robust zero placement for motion control of lightly damped systems

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 9, SEPTEMBER 2013 3857

Robust Zero Placement for Motion Control ofLightly Damped Systems

Chang-Wan Ha, Student Member, IEEE, Keun-Ho Rew, Member, IEEE, and Kyung-Soo Kim, Member, IEEE

Abstract—A unified tuning rule for the motion profiles is newlyproposed to reduce the residual vibration. For several motion pro-files, including trapezoidal velocity profile, S-curve, and asymmet-ric S-curve, the characteristics and tuning rules are analyticallyformulated using the time- and Laplace-domain approaches. Byadjusting the tuning parameters to place the zeros of the motionprofile on a vibratory pole of the system, the residual vibrationcan be minimized. In particular, the placement of multiple zerossignificantly improves the robust performance against modelingerrors, owing to robust pole-zero cancellation. By experiments, theefficacy of the proposed approach is validated.

Index Terms—Asymmetric S-curve (AS-curve), asymmetric-ity, motion profiles, residual vibration, robust zero placement,S-curve, smoothness, trapezoidal velocity profile.

I. INTRODUCTION

R EDUCING residual vibration for fast and accurate move-ment has been one of the major issues in motion control

engineering for the last several decades. Ranging from con-sumer electronic devices to manufacturing equipment, motioncontrol techniques are widely adopted for enhancing the perfor-mance and the productivity. For example, hard disk drives relyon the motion control profile for fast operation [1]. In addition,semiconductor manufacturing equipment requires pick-and-place motions having high speed and accuracy for enhancedproductivity and quality.

In view of the practical implementation, the trapezoidalvelocity profile is one of the simplest motion profiles. However,compared with S-curve, it may induce more residual vibration,which has an adverse effect on the settling time [2], [3].

S-curve has been widely applied to motion control field (see,e.g., [4]–[8]) and implemented using many methods, such asdigital finite-impulse-response filter [9], quintic-spline differ-ential analyzer [10], buttered digital differential analyzer [11],and optimal control approach [12]. The major challenge with itis to reduce the residual vibration in high-speed operations. Totackle this, important progresses have been made in literature.

Manuscript received September 26, 2011; revised April 8, 2012; acceptedJune 18, 2012. Date of publication July 6, 2012; date of current version May 2,2013. This work was supported in part by the Ministry of Knowledge Economyand Korea Institute for Advancement of Technology under the Human ResourceTraining Project for Strategic Technology, and by the Ministry of Education andScience Technology under BK21 Program, Republic of Korea.

C.-W. Ha and K.-S. Kim are with the Department of Mechanical Engineer-ing, Korea Advanced Institute of Science and Technology, Daejeon 305-701,Korea (e-mail: [email protected]; [email protected]).

K.-H. Rew is with the School of Mechanical Engineering, Depart-ment of Robotics Engineering, Hoseo University, Asan 336-795, Korea(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2012.2206334

In particular, an optimized S-curve was proposed based on afrequency analysis of the forcing function input [13], [14]. Inaddition, a nonlinear filter satisfying the velocity and acceler-ation constraints was proposed for smooth motion [15]–[17].Recently, the iterative learning control theory was adopted forgenerating a position trajectory with minimized jerk [6], [18].

To enhance the performance of S-curve, other speciallydesigned motion profiles were suggested [19]–[21]. One ofthe most practical and promising approaches is introducingasymmetricity between acceleration and deceleration periodsfor effectively adjusting motion profile [22]. Owing to theadditional degree of freedom, it is possible to generate afast-starting and slow-arriving motion (or slow-starting andfast-arriving motion). A complete closed-form solution toasymmetric S-curve (the so-called AS-curve) was derived andapplied to many applications [23]–[25]. It turned out that, byselecting appropriate parameters of AS-curve, the settling time,as well as the residual vibration, can be remarkably reduced[26], while the tuning was heuristically conducted by a trial-and-error method. More recently, in [27], a simple but analyticmethod has been proposed to tune the design parameters of theAS-curve, based on the pole-zero cancellation. After convertingthe AS-curve to a Laplace-transformed signal, the zeros of itcan be placed on the vibratory pole of the system in the absenceof modeling error. Unfortunately, however, the approach in [27]suffers from the lack of robustness against the modeling error,in particular, such as the natural frequency variation.

To tackle the robustness issue, this paper aims at newlypresenting a robust tuning rule against modeling errors. Theprinciple of robust pole-zero cancellation is analytically illus-trated, and the physical characteristics are discussed. Moreover,it will be shown that the proposed tuning rule can be applied tothe S-curve and the trapezoidal velocity profile, as well as theAS-curve.

In Section II, motion profiles introduced in literature (see[23]–[26]) are revisited. Then, in Section III, the characteristicsof the residual vibration are discussed. In Sections IV–VI,a unified tuning rule is derived and verified by experiments.Finally, the conclusion follows in Section VII.

II. REVIEW OF THE MOTION PROFILES: TRAPEZOIDAL

VELOCITY PROFILE, S-CURVE, AND AS-CURVE

Recently, AS-curve, which is a superset of trapezoidal veloc-ity profile and S-curve, has been introduced in a closed-formsolution, and its characteristics have been widely investigated(see [23]–[26]). AS-curve has two tuning parameters foradjusting the motion profile: smoothness and asymmetricity.

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3858 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 9, SEPTEMBER 2013

Fig. 1. Notations of AS-curve [26].

Fig. 2. Effects of the tuning parameters of AS-curve: (a) Smoothness and(b) asymmetricity [26].

Notations of AS-curve and the effects of the two tuning param-eters are shown in Figs. 1 and 2, respectively.

The smoothness β ∈ (0, 1] determines the magnitude of theapplied jerk under the actuator limitations such as the maxi-mum velocity Vmax and the maximum acceleration Amax. Theapplied jerk is defined as

J =Amax

βτm(1)

where

τm :=Vmax

Amax. (2)

As shown illustrated in Fig. 2(a), if the smoothness goes to zeroβ → 0 (i.e., the jerk goes to infinity), the shape of the velocityprofile approaches the trapezoidal shape. On the contrary, asthe smoothness goes to one β → 1, the shape of velocity profileapproaches the bell shape.

Fig. 3. Modeling of a lightly damped system.

The other tuning parameter γ may introduce the asymmetric-ity between acceleration and deceleration periods. When γ = 1,AS-curve turns out to be the S-curve. However, when theasymmetricity is not equal to one, the fast-starting and slow-arriving (or the slow-starting and fast-arriving) movements canbe obtained. This provides the flexibility in adjusting the motionprofile, as shown in Fig. 2(b).

Based on [26], given a target distance δtarget, if it holds that

δtarget ≥ δl :=(1 + β)(1 + γ)τmVmax

2(3)

the profile is classified as a long distance case. In this case, onemay have the motion parameters as follows:⎧⎨

⎩tj = βτmta = (1− β)τmtv =

(δtarget−δl)Vmax

(4)

where tj , ta, and tv are the periods of constant jerk, constantacceleration, and constant velocity, respectively. Interestingly,there is a generic constraint between tj and ta such that

tj + ta = τm (5)

which implies that β plays a role of distributing the periods ofconstant jerk and acceleration to achieve the maximum velocityduring the motion.

In this paper, only the long distance case will be discussedfor brevity. In addition, let us assume the following.

A1) Constant velocity period tv is long enough.A2) The damping ratio ζ is negligible.Assumption A1) implies that only the vibration induced at

the deceleration period will be considered.

III. CHARACTERISTICS OF RESIDUAL VIBRATION

The lightly damped system may be simplified as shown inFig. 3. The dynamic relationship between the base and thesecond mass is described by the transfer function

X2

X1=

−s2

s2 + ω2n

(6)

where X1, X2, and ωn are the absolute position of the base, therelative position of the second mass with respect to the base,and the natural frequency of the lightly damped system (ωn =√k/m), respectively.By controlling the base, suppose that x1(t) exactly follows

the motion profile shown in Fig. 1. Then, from the accelerationprofile for t ≥ t4, it may be shown that

s2X1(s) = − J

γ2

(1− e−βγτms)(1− e−γτms)

s2. (7)

HA et al.: ROBUST ZERO PLACEMENT FOR MOTION CONTROL OF LIGHTLY DAMPED SYSTEMS 3859

Fig. 4. (Top) Acceleration profile of the base movement and (bottom) theinduced vibration of the second mass during deceleration period.

Hence, the relative motion of the second mass is given by thefollowing, from (6):

s2X2(s) =J

γ2

(1− e−βγτms)(1− e−γτms)

(s2 + ω2n)

. (8)

Using the inverse Laplace transform, it is straightforward tohave

x2(t)=J

γ2ωn[sin(ωnt)− sin (ωn(t− βγτm))U(t− βγτm)

− sin (ωn(t− γτm))U(t− γτm)

+ sin (ωn (t− (1 + β)γτm))

× U (t− (1 + β)γτm)] (9)

where U(t) is a unit step function.This shows that the relative acceleration of the second mass

in the deceleration period can be expressed with four sine waveswith the same oscillating frequency but different phases, asshown in Fig. 4. The amplitude of the sine waves inducedby an abrupt change of the applied jerk is determined by themagnitude of the applied jerk and the natural frequency of thelightly damped system.

Our concern is to investigate the residual vibration of thesecond mass after the motion. To this end, let us define a timeshift as

t∗ := t− (1 + β)γτm. (10)

Then, when t∗ ≥ 0 (i.e., t ≥ t7), we have

x2(t∗) =

J

γ2ωn[sin (ωnt

∗ + (1 + β)γωnτm)

− sin(ωnt∗ + γωnτm)

− sin(ωnt∗ + βγωnτm) + sin(ωnt

∗)]

= − 4J

γ2ωnsin

(βγωnτm

2

)sin

(γωnτm2

)

× sin

(ωnt

∗ +(1 + β)γωnτm

2

). (11)

Fig. 5. Contour maps of the dimensionless residual vibration amplitude in theγ−β plane with fnτm = 1.587.

Hence, a dimensionless residual vibration amplitude, definedas the ratio of the amplitude of the residual vibration to themaximum acceleration, is expressed by the following, fromAmax = Jβτm and ωn = 2πfn:

a∗ :=

∣∣∣∣ x2(t∗)

Amax

∣∣∣∣=

∣∣∣∣ 2

πβγ2fnτmsin(πβγfnτm) sin(πγfnτm)

∣∣∣∣ . (12)

Fig. 5 shows the contour maps of the dimensionless residualvibration amplitude drawn in the γ−β plane with fnτm =1.587 as example. The brightness of the color is proportionalto the amplitude of the residual vibration (i.e., darker colordenotes less residual vibration). Note that there exist multiplechasms where the residual vibration becomes much smallerthan the neighborhood. That is, adjusting the motion profile tobe in a chasm, the residual vibration can be minimized. Theconditions for chasm and its characteristics will be discussed inSection IV.

IV. FREE-VIBRATION CONDITIONS

The free-vibration conditions can be derived from (12) asreported in [27]. In particular, when either of the sine termsgoes to zero, the residual vibration becomes zero. Hence, thefree-vibration conditions are as follows:

C1. βγfnτ = k, k = 1, 2, 3, . . .

C2. γfnτm =m, m = 1, 2, 3, . . . .

A. Physical Meaning of the Free-Vibration Conditions

1) C1: As shown in Fig. 6, when the time period betweent4 and t5 (or t6 and t7), i.e., βγτm, is the integer multiplesof the period of oscillation (T = 1/fn), the residual vibrationis canceled out after the end of the base movement due to thedestructive interference between four sine waves. The first sinewave is canceled out by the second one. Similarly, the third sine


Fig. 6. (Top) Acceleration profile of the base and (bottom) response of thesecond mass when free-vibration condition 1 is satisfied: βγτm = T .

Fig. 7. (Top) Acceleration profile of the base and (bottom) response of thesecond mass when free-vibration condition 2 is satisfied: γτm = T .

wave is canceled out by the fourth one. k implies the numberof oscillations within the time period between t4 and t5 (or t6and t7). The maximum vibration during deceleration period isfixed at J/(γ2ωn), while residual vibration is eliminated afterthe end of the motion.

2) C2: As shown in Fig. 7, when the time period betweent4 and t6 (or t5 and t7), i.e., γτm, is the integer multiples of theperiod of oscillation, the residual vibration is canceled out afterthe end of the motion due to the destructive interference. Thefirst and second sine waves are canceled out by the third andfourth ones, respectively. m implies the number of oscillationswithin the time period between t4 and t6 (or t5 and t7).The maximum vibration during deceleration period is boundedwithin 2J/(γ2ωn), while the residual vibration is eliminatedafter the end of the motion.

B. Pole-Zero Cancellation

As a matter of fact, the free-vibration conditions C1 and C2are to have the pole-zero cancellations in (8). Observe that the

Fig. 8. Pole-zero cancellations when (a) C1 and (b) C2 are satisfied with k =m = 1, respectively.

poles of the relative vibration are

sp = ±jωn = ±j2πfn. (13)

To find the zeros in (8), one may have 1− e−βγτms = 0 at

sz1 = ±j2πk

βγτm, k = 0, 1, 2, . . . . (14)

Also, the term 1− e−γτms gives the other set of zeros asfollows:

sz2 = ±j2πm

γτm, m = 0, 1, 2, . . . . (15)

Therefore, in the case of AS-curve, there are one pair ofpoles and two sets of zeros, determined by the characteristicsof the system and the designed motion control profile, respec-tively. From (13) and (14), the nontrivial pole-zero cancellationhappens under C1. Also, from (13) and (15), C2 guaranteesthe other pole-zero cancellation. Fig. 8 shows the pole-zerocancellation when C1 and C2 are satisfied with k = m = 1. Infact, the system poles are canceled by the zeros of the motionprofile, which effectively eliminates the vibratory mode of thesystem.

C. Free-Vibration Conditions in the Special Cases

1) S-Curve: When γ = 1, S-curve is the special case ofAS-curve [26]. The free-vibration conditions for S-curve areobtained by substituting γ = 1 into C1 and C2. Then, we have

βfnτm = k (16)

fnτm =m. (17)

Similar to the results of AS-curve, there are two free-vibrationconditions. However, contrary to the first free-vibration condi-tion (16), the second one (17) may not be met with ease sincefn and τm are not the free design parameters.

2) Trapezoidal Velocity Profile: When the acceleration anddeceleration periods are symmetric and the magnitude of theapplied jerk is infinite, the motion profile turns out to be atrapezoidal velocity profile [26]. The free-vibration conditionsfor trapezoidal velocity profile are obtained by substitutingβ = 0 and γ = 1 into C1 and C2. Then, we have

fnτm = m. (18)


Fig. 9. Sensitivity curves of the dimensionless residual vibration amplitudewith respect to the variation of the natural frequency with fnτm = 1.587.

Contrary to S-curve and AS-curve, the trapezoidal velocityprofile has only one free-vibration condition because the free-vibration condition C1 is meaningless when β = 0. Similar to(17), the free-vibration condition for trapezoidal velocity profile(18) may not be met with ease.

In summary, the trapezoidal velocity profile has only afree-vibration condition determined by the characteristics ofthe system and the actuator specification. Contrary to trape-zoidal velocity profile, S-curve and AS-curve have two free-vibration conditions. In particular, AS-curve provides moreflexibility in adjusting the motion profile to minimize theresidual vibration than the others due to the additional tuningparameters.

V. ROBUSTNESS AGAINST MODELING ERRORS

The free-vibration conditions C1 and C2 depend on thesystem parameter (i.e., fn). If there is a modeling error which isinevitable in practice, then the free-vibration conditions will notresult in zero vibration. Therefore, the motion control designershave to consider not only minimizing the residual vibration butalso the robustness against modeling errors.

In general, slower movement (i.e., with large values of β andγ) induces smaller residual vibration. As shown in Fig. 5, asthe value of β or γ increases, the region with small residualvibration gets wider. In other words, as β and γ increase, theresidual vibration becomes more robust against the variation ofthe system parameter. However, it is undesirable to increase βand γ much since this leads to slow movement.

The robust conditions against the modeling errors with fastmovement are investigated. The conditions satisfying both free-vibration conditions C1 and C2 are defined as a crater. Dif-ferent from the crater, the conditions satisfying only one ofthe free-vibration conditions are defined as a chasm. As shownin Fig. 5, compared to a chasm, a crater has wider range ofsmall residual vibration. This can be visualized by plotting asensitivity curve to show the dimensionless residual vibrationwith respect to the natural frequency variation. Sensitivitycurves for a crater (i.e., β = 0.5 and γ = 1.26) and a chasm(i.e., β = 0.75 and γ = 1.26) are shown in Fig. 9. Note that,

TABLE ICOMPARISON OF INSENSITIVITIES UNDER fnτm = 1.587

as the actual natural frequency deviates from the modeled one,the amount of the residual vibration increases. In particular, it isnoted that the motion profile tuned at a crater keeps the residualvibration (much) lower than at a chasm when the modelingerror is less than a certain range (i.e., 16% in this case).Considering that the modeling error in practice may not be solarge, tuning the motion profile to be at a crater is advantageousfor enhancing the robustness. For rigorous comparison, let usdefine a measure for robustness—insensitivity. The insensitivityis defined as the range of normalized natural frequency in whichthe dimensionless residual vibration (i.e., a∗) is less than atolerance of 0.05 (i.e., 5%). From Fig. 9, it is easy to have theinsensitivities as summarized in Table I. Observe that the cratercondition produces the AS-curve three times more robust thanthe chasm does.

This phenomenon can be analytically demonstrated by therobust pole-zero cancellation. Contrary to a chasm having azero, a crater condition places double zeros at the pole of thesystem. Thus, at a crater, it holds that

s2X1(s)∣∣s=sp

=0

d

ds

{s2X1(s)

} ∣∣∣s=sp

=0. (19)

In fact, (19) implies that s2X1(s) is zero not only at s = sp butalso at approximately zero around s = sp. That is, observe that

s2X1(s+Δs)∣∣s=sp

∼=s2X1(s)∣∣s=sp

+d

ds

{s2X1(s)

}∣∣∣s=sp

·Δs.

(20)

Hence, (19) guarantees that s2X1(s+Δs)|s=sp∼= 0 for small

Δs (i.e., around s = sp), which means that the pole is robustlycanceled out by the zeros even though there exists a modelingerror to some extent.

In the case of trapezoidal velocity profile having a set ofzeros, it is impossible to achieve the crater condition. However,the S-curve and the AS-cure have two sets of zeros which mayallow the crater condition. In particular, in the case of the AS-curve, the crater can be met with ease by adjusting two designparameters (i.e., β and γ). This is the reason why the AS-curveis advantageous over the other profiles in terms of robustness.

The algorithm to design the AS-curve to be in a cratercondition is presented in the following.

Algorithm: Suppose that Vmax and Amax are fixed by theactuator specification and the nominal system parameters suchas fn and ζ are known

S1. Based on Vmax and Amax, τm is calculated from (2).S2. For fast movement, choose a γ from C2 with the smallest

positive integer value of m satisfying γ ≥ 1.S3. Based on C1, β is selected as 1/m with k = 1.


Fig. 10. Experimental setup.

TABLE IIPARAMETERS FOR EXPERIMENTAL SETUP

VI. EXPERIMENTS

In this section, the proposed tuning method is validatedthrough experiments with an XY stage with a flexible structure.In practice, many of the automated machineries are built on (orwith) XY stages. For examples, XY stages are widely adoptedin industrial applications such as a crane, an inspection robot,and chip assembly equipment [26]. For achieving high bondingaccuracy with highly integrated electronic components, theresidual vibration is a challenge to chip assembly equipmentwith flexible tool tip [28]. Moreover, in the case of fine-pitchassembly processes, the residual vibration of a tool tip causedby the stage motion may induce the misalignment of specimensand low working speed.

To verify the effectiveness of the proposed robust tuning ruleof the motion profile, an experimental system with an XY stageis adopted as shown in Fig. 10. To realize the flexible mode in atool tip or an end effector of a manufacturing machine, a flexiblebeam is attached to the base driven by a belt-type transmissionpowered by an ac motor (HC-KFS13, MITSUBISHI Inc.).

For achieving the prescribed motion profile, a motion controlboard (PCI-7345, NI Corporation) is used to generate drivingsignals with a sampling period of 3 ms. The ac motor is con-trolled by a servomotor driver (MR-J2S-10A, MITSUBISHIInc.) using a proportional–integral–derivative control with anencoder. The physical parameters used for the experiments aresummarized in Table II. To test the robustness of the proposedmethod, an additional mass is attached to vary the naturalfrequency of the flexible beam from 15.87 to 14.28 Hz. Thedamping ratio is 0.015. Considering the actuator specification,the maximum velocity and maximum acceleration are chosenas 0.14 m/s and 1.4 m/s2, respectively. The target distance is270 mm. The absolute position of the base and the relativevibration of the flexible beam are measured using a laser

Fig. 11. Experimental measurements of the base movement. (Top) Accelera-tion, (middle) velocity, and (bottom) position profiles of the base.

Fig. 12. (Top) Acceleration profile of the base and (middle and bottom)induced vibration of the flexible beam.

TABLE IIICOMPARISON OF SETTLING TIMES

displacement sensor (LK-G505, KEYENCE Corporation) andan accelerometer (MMA-7260Q, FREESCALE Inc.), respec-tively. The measured sensor signals are recorded by the dataacquisition board (DAQ-6251, NI Corporation).


Fig. 13. Acceleration profile of the base and induced vibration of the flexible beam: AS-curve tuned at (a) a crater and (b) a chasm with/without modeling error.(a) AS-curve tuned as crater (β = 0.5; γ = 1.26). (b) AS-curve tuned as chasm (β = 0.75; γ = 1.26).

TABLE IVCOMPARISON OF SETTLING TIMES

Using the described system, we performed the experimentswith (β, γ) = (0.5, 1) and (β, γ) = (0.5, 1.26). Note that thecase of (β, γ) = (0.5, 1) corresponds to a conventional S-curve.Moreover, as shown in Fig. 5, the case of (β, γ) = (0.5, 1.26)is the AS-curve tuned for placing double zeros on the pole (i.e.,crater). As can be seen in the Fig. 11, the velocity profiles (ofthe base) actually obtained are close to the nominal design butnot exactly due to mechanical nonlinearities such as friction andthe belt fluctuation. These nonlinearities may cause additionalvibration during motion. Fig. 12 shows the acceleration profileof the base x1 and the induced vibration of the flexible beamx2. Observe that the residual vibration of the AS-curve (ina crater condition) significantly reduces, compared with theS-curve case. For comparison, the settling times within whichthe relative acceleration of the flexible beam gets smaller thana tolerance (i.e., ±0.1 m/s2) are measured and summarized inTable III. The settling time of the AS-curve at a crater reducesby 28.7% compared with the S-curve. Note that the perfectattenuation of the residual vibration under the proposed tuningconditions could not be attained because of the system nonlin-earities. Nevertheless, it turns out that the proposed approachcan improve the motion control performance effectively.

For demonstrating an advantage of robust pole-zero cancel-lation, we performed the experiments with the motion profilestuned at a crater and a chasm in the presence of 10% modelingerror. As described in the previous section, placing double zeroson the pole (i.e., to be a crater) is more robust against modelingerrors than placing a single zero on the pole (i.e., to be a chasm).As shown in Fig. 13, in the absence of modeling error, theresidual vibration is minimized in both cases. However, as thenatural frequency varies by 10%, the amount of the residualvibration increases. Note that the increment of the residualvibration at a crater is much smaller than that at a chasm. Itcan be also compared with ease by measuring the setting times.

The settling times are measured and summarized in Table IV.The increments of settling time due to the modeling errorfor the motion profile tuned at a crater and a chasm are 22.01%and 43.44%, respectively. It is obvious that the motion profiletuned at a crater is desirable in the presence of modeling error.

As a remark, through the experiments, we observed thatthe proposed approach is an effective and efficient method forreducing the residual vibration when the damping ratio is verysmall. Moreover, the analysis presents a unified framework thatcan suggest a systematic design guideline for generating themotion profiles, including AS-curve, S-curve, and trapezoidalvelocity profile.

VII. CONCLUSION

In this paper, a unified framework to minimize the resid-ual vibration of the motion profiles, including AS-curve,S-curve, and trapezoidal velocity profile, has been derived usingLaplace- and time-domain approaches. The proposed frame-work can illustrate the physical characteristics of the motionprofiles analytically. In particular, the robust tuning rule againstthe modeling error was newly discussed using the robust pole-zero cancellation concept, which is effective in practice havingmodeling errors inevitably. Through the experiments, we havedemonstrated the effectiveness of the proposed approach.

The proposed approach is expected to suggest an efficientdesign guideline for generating motion control profiles with theenhanced motion control performance.

REFERENCES

[1] T. Atsumi and W. C. Messner, “Optimization of head-positioning controlin a hard disk drive using the RBode plot,” IEEE Trans. Ind. Electron,vol. 59, no. 1, pp. 521–529, Jan. 2012.

[2] T. R. Kurfess, Robotics and Automation Handbook. Boca Raton, FL:CRC Press, 2005.


[3] M. Brady, J. M. Hollerbach, T. L. Johnson, T. L. Perez, andM. T. Mason, Robot Motion: Planning and Control. Cambridge, MA:MIT Press, 1979.

[4] PMAC Manual, Delta Tau Data Systems, Inc., Chatsworth, CA, 1996.[5] K. D. Nguyen, I. M. Chen, and T. C. Ng, “Planning algorithms for S-curve

trajectories,” in Proc. IEEE/ASME Int. Conf. Adv. Intell. Mechatronics,ETH Zurich, Switzerland, 2007, pp. 1–6.

[6] J. Wu, Z. Xiong, K.-M. Lee, and H. Ding, “High-acceleration precisionpoint-to-point motion control with look-ahead properties,” IEEE Trans.Ind. Electron, vol. 58, no. 9, pp. 4343–4352, Sep. 2011.

[7] H. Ding and J. Wu, “Point-to-point motion control for a high-accelerationpositioning table via cascaded learning schemes,” IEEE Trans. Ind.Electron, vol. 54, no. 5, pp. 2735–2744, Oct. 2007.

[8] Y-.S. Kung, C-.C. Huang, and M-.H. Tasi, “FPGA realization of anadaptive fuzzy controller for PMLSM drive,” IEEE Trans. Ind. Electron,vol. 56, no. 8, pp. 2923–2932, Aug. 2009.

[9] C. S. Chen and A. C. Lee, “Design of acceleration/deceleration profiles inmotion control based on digital FIR filters,” Int. J. Mach. Tools Manuf.,vol. 38, no. 7, pp. 799–825, Jul. 1998.

[10] K. Erkokmaz and Y. Altintas, “High speed CNC system design. Part 1:Jerk limited trajectory generation and quintic spline interpolation,” Int. J.Mach. Tools Manuf., vol. 41, no. 9, pp. 1323–1345, Jul. 2001.

[11] R. C. Hong and Y. F. Chang, “Parametric curve machining of a CNCmilling EDM,” Int. J. Mach. Tools Manuf., vol. 45, no. 7/8, pp. 941–948,Jun. 2005.

[12] B.-H. Chang and Y. Hori, “Trajectory design considering derivative ofjerk for head-positioning of disk drive system with mechanical vibra-tion,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 3, pp. 273–279,Jun. 2006.

[13] P. H. Meckl and P. B. Arestides, “Optimized S-curve motion profiles forminimum residual vibration,” in Proc. Amer. Control Conf., Philadelphia,PA, 1998, pp. 2627–2631.

[14] P. H. Meckl and R. Kinceler, “Robust motion control of flexible systemsusing feedforward forcing functions,” IEEE T. Contr. Syst. T., vol. 2, no. 3,pp. 245–254, Sep. 1994.

[15] R. Zanasi, C. G. L. Bianco, and A. Tonielli, “Nonlinear filters forthe generation of smooth trajectories,” Automatica, vol. 36, no. 3, pp. 439–448, Mar. 2000.

[16] C. G. L. Bianco and R. Zanasi, “Smooth profile generation for a tileprinting machine,” IEEE Trans. Ind. Electron, vol. 50, no. 3, pp. 471–477,Jun. 2003.

[17] W. M. Tao, M. J. Zhang, O. Ma, and X. P. Yun, “Industrial robottrack modeling and vibration suppression,” Ind. Robot., vol. 34, no. 4,pp. 317–325, Jun. 2007.

[18] C. T. Freeman, Z. Cai, E. Rogers, and P. L. Lewin, “Iterative learningcontrol for multiple point-to-point tracking application,” IEEE Trans. Ind.Electron, vol. 19, no. 3, pp. 590–600, May 2011.

[19] H. Li, M. D. Le, Z. M. Gong, and W. Lin, “Motion profile design toreduce residual vibration of high-speed positioning stages,” IEEE/ASMET. Mech., vol. 14, no. 2, pp. 264–269, Apr. 2009.

[20] Y. J. Shin and P. H. Meckl, “Application of combined feedforwardand feedback controller with shaped input to benchmark problem,”J. Dyn. Syst., Meas., Control, vol. 132, no. 2, pp. 021001-1–021001-8,Mar. 2010.

[21] A. Abe, “Trajectory planning for flexible Cartesian robot manipulator byusing artificial neural network: Numerical simulation and experimentalverification,” Robotica, vol. 29, no. 5, pp. 797–804, Sep. 2010.

[22] D. M. Tsay and C. F. Lin, “Asymmetrical inputs for minimizing residualresponse,” in Proc. IEEE Int. Conf. Mechatronics, Taipei, Taiwan, 2005,pp. 235–240.

[23] K-.H. Rew and K-.S. Kim, “Using asymmetric S-curve profile for fast andvibrationless motion,” in Proc. Int. Conf. Control, Autom. Syst., Seoul,Korea, 2007, pp. 500–504.

[24] C-. W. Ha, K-. H. Rew, and K-. S. Kim, “A complete solution to asym-metric S-curve motion profile: Theory & experiments,” in Proc. Int. Conf.Control, Autom. Syst., Seoul, Korea, 2008, pp. 2845–2849.

[25] K-.H. Rew, C-.W. Ha, and K-.S. Kim, “A practically efficient method formotion control based on asymmetric velocity profile,” Int. J. Mach. ToolsManuf., vol. 49, no. 7/8, pp. 678–682, Jun. 2009.

[26] K-.H. Rew and K-.S. Kim, “A closed-form solution to asymmetric motionprofile allowing acceleration manipulation,” IEEE Trans. Ind. Electron,vol. 57, no. 7, pp. 2499–2506, Jul. 2010.

[27] C-.W. Ha, K-.H. Rew, and K-.S. Kim, “Formulating a Laplace domainapproach for tuning motion profile,” in IEEE/ASME Int. Conf. Adv. Intell.Mechatronics, Budapest, Hungary, 2011, pp. 356–360.

[28] N. C. Singer, Residual Vibration Reduction in Computer ControlledMachines. Cambridge, MA: MIT Press, 1989.

Chang-Wan Ha (S’09) received the B.S. degree inmechanical engineering from the Department of Me-chanical and Control Engineering, Handong GlobalUniversity, Pohang, Korea, in 2008 and the M.S.degree in mechanical engineering from the Depart-ment of Mechanical Engineering, Korea AdvancedInstitute of Science and Technology, Daejeon, Korea,in 2010, where he is currently working toward thePh.D. degree.

His research interests include motion control, in-put shaping, bonding technology, and semiconductor

manufacturing equipment.

Keun-Ho Rew (M’10) received the B.S. and M.S.degrees in mechanical engineering and the Ph.D.degree in aerospace engineering from the KoreaAdvanced Institute of Science and Technology,Daejeon, in 1994, 1996, and 2001, respectively.

He was a Senior Engineer with Mirae IndustryCompany, Ltd., Cheonan, Korea, from 2001 to 2003and a Chief Engineer with Fine D&C Company, Ltd.,from 2003 to 2005. In 2005, he joined the Depart-ment of Mechanical Engineering, Hoseo University,Asan, Korea, as a Faculty Member, where he has

been with the Department of Robotics Engineering since 2007. His researchinterests include motion control, robotics, semiconductor manufacturing equip-ment, and digital image processing techniques.

Kyung-Soo Kim (M’00) received the B.S., M.S.,and Ph.D. degrees in mechanical engineeringfrom the Department of Mechanical Engineering,Korea Advanced Institute of Science and Technology(KAIST), Daejeon, Korea, in 1993, 1995, and 1999,respectively.

He was a Chief Researcher with LG Electronics,Inc., from 1999 to 2003 and as a DVD Group Man-ager with STMicroelectronics Company, Ltd., from2003 to 2005. In 2005, he joined the Department ofMechanical Engineering, Korea Polytechnic Univer-

sity, Seoul, Korea, as a Faculty Member. Since 2007, he has been with theDepartment of Mechanical Engineering, KAIST. He serves as an AssociateEditor of Automatica and Journal of Mechanical Science and Technology. Hisresearch interests include digital system design for controlled mechatronics,sensor and actuator design, robot manipulator design, and control theories suchas robust control and sliding mode control.

robust zero placement for motion control of lightly damped systems

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