araki. ∗ Corresponding address: Georgia Institute of Technology, The George W. Woodruff School of Mechanical Engineering, 813 Ferst Street, 30332-0405, Atlanta, GA, USA. Tel.: +1 404 385 0668; fax: +1 404 894 9342. E-mail addresses: [email protected] (K.L. Sorensen), [email protected] (W.E. Singhose).

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.01.029

command filtering method used to reduce the oscillatory dynamics of flexible systems. It has been used to mitigate unwanted oscillation in cranes (Singer, Singhose, & Kriikku, 1997; Sorensen, Singhose, & Dickerson, 2007), coordinate measuring machines (Jones & Galip Ulsoy, 1999; Singhose, Singer, & Seering, 1996), satellites (Singhose, Porter, Tuttle, & Singer, 1997; Tuttle & Seering, 1996), micro-milling machines (Fortgang, Singhose, de Juanes

K.L. Sorensen, W.E. Singhose / Automatica 44 (2008) 2392–2397

Fig. 2. Correctly shaped command inducing zero residual oscillation.

Marquez, & Perez, 2005), and long-reach manipulators (Magee & Book, 1995). Input shaping is implemented by convolving a sequence of impulses, known as an input shaper, with a baseline command (Singer & Seering, 1990; Smith, 1957). The convolution product, instead of the original baseline command, is then issued to the plant. For baseline commands that reach a steady-state value, and for correctly designed input shapers, a linear system will exhibit zero residual oscillation in response to the modified command. This scenario is illustrated in Fig. 2. The original command (in this case a step) is modified by a correctly designed two-impulse input shaper. The response of G to the shaped command contains no residual oscillation. This procedure was used in real-time to create the velocity command in Fig. 1(b), which moved the industrial crane without significant payload swing. If an incorrectly designed input shaper is utilized during the convolution process, then the vibration-reducing properties of the shaped command are reduced. In this way, the influence that shaped commands have on residual oscillation depends fundamentally on the input shaper. A more definitive statement of this relationship is: The response of a linear plant to a shaped command is proportional to the response that would be induced by actuating the plant with the impulse sequence comprising the input shaper (Eloundou & Singhose, 2002; Singhose, Seering, & Singer, 1994). Therefore, a system’s oscillatory response to a shaped command can be determined by considering only the input shaper, not the shaped command that actually drives the system. A vector diagram approach has been developed for analyzing residual system vibration induced by input shapers (Singhose et al., 1994). This analytical tool provides a graphical, intuitive, and computationally simple method for predicting vibratory system response. The limitation that this tool can only be used with input-shaped systems serves as a motivation for the two primary objectives of this paper: (i) Interpret arbitrary commands as input-shaped commands. By so doing, one may: (ii) Determine the oscillatory effects of arbitrary commands on linear systems by using vector diagram principles. Eloundou et al. completed an initial investigation addressing these objectives (Eloundou & Singhose, 2002). Common “smooth” transitionary profiles were interpreted as shaped commands. However, this study was limited to S-curves, versines, and transition sine functions. This paper expands the previous work to include any function, and thereby demonstrates that arbitrary commands and inputshaped commands can be analyzed in a universal manner. To achieve the primary objectives, it will be shown that arbitrary signals can be deconvolved into both (1) a specified input shaper and baseline command, and (2) a specified baseline command and an input shaper. The former deconvolution is used to qualitatively describe the oscillatory effects of arbitrary signals. The latter deconvolution, in conjunction with vector diagram principles, is used to quantitatively describe the oscillatory effects of arbitrary signals.

2393

Section 2 provides additional information about input shaping and vector diagrams. Section 3 demonstrates how arbitrary commands can be resolved into input shapers and baseline commands. In Section 4, a case study is presented in which the oscillatory effects of rate-limited step commands on a harmonic oscillator are examined. Experimental results from an industrial bridge crane are used for validation. 2. Input shaping and vector diagrams Vibration suppression can be accomplished with a reference command that causes a system to cancel out its own motioninduced vibration. This was demonstrated by the experimental results in Fig. 1(b). Fig. 2 illustrated that this type of command can be generated by convolving a correctly designed input shaper with a baseline command. 2.1. Input shaping theory To understand how input shapers reduce oscillation, consider the impulse response of an undamped second-order harmonic oscillator: yo (t) =

ωn Ao sin (ωn (t − to )) · 1(t − to ),

(1)

where Ao is the strength of the impulse, to is the time the impulse is applied, ωn is the natural frequency of the system, and 1(t) is the Heaviside function. The response from a sequence of impulses is a superposition of the response given in (1). After application of the nth impulse, the steady-state response of the system can be described by: p yss (t) = ωn C 2 + S2 sin(ωn t − Ψ ), where   n n X X S C= Ai cos(ωn ti ), S = Ai sin(ωn ti ), Ψ = atan . (2) i=1

i=1

C

In order to reduce the magnitude of steady-state vibration, the amplitudes, Ai , and time locations, ti , of the impulse sequence can be determined by solving a set of constraint equations (Grosser & Singhose, 2000). If the amount of residual oscillation produced by the impulse sequence is set equal to zero, then an input shaper that satisfies the constraint equations is called a Zero Vibration (ZV) shaper (Singer & Seering, 1990; Smith, 1957). The steady-state amplitude of vibration induced by a shaped command is proportional to the oscillation caused by the impulse sequence (provided that the baseline command achieves a steadystate value) (Eloundou & Singhose, 2002). In this way, the amplitudes and time locations of the impulses comprising the input shaper determine the vibration-inducing properties of the shaped command. 2.2. Vector diagrams A vector diagram graphically represents an input shaper by using polar coordinates (r–θ space) in the phase plane (Singer, 1989; Singhose et al., 1994). Fig. 3 shows how a vector diagram is formed. The magnitude of each vector is created by setting ri equal to the ith amplitude, Ai , of an impulse sequence. The angle of each vector, θi , is equal to ωn ti , where ωn is the undamped frequency of the system to which the input shaper is being applied, and ti is the time of the ith impulse. When a vector diagram is created in this manner, it becomes a useful tool for analyzing the oscillation-inducing properties of an input shaper. This is because the resultant vector sum, R, is closely

K.L. Sorensen, W.E. Singhose / Automatica 44 (2008) 2392–2397

2394

Fig. 3. Input shaper (left); equivalent vector diagram (center); resultant vector (right).

related to the steady-state impulse response in (2). This can be easily seen from the analytical expression for R:   q Ry , where R = R2x + R2y 6 atan Rx

Rx =

n X

Ai cos(ωn ti ),

i=1

Ry =

n X

Ai sin(ωn ti ).

(3)

i=1

The magnitude of R is proportional to the steady-state amplitude in (2), which, in turn, is proportional to the oscillatory system response induced by a command shaped by the given impulse sequence. The angle of R is equal to the phase shift in (2). Therefore, R can be used as an indicator of anticipated system vibration in response to shaped commands. Another important attribute of the resultant vector is that it produces a steady-state oscillation in a system, equivalent to the steady-state oscillation caused by the original sequence of impulses. That is, the response of a second-order system to an impulse with the same magnitude and phase as R is identical to the response obtained from the original sequence of impulses after application of the last impulse.

In order to interpret a signal as a shaped signal, it will be shown that a class of commands can always be resolved into an input shaper and a baseline command. In this way, an arbitrary signal, x(t), can be expressed as the convolution of a baseline command, w(t), and an input shaper, IS(t). Section 3.1 shows how this decomposition may be made when the input shaper is specified. Section 3.2 shows how the decomposition may be made when the baseline command is specified. Special attention is given to the resulting input shaper when the specified command is a step input. By analyzing this shaper with vector diagram techniques, it is shown that a simple relationship exists for determining the oscillatory effects of arbitrary signals on linear plants. 3.1. Specified input shaper Any signal, x(t), may be created from the convolution of a baseline command, w(t), and an n-impulse input shaper having the form: IS(t) =

n X

Ai δ(t − ti ) :

0 ≤ ti < ti+1 , Ai 6= 0,

(6)

i=1

where δ(t) is the Dirac delta function. If w(t) is constrained to equal zero for t < 0, then the convolution relationship can be expressed as2 : Z ∞ w(τ)IS(t − τ)dτ. (7) x(t) = 0

Given the form of the input shaper in (6), the expression for x(t) may be simplified as: x(t) =

n X

Ai w(t − ti ).

(8)

i=1

2.3. Effects of damping When a harmonic oscillator has viscous damping, the vector diagram representation of vibration must be modified in two ways (Singhose et al., 1994). First, the damped natural frequency, ωd , rather than the undamped frequency, ωn , must be used to determine the angles of the impulse vectors. This corresponds to using:

θi = ωd ti .

(4)

Second, the amplitudes of the phase plane vectors must be scaled to account for the decay of the system response. Therefore, one should use a scaled version of the impulse amplitudes defined as: e Ai = Ai e−ζωn (tn −ti ) ,

(5)

where ζ is the damping ratio, and tn is the time of the last impulse.1 Using these modifications, the vector diagram technique for analyzing oscillatory properties of systems with input shaping may be applied generally to damped second-order systems. 3. Deconvolution analysis of arbitrary signals

Solving (8) for w(t − t1 ), and accounting for the t1 time shift, w(t) can be expressed in the form of a delay equation: " # n X 1 w(t) = x(t + t1 ) − Ai w(t + t1 − ti ) . (9) A1

i=2

The significance of (9) is that it verifies the existence of a baseline command for any shaper of the form described in (6), and the original signal, x(t). The utility of (9) is that it provides a framework for a forward-time-marching numerical solution of w(t), given x(t) and the specified input shaper. This equation states that at time t, the value of w(t) is proportional to the value of x(t + t1 ), and previous values of w(t). Therefore, because w(t) = 0 for t < 0, the signal x(t) and IS(t) completely determine w(t) for all t ≥ 0. Fig. 4 depicts an example deconvolution of a rate-limited step command, x(t) (solid line). The specified input shaper for this example is shown in the subplot. Using a forward-time-marching numerical solution with (9), w(t) was found (dotted line). The following summarizes the key points of this subsection: Assumption. w(t) = 0 for t < 0. Assumption. The specified input shaper has the form in (6).

This section discusses how arbitrary reference commands can be interpreted as input-shaped signals. When the commands are interpreted in this way, input-shaping principles can be used to determine the vibration-inducing properties of these commands.

1 An alternative scaling defined as e Ai = Ai eζωn ti may be used. The latter produces a resultant vector that causes equivalent oscillation if applied at time t = 0. The former produces a resultant vector that causes equivalent oscillation when applied at time t = tn .

Corollary. x(t) = 0 for t < t1 . Conclusion. A baseline command, w(t), exists that is completely determined by the signal x(t) and the specified input shaper.

2 A corollary to the assumed form of IS(t) and the constraint on w(t) is that x(t) will equal zero for t < t1 . This fact does not restrict x(t) because the choice of t1 is arbitrary and usually set equal to zero.

K.L. Sorensen, W.E. Singhose / Automatica 44 (2008) 2392–2397

2395

Fig. 5. Response of a harmonic oscillator to a rate-limited step command.

Thus,

|R| = e−ζωn tf |sX (s)|s=−ζωn −jωd .

Fig. 4. Deconvolution of x(t) into w(t) and a two-impulse shaper.

3.2. Specified baseline command The general convolution relationship between x(t), w(t), and

IS(t) in the Laplace domain is: IS(s) =

X (s) W (s)

.

(10)

If the baseline command, w(t), is specified to be a unit step, and x(t) is constrained to equal zero for t ≤ 0, then the inverse Laplace transform of (10) yields: IS(t) = x˙ (t).

(11)

If x(t) is further constrained such that x(t) is equal to a constant for t ≥ tf , then the time derivative of x(t) can be expressed as an infinite sequence of impulses: Z tf x˙ (t) = x˙ (τ)δ(t − τ)dτ. (12) 0

The input shaper, IS(t), is comprised of this sequence. Each impulse can be assigned an angle according to (4), and be scaled according to (5). Then, from (12) the following vector relationship can be established: scaling term

0

tf

z }| { x˙ (τ)δ(t − τ)dτ ejωd t .

Assumption. x(t) = 0 for t ≤ 0. Assumption. x(t) = C for t ≥ tf , C ∈ R. Corollary. x˙ (t) = 0 for t < 0. Corollary. x˙ (t) = 0 for t > tf . Conclusion. R represents the oscillatory response of a damped harmonic system to an arbitrary command, x(t). |R| is proportional to the amplitude of the response. The phase of R is equal to the phase shift of the response.

(13)

The vector sum of this quantity may be obtained by summing the individual impulse vectors. After simplifying the summation, the resultant vector, R, is: Z tf (14) R = e−ζωn tf x˙ (t)et(ωn ζ+ωd j) dt. 0

This resultant vector has the same physical meaning and significance as the resultant vector in Eq. (3). It represents the geometric summation of impulse vectors in the phase plane. The magnitude and angle of R represent the oscillatory response of a harmonic system actuated by a signal derived from the convolution of IS(t) with a unit step. A useful form of (14) can be derived by examining x˙ (t) in the Laplace domain: Z ∞ L[˙x(t)] = x˙ (t)e−st dt ≡ sX (s) − x(0). (15) −∞

By applying the constraints that x(t) is equal to zero for t ≤ 0, and x(t) is a constant for t ≥ tf , then evaluating (15) when s equals −ζωn − jωd , one obtains: Z tf x˙ (t)et(ωn ζ+ωd j) dt, (16) [sX (s)]s=−ζωn −jωd = 0

which is precisely the integral portion of (14). Therefore, R = e−ζωn tf [sX (s)]s=−ζωn −jωd .

The significance of (14), (17) and (18) is that the magnitude of residual oscillation induced into a linear system by an arbitrary signal (limited by the aforementioned constraints) may be ascertained from either the integral of the signal’s derivative, or the Laplace representation of the signal evaluated when s is set equal to the system poles. This result is related to an analysis conducted by Bhat and Miu in the time domain (Bhat & Miu, 1990), and later by Park et al. in the digital domain (Park, Lee, Lim, & Sung, 2001). They demonstrated that a system would exhibit zero residual vibration if the command signal had zeros at the system’s flexible poles. The following summarizes the key points of this subsection:

4. Case study: Oscillatory effects of a rate-limited step command

angle term

z }| { Z x˙ (t) = e−ζωn (tf −t)

(18)

(17)

This section demonstrates the utility and efficacy of the described deconvolution analysis through an elementary case study. The simplicity of the example is purposeful in order that the approach for analyzing more complex systems is clear. This example is concerned with the system illustrated in Fig. 5. A rate-limited step command, x(t), excites a damped harmonic oscillator, G, which responds with y(t). The rising slew rate of the command is defined by the slope, m. The deconvolution analysis technique can determine the oscillatory effects of the rate-limited command on the plant by interpreting the signal, x(t), as an input-shaped signal. Therefore, the rate-limited step command is considered here to be the product of an input shaper, IS(t), convolved with a baseline command, w(t): rate-limited step

z

}|

 { 1 = w(t) ∗ IS(t). x(t) = mt − (mt − 1) · 1 t − m

(19)

Section 4.1 addresses the problem when the baseline command is specified, but the input shaper is unknown. The resultant vector of the unknown shaper quantitatively describes the magnitude of residual oscillation. Section 4.2 interprets x(t) when the input shaper is specified, but the baseline command is unknown. The resulting baseline command qualitatively describes the residual oscillation. Section 4.3 compares experimentally measured residual oscillation with theoretical predictions.

K.L. Sorensen, W.E. Singhose / Automatica 44 (2008) 2392–2397

2396

Fig. 6. Resultant vector amplitudes for a rate-limited step command.

4.1. Quantitative oscillation description If a signal is deconvolved into a unit step and an unknown input shaper, then, according to (14), the resultant vector of the unknown shaper can be determined from an integration of x˙ (t) multiplied by a scaling factor. Since this approach is easily implemented numerically, it is suitable for use with complicated forms of x(t). For the simple signal of this example, (18) will be used. This equation determines the resultant vector of the unknown input shaper from the Laplace transform of x(t): i mh (20) X (s) = 2 1 − e−s/m . s

The magnitude of the resultant vector is obtained by substituting (20) into (18) and simplifying to obtain:

|R| =

 1   −2ζωn −ζωn ωd 2 1 + e m − 2e m cos . ωn m m

(21)

In Fig. 6, the magnitude of the resultant vector is plotted for several different damping ratios as a function of the slew-rate limit, m. The horizontal axis has been normalized by a value of mc = ωn /2π. The value ωn /2π is significant; it is equal to the natural frequency (in Hz) of the oscillator, G. A large value of m is indicative of a fast rise time for the command, x(t); as m approaches infinity, x(t) approaches a step command. Since the amplitude of residual oscillation varies directly with the magnitude of R, Fig. 6 provides a concise quantitative description of the oscillatory effects of the rate-limited command on the harmonic oscillator. As the command becomes more aggressive, the magnitude of R asymptotically approaches 1. Therefore, a value of 1 on the vertical axis indicates that the system will respond to x(t) with the same oscillation amplitude induced by a unit step command. Similarly, a value of 0.25 on the vertical axis indicates that the system will respond to x(t) with a quarter of the oscillation amplitude induced by a unit step. When the R-value is zero, x(t) causes no residual oscillation. Given this physical interpretation, an R-value plot becomes a useful tool for concisely describing the oscillatory effects of arbitrary commands on a harmonic oscillator. 4.2. Qualitative oscillation description An alternative to using the R-value analysis for describing residual oscillation is to interpret an arbitrary signal as the convolution of a specified input shaper and an unknown baseline command. Then, the baseline command can be obtained by using (9). The utility of this approach is that if the specified input shaper is correctly selected, then the behavior of the resulting baseline command is meaningful. It provides a qualitative description of oscillation insofar as it resembles the response of a plant to the arbitrary command. If the baseline command is oscillatory, then the system response to the arbitrary command will also be

Fig. 7. Baseline commands deconvolved from a rate-limited step and a ZV input shaper (solid). System response to rate-limited step commands (dotted).

oscillatory. Conversely, if the baseline command quickly obtains a steady-state value, so will the system response. This result is related to a fundamental property of input shaping. When a correctly designed shaper is convolved with a baseline command, the resulting shaped command produces zero residual oscillation in the plant after the baseline command has reached a steady-state value. This was previously illustrated in Fig. 2. G exhibited zero oscillation after the baseline command reached a steady-state value. If the baseline command was periodic, instead of obtaining a steady-state value, then the response of G would also oscillate periodically. To demonstrate the baseline command analysis, consider again the example of the rate-limited step. This command can be deconvolved into a baseline command and a two-impulse ZV shaper. The amplitudes and time locations of the shaper impulses are welldocumented functions of the natural frequency and damping ratio of G (Singer & Seering, 1990; Smith, 1957). Several of the baseline commands obtained from (9) are plotted with the solid lines in Fig. 7 for different values of m, when ζ was equal to 0.1. When convolved with the ZV shaper, these baseline commands exactly reproduce the original rate-limited commands. The response of G to the rate-limited step commands is shown with the dotted lines in Fig. 7. Notice that the response signals closely resemble the baseline command signals in both amplitude and phase. Therefore, the utility of the baseline analysis is that it yields a baseline command closely resembling the response of the given system to the given arbitrary command. If the analysis yields a baseline command that quickly obtains a steady-state value, then the plant will exhibit little residual oscillation. Conversely, oscillatory baseline commands indicate an oscillatory plant response. 4.3. Experimental verification Fig. 8 shows a 10-ton bridge crane. A linear model of the relationship between the velocity of the overhead trolley and the velocity of the suspended payload is (Sorensen et al., 2007): Vout Vin

=

2ζωn s + ω2n

s2 + 2ζωn s + ω2n

.

(22)

K.L. Sorensen, W.E. Singhose / Automatica 44 (2008) 2392–2397

2397

Acknowledgements This project would not have been possible without the generous support of CAMotion Inc. and Siemens Energy & Automation. References

Fig. 8. 10-ton bridge crane.

Fig. 9. Experimental and theoretical resultant vector amplitudes for a rate-limited step command.

The model reveals that the system behaves like a damped harmonic oscillator. The damping ratio is approximately 0.012 and the natural frequency is: q ωn = g/L, (23) where g is the acceleration due to gravity, and L is the length of the cable supporting the load. A series of rate-limited step commands (in velocity) were issued to the crane. The slew-rate limit of the commands varied from m = 0.7mc to m = 2.8mc . For each command, the residual oscillation amplitude of the suspended payload was measured. Each measured amplitude was divided by the oscillation amplitude occurring when a command corresponding to m = 10mc was issued to the crane. The resulting normalized amplitudes are plotted with the solid circles in Fig. 9. The resultant vector plots from Fig. 6 are superposed over the experimental results. This provides a comparison between the predicted and measured oscillation amplitudes. Given the low damping ratio of the crane, the measured amplitudes align closely with the predictions obtained using the R-value analysis technique for the undamped case. 5. Conclusion It is possible to deconvolve an arbitrary signal into both 1) a specified input shaper and baseline command, and 2) a specified baseline command and an input shaper. The utility of the deconvolution is that arbitrary signals can be analyzed from the perspective of input-shaped functions. This ability allows analytical tools, such as the vector diagram, to be utilized with many types of signals. By so doing, simple equations can be used to calculate the oscillatory effects of an arbitrary command on a harmonic oscillator. The efficacy of the deconvolution analysis technique was demonstrated by a case study in which the oscillatory response of rate-limited step commands on a harmonic oscillator was predicted. Experiments on a 10-ton industrial bridge crane confirmed the theoretical results.

Bhat, Sudarshan P., & Miu, Denny K. (1990). Precise point-to-point positioning control of flexible structures. Journal of Dynamic Systems, Measurement, and Control, 112(4), 667–674. Eloundou, Raynald, & Singhose, William (2002). Interpretation of smooth reference commands as input-shaped functions. In American control conf. (pp. 4948–4953). Fortgang, Joel, Singhose, William, de Juanes Marquez, Juan, & Perez, Jesus (2005). Command shaping for micro-mills and cnc controllers. In American control conf. (pp. 4531–4536). Grosser, Karen, & Singhose, William (2000). Command generation for reducing perceived lag in flexible telerobotic arms. JSME International Journal, 43(3), 755–761. Jones, Stephen, & Galip Ulsoy, A. (1999). An approach to control input shaping with application to coordinate measuring machines. Journal of Dynamic Systems, Measurement, and Control, 121(June), 242–247. Magee, David P., & Book, Wayne J. 1995. Filtering micro-manipulator wrist commands to prevent flexible base motion. In American control conf. (pp. 924–928). Park, U. H., Lee, J. W., Lim, B. D., & Sung, Y. G. (2001). Design and sensitivity analysis of an input shaping filter in the z-plane. Journal of Sound and Vibration, 243(1), 157–171. Singer, Neil, Singhose, William, & Kriikku, Eric (1997). An input shaping controller enabling cranes to move without sway. In ANS 7th topical meeting on robotics and remote systems. Singer, Neil C., & Seering, Warren P. (1990). Preshaping command inputs to reduce system vibration. Journal of Dynamic Systems, Measurement, and Control, 112(March), 76–82. Singer, Neil Cooper (1989). Redisual vibration reduction in computer controlled machines. Ph.D. thesis. Massachusetts Institute of Technology. Singhose, William, Seering, Warren, & Singer, Neil (1994). Residual vibration reduction using vector diagrams to generate shaped inputs. Journal of Mechanical Design, 116(June), 654–659. Singhose, William, Singer, Neil, & Seering, Warren (1996). Improving repeatability of coordinate measuring machines with shaped command signals. Precision Engineering, 18(April), 138–146. Singhose, William E., Porter, Lisa J., Tuttle, Timothy D., & Singer, Neil C. (1997). Vibration reduction using multi-hump input shapers. Journal of Dynamic Systems, Measurement, and Control, 119(June), 320–326. Smith, Otto J. M. (1957). Posicast control of damped oscillatory systems. Proceedings of the IRE, 45(September), 1249–1255. Sorensen, Khalid L., Singhose, William E., & Dickerson, Stephen (2007). A controller enabling precise positioning and sway reduction in bridge and gantry cranes. Control Engineering Practice, 15(7), 825–837. Tuttle, T. D., & Seering, W. P. (1996). Vibration reduction in flexible space structures using input shaping on mace: Mission results. In IFAC world congress.

Khalid L. Sorensen received the B.S. degree in mechanical engineering in 2002 from Walla Walla College, Washington, USA. In 2005, Khalid received his M.S. in mechanical engineering from the Georgia Institute of Technology, USA, where he is currently pursuing a Ph.D. He is a former NSF STEP Fellow, USAA National Collegiate Engineer, ARCS fellowship recipient, and current member of the ASME.

William E. Singhose received a Ph.D. from the Massachusetts Institute of Technology in June 1997. He then joined the faculty of the Woodruff School of Mechanical Engineering at the Georgia Institute of Technology. Dr. Singhose worked in industry before getting his Ph.D. He developed and installed control systems on industrial machines such as silicon-handling robots, coordinate measuring machines, and high precision air bearing positioning stages. In 2005 and 2006, Dr. Singhose was a visiting professor at the Tokyo Institute of Technology and worked in the Center of Excellence in Robotics. His research interests are dynamics, controls, active seats, and air traffic flow management.