Reference Command Shaping Using Specified-Negative-Amplitude Input Shapers for Vibration Reduction

Transcription

1 15 Ishitobi, M., 2000, A Stability Coditio of Zeros of Sampled Multivariable Systems, IEEE Tras. Autom. Cotrol, AC-452, pp Laub, A. J., ad Arold, W. F., 1984, Cotrollability ad Observability Criteria for Multivariable Liear Secod-Order Models, IEEE Tras. Autom. Cotrol, 292, pp M. Ikeda, 1990, Zeros ad Their Relevace to Cotrol-III; System Structure ad Zeros i Japaese, Joural of the Society of Istrumet ad Cotrol Egieers, 295, pp Gatmacher, F. R., 1959, The Theory of Matrices, Vols. I ad II, Chelsea, New York. 19 Rosebrock, H. H., 1970, State-space ad Multivariable Theory. Nelso, Lodo. 20 Suda, N., ad Mutsuyoshi, E., 1978, Ivariat Zeros ad Iput-Output Structure of Liear, Time-Ivariat Systems, It. J. Cotrol, 284, pp Suda, N., 1993, Liear Systems Theory i Japaese. Asakura, Tokyo. Referece Commad Shapig Usig Specified-Negative-Amplitude Iput Shapers for Vibratio Reductio William Sighose Erika Oote Biediger Ye-Hwa Che commad. The shapig process icreases the commad rise time by the duratio of the shaper,. This paper presets ew iput shapers that allow the desiger to trade off rise time agaist robustess to modelig errors, actuator effort, ad excitatio of umodeled high modes. Iput shapig schemes first appeared i the 1950s 1. Ufortuately, the methods were sesitive to modelig errors; small errors caused sigificat residual vibratio. Iput shapig became useful for may real systems whe a robust method was developed 2. Methods for further icreasig the robustess have sice bee preseted 3. Give its robustess, iput shapig has bee implemeted o a variety of systems icludig robotic maipulators 4,5, craes 6,7, ad wafer steppers 8. 2 Previously Developed Iput Shapers ad Their Effects o Nth Order Systems The amplitudes ad time locatios of the impulses i a iput shaper are determied by satisfyig costrait equatios. Most costraits ca be categorized as residual vibratio costraits, robustess costraits, costraits o the impulse amplitudes, or time optimality requiremets. The costrait o residual vibratio amplitude ca be coveietly expressed as the ratio of residual vibratio amplitude with iput shapig to that withouput shapig. For a mode of atural frequecy,, ad dampig ratio,, this percetage residual vibratio is give by 2: Departmet of Mechaical Egieerig, Georgia Istitute of Techology, Atlata, GA USA Bart Mills Soy Corporatio, Sprigfield, OR V,e t C, 2 S, 2 (1) C, i1 A i e cos 1 2 (2) S, i1 A i e si 1 2, (3) Residual vibratios ca be greatly reduced by usig speciallyshaped referece commad sigals. Iput shapig is oe such techique that reduces vibratio by covolvig a sequece of impulses with ay desired referece commad. Several types of useful impulse sequeces have bee developed. Most of these have cotaied oly positively valued impulses. However, rise time ca be improved by usig some egative impulses i the sequece. Ufortuately, the use of egative impulses ca excite umodeled high modes. A ew type of impulse sequece cotaiig egative impulses is proposed. These sequeces are desiged to fill the performace gap betwee all-positive impulse sequeces ad the egative sequeces previously developed. A proof goverig the worst case sceario provides a upper boud o high-mode excitatio. The resultig class of impulse sequeces allows the desiger to make a precise trade off betwee rise time ad vibratio reductio. DOI: / A i ad are the amplitudes ad time locatios of the ith impulse, ad is the umber of impulses i the iput shaper. If V is set 1 Itroductio Iput shapig is a commad geeratio scheme that reduces vibratio of flexible systems. Iput shapig is implemeted by covolvig a sequece of impulses, called the iput shaper, with ay system commad. The covolved sigal is the used as the referece commad. Figure 1a demostrates the process with a ushaped step iput ad a iput shaper that cosists of two positive impulses. The resultig shaped ipus a two-step staircase Cotributed by the Dyamic Systems, Measuremet, ad Cotrol Divisio of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publicatio i the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Mauscript received by the ASME Dyamic Systems ad Cotrol Divisio Jauary 21, 2002; fial revisio, July 16, Associate Editor: R. Ghulil. Fig. 1 Iput shapig a step iput 210 Õ Vol. 126, MARCH 2004 Copyright 2004 by ASME Trasactios of the ASME

2 The ZVD shaper cotais three impulses ad the duratio of the ZVD shaper is twice the duratio of the ZV shaper oe full period of the vibratio. Ay shaped commad will have its rise time icreased by the duratio of the shaper; therefore, the ZVD shaper icreases the rise time more tha the ZV shaper. This icreased rise time is the cost of the icreased robustess to modelig errors. Fig. 2 Iput shaper sesitivity curves equal to zero at the modelig parameters, ( m, m ) ad the iput shaper is required to satisfy 1, the the resuls called a Zero Vibratio ZV shaper. To esure that the shaped commad produces the same rigidbody motio as the ushaped commad, the impulse amplitudes must sum to oe: A i1. (4) The reaso for this costrais see i Fig. 1. If the impulses do ot sum to oe, the the shaped commad will ot reach the same fial value as the ushaped commad. I practice, ZV shapers ca be very sesitive to modelig errors. To demostrate this effect, the amplitude of residual vibratio ca be plotted as a fuctio of the modelig error. Figure 2 shows such a sesitivity curve for the ZV shaper. The vertical axis is the percetage vibratio give by 1, while the horizotal axis is the ormalized frequecy formed by dividig the actual frequecy of the system, a, by the modelig frequecy, m. Note that the residual vibratio icreases rapidly with modelig error. I order to icrease robustess, the shaper must satisfy additioal costraits. Oe such costrait takes the derivative of the residual vibratio equatio, 1, with respect to frequecy, ad sets it equal to zero 2. Thas: 0 d V,. (5) d Whe 1, 4, ad 5 are satisfied with V0, the resuls a Zero Vibratio ad Derivative ZVD shaper 2. Figure 2 shows the ZVD shaper is sigificatly more robust tha the ZV shaper. The trade-off is that the ZVD shaper has a loger duratio ad, cosequetly, degrades rise time. Other types of robustess costraits have bee proposed 3; however, we will limit our discussio here to ZV ad ZVD iput shapers. Due to the trascedetal ature of 1 ad 5, there will be multiple solutios. To obtai the fastest rise time, the shaper duratio must be made as short as possible therefore it must satisfy a miimum time costrait. Is ecessary to place a limit o the magitude of the impulses as miimizig the time duratio will drive the impulse amplitudes to positive ad egative ifiity. The ext subsectios discuss previously proposed magitude costraits. The remaider of the paper is the devoted to a ew type of magitude costrait that produces iput shapers that spa the performace gap betwee the previously proposed shapers. 2.1 Positive Iput Shapers. Oe way to avoid the problem of large amplitude impulses is to require all of the impulses to have positive values 9. If the amplitudes are all positive, the each idividual impulse must be less tha oe so that 4 ca be satisfied. The positive ZV 1,2 ad ZVD 2 shapers have bee derived i the literature previously. The duratio of the ZV shaper is equal to oe-half period of the damped vibratio. The ZV shaper is show schematically i Fig. 1a. 2.2 Effects of Positive Shapers o Nth-Order Systems. Although iput shapers cotaiig positive impulses elimiate the vibratio at the frequecies they are desiged to suppress, the questio arises as to what their effecs o a complicated system with umodeled modes. To address this questio, cosider a thorder system whose trasfer fuctio is G(s). Suppose its uitmagitude impulse respose is give by c(t),. Assume that the maximum of c(t) for all exists at a uique fiite time, T p 0. Deote this maximum by: max ctct p Now cosider the system subjected to a sequece of impulses: A 1 tt 1, A 2 tt 2,...,A tt, T T 1...T 1 0 ad, i1 A i 1, A i 0 The respose of the system is give by superpositio as: ĉ1 A i ctt i htt i where h(tt i ) is the Heavyside step fuctio. Theorem 1: Assume there is a time T 0 such that c(t )0. The maximum of the respose to the impulse sequece, ĉ(t), is less tha the maximum of the uit-magitude impulse resposes: max ĉtct p Proof: max ĉtmax i1 A i ctt i htt i i1 maxa i ctt i htt i We first ote that, uder the assumptio of the theorem: max A i ctt i htt i maxa i ctt i tt i This is because: A i ctt i htt i 0 for all tt i. Next, by the property of the traslatio fuctios ad sice A i 0: max A i ctt i A i maxctt i A i ct p tt i tt i This occurs whe tt p T i. This leads to: max ĉ1 where the equality occurs as: A i ct p Joural of Dyamic Systems, Measuremet, ad Cotrol MARCH 2004, Vol. 126 Õ 211

3 Fig. 3 tt p T 1 ad tt p T 2 ad... tt p T But, this ever happes because: T 1 T 2...T Therefore we coclude that max ĉ1 High-mode sesitivity curves A i ct p ct p i1 A i ct p ad the theorem is proved. The mai result of the theorem that cocers iput shapig is that a shaper cotaiig all positive impulses caocrease the vibratio of a th-order system over what would exist without iput shapig, eve if there are modes that are completely igored or ukow durig the shaper desig. 2.3 Uity-Magitude UM Shapers. I a effort to improve rise time, the shaper duratio ca be shorteed by allowig egative impulse amplitudes. The occurrece of egative impulses ca occur durig the process of desigig shapers i the z-domai 10. Oe method that allows egative impulses to be used i a systematic way requires the impulse amplitudes to switch betwee 1 ad 1 11: A i 1 i1 i1,..., (6) These uity-magitude UM shapers are useful because they ca be used with a wide variety of ushaped iputs without causig actuator saturatio. Figure 1c shows a step iput shaped with a UM ZV shaper. The duratio of the UM ZV shaper is oly oe-third of the vibratio period, as compared to oe-half period for the positive ZV shaper. The time savigs provided by usig the egative impulse is ot without pealty. Shapers cotaiig egative impulses have a tedecy to excite umodeled high modes 11. They also require larger swigs i the actuator effort because the chage i commad at each ew impulse is much larger. To see this, compare the shaped iputs show i Figs. 1a ad 1c. To demostrate the effect of high-mode excitatio, the sesitivity curve for a shaper ca be plotted over a rage of high frequecies. The value of the curve at ay high frequecy idicates the degree to which the high frequecy will be excited. Figure 3 compares the sesitivity curves for the positive ZV ad UM ZV shapers for frequecies up to te times the modelig frequecy. Note that the curve for the positive ZV shaper ever exceeds 100%. This idicates that the shaper will ever cause high frequecies to be excited more tha they are withouput shapig, ad demostrates the key result of Theorem 1. O the other had, the sesitivity curve for the UM ZV shaper reaches a maximum value of 300% at frequecies 3 ad 9 times greater tha the modelig frequecy. This meas thaf the system happes to have umodeled modes at these frequecies, the they will be excited to 3 times their level withouput shapig. 3 Specified-Negative-Amplitude Iput Shapers The larger the magitude of the egative impulses, the shorter the shaper duratio. However, the shaper iduces more high-mode excitatio. Give the tradeoffs that occur with egative impulses, is desirable to develop a iput shaper whose egative amplitudes ca be set to ay value. By specifyig the egative impulse amplitudes, the desiger ca set the performace tradeoffs. The goal is to develop a shaper that spas the performace gap betwee positive shapers ad UM shapers. Such a shaper is show schematically i Fig. 1b. The duratio is shorter tha the positive ZV shaper, but loger tha a UM ZV shaper. The key feature of the proposed specified-egative-amplitude SNA shapers is that the amplitude of the egative impulses ca be set to ay egative value, b. The SNA ZV shaper ca be determied i closed form if we assume a udamped system usig the amplitude summatio costrait, the zero vibratio costrait, ad a trigoometric idetity. The resultig udamped SNA ZV shaper is give by: b a A i t a 1 b 1 b2 i 0 cos1 2a cos1 (7) 2a 21 a 1b (8) 2 The SNA ZVD iput shaper must satisfy the additioal costrait of 5. Therefore, it must cotai five impulses. I additio to specifyig the amplitude of the egative impulses the secod ad fourth impulses, the amplitude of the middle impulse, c, must also be specified. The ZVD shaper caot be solved i closed form, so a umerical optimizatio is performed to miimize the shaper duratio. This is a relatively easy optimizatio ad the shapers show i this paper were geerated usig the MATLAB Optimizatio Toolbox. 3.1 Effects of SNA Shapers o th-order Systems. I Sectio 2.2 it was show that shapers cotaiig all positive impulses caocrease vibratio, eve whe there are umodeled modes. The result for shapers cotaiig egative impulses is ot the same, as was demostrated i Fig. 3. This sectio determies the worst case sceario. As i Sec. 2.2, cosider a th order system whose trasfer fuctio is G(s), ad the respose to a impulse respose is give by c(t),. The maximum of c(t) for all exists at a fiite time, T p 0. The maximum is the same as i Sec. 2.2, however, i this case A i ca be egative, but the sum of the magitudes is limited by: i1 A i i1 A i The respose of the system is give by superpositio as: ĉ1 A i ctt i htt i where the Heavyside step fuctio is: htt i Theorem 2: Assume there is a time T 0 such that ĉ(t )0. The maximum of the respose to the sequece of impulses, is less tha the maximum of the uit-magitude impulse respose multiplied by the sum of the absolute values of the amplitudes: max ĉ1 A i ct p 212 Õ Vol. 126, MARCH 2004 Trasactios of the ASME

4 Fig. 4 Duratio of the SNA ZV shaper Fig. 7 Impulse time locatios of the SNA ZVD shaper bäà1 Proof: The proof for Theorem 2 follows the same logic as the proof for Theorem 1. However for Theorem 2, the sum of the maximum amplitudes must equal the absolute values of each amplitude as seem below. i1 max A i i1 A i Fig. 5 SNA ZV high-mode excitatio 3.2 SNA Shaper Properties. At this pois of iterest to examie the shaper properties as a fuctio of the amplitude of the egative impulse ad the middle impulse for the ZVD shaper. It ca be see from 7 that as b becomes more egative the duratio of the SNA ZV shaper decreases. Whe the egative impulse is oexistet b0, the the shaper is the same as the positive ZV shaper. Whe b1, the the shaper is equivalet to the UM ZV shaper. Figure 4 shows the duratio of the SNA ZV shaper over a rage of dampig ratios. As the magitude of the egative impulse icreases, the shaper duratio decreases sigificatly. By usig a shaper with a egative amplitude of 1.0 the shaper duratio is reduced by approximately 30%. Although this time savigs is sigificat, it may ot be justified if there are sigificat umodeled high modes. These trade-offs are examied i the followig paragraphs. Although Theorem 2 provides a upper boud o the vibratio iduced ito ay umodeled modes, is of iterest to kow how close to the upper boud a typical system will be. Oe way to characterize the effect o high modes is to determie the average value of the sesitivity curve over a rage of high frequecies. High-mode excitatio HME will refer to the average value of the sesitivity curve from 2 to 10 times the modelig frequecy. Figure 5 shows the HME as a fuctio of the egative amplitude b. As b gets more egative, the HME icreases as expected. Note, however, that these average values lie well below the upper boud provided by Theorem 2. Figure 6 shows the duratio of the SNA ZVD shaper as a fuctio of the magitude of the egative impulses ad the middle impulse for a udamped system. The ZVD shaper duratio is miimized by some fiite value of the middle impulse assumig a give value for the egative impulse. For a give value of b, the value of c that miimizes the shaper duratio is: c b0.1019b 2. (9) Note that the middle impulse caot be smaller tha 0.5. Whe the middle amplitude exactly equals 0.5, the shaper duratio is equal to oe period ad the shaper is equivalet to the positive ZVD shaper. This pheomeo ca be see i Fig. 7 where the impulse time locatios are plotted as a fuctio of the middle impulse whe b1. Also oted o the figure is the case whe the shaper is equivalet to the UM ZVD shaper 11. Figure 8 shows the high-mode excitatio for the SNA ZVD Fig. 6 Duratio of the SNA ZVD shaper Fig. 8 SNA ZVD high-mode excitatio Joural of Dyamic Systems, Measuremet, ad Cotrol MARCH 2004, Vol. 126 Õ 213

5 Fig. 10 Experimetal results from robotic maipulator shapers. As the middle amplitudes icrease, the HME teds to icrease. However, is clear there are some miima that would be prudet choices whe selectig the middle amplitudes. For example, if a desiger determies that a shaper duratio of 0.75 is acceptable ad the chooses b1.0, this would lead to a value of for c ad a value of 2.32 for the HME excitatio. By examiig Figs. 7 ad 8 the egieer could itelligetly refie the desig by slightly reducig c, say to This would decrease the shaper duratio by 1.7% ad also decrease the HME by 14.5%. 4 Experimetal Results Is possible that the best shaper for a give system does ot always correspod to either a UM shaper or a positive shaper. By utilizig SNA shapers, better overall results ca be obtaied. This is especially true whe the system has higher modes that ca be easily excited by usig a UM shaper. Is possible for the SNA shaper to have a much faster rise time tha the positive shaper ad yet ot sigificatly excite the umodeled high modes. To experimetally verify the key results i this paper, tests were coducted o a large robot arm used to move a suspeded payload as see i Fig While the edpoit of the robot arm was moved i the horizotal directio, the positio of the payload was captured usig a video camera. The UM ZV, SNA ZV b0.5, ad Positive ZV shapers were used to modify the commads used to drive the robot. The shapers were desiged for the hagig pedulum s frequecy of Hz, ad the amplitudes ad time locatios for the shapers are: A i UM ZV (10) SNA ZV (11) A i A i Fig. 9 Robotic maipulator Positive ZV (12) The vibratio modes of the robotic arm the domiat robot mode is at approximately 1.8 Hz were ot take ito accout whe desigig the shapers. I this case, the umodeled high frequecy is approximately 5 times the pedulum frequecy. The positio of the payload for the three cases is show i Fig. 10. All three shapers elimiate most of the residual vibratio of the pedulum mode. I the ushaped case, the pedulum oscillatio amplitude would be off the scale 22 i. The ZV shaper does ot excite the higher frequecies of the robotic arm ad moves the hagig pedulum with a miimal amout of residual vibratio. O the other had, the UM ZV excites the higher modes of the robot. However, it does produce the fastest rise time. The payload of the robot reaches the desired locatio much faster with the SNA ZV shaper tha with the positive ZV shaper, ad the robot modes are ot excited. These results cofirm that the flexibility of the SNA shaper provides a meas for achievig excellet performace thas ot possible with either the positive ZV of UM ZV shapers. 5 Coclusios A ew type of iput shaper has bee developed to allow the desiger to specify the magitude of the egative impulses. These shapers allow the desiger to set the tradeoffs betwee rise time, robustess, ad high-mode excitatio. The shapers preseted here provide a wide rage of possible performace levels. A proof has bee give that shows that positive iput shapers caocrease vibratio eve whe there are ukow modes. This proof was exteded to obtai a upper boud o the amout of vibratio that ca occur from the ew type of egative iput shaper. Both simulatios ad hardware experimets demostrated the usefuless of the ew iput shapers. Refereces 1 Smith, O. J. M., 1958, Feedback Cotrol Systems, New York: McGraw-Hill Book Co., Ic. 2 Siger, N. C., ad Seerig, W. P., 1990, Preshapig Commad Iputs to Reduce System Vibratio, J. Dy. Sys., Meas. Cotrol, 112, pp Sighose, W., Seerig, W., ad Siger, N., 1994, Residual Vibratio Reductio Usig Vector Diagrams to Geerate Shaped Iputs, J. Mech. Des., 116, pp Drapeau, V., ad Wag, D., 1993, Verificatio of a Closed-Loop Shaped- Iput Cotroller for a Five-Bar-Likage Maipulator, preseted at IEEE It. Cof. o Robotics ad Automatio, Atlata, GA. 5 Magee, D. P., ad Book, W. J., 1995, Filterig Micro-Maipulator Wrist Commads to Prevet Flexible Base Motio, preseted at America Cotrol Cof., Seattle, WA. 6 Kress, R. L., Jase, J. F., ad Noakes, M. W., 1994, Experimetal Implemetatio of a Robust Damped-Oscillatio Cotrol Algorithm o a Full Sized, Two-DOF, AC Iductio Motor-Drive Crae, preseted at 5th ISRAM, Maui, HA. 7 Siger, N., Sighose, W., ad Kriikku, E., 1997, A Iput Shapig Cotroller Eablig Craes to Move Without Sway, preseted at ANS 7th Topical Meetig o Robotics ad Remote Systems, Augusta, GA. 8 deroover, D., Sperlig, F. B., ad Bosgra, O. H., 1998, Poit-to-Poit Cotrol of a MIMO Servomechaism, preseted at America Cotrol Coferece, Philadelphia, PA. 9 Tuttle, T. D., ad Seerig, W. P., 1994, A Zero-Placemet Techique for Desigig Shaped Iputs to Suppress Multiple-Mode Vibratio, preseted at America Cotrol Cof., Baltimore, MD. 10 Murphy, B. R., ad Wataabe, I., 1992, Digital Shapig Filters for Reducig Machie Vibratio, IEEE Tras. Rob. Autom., 8, pp Sighose, W., Siger, N., ad Seerig, W., 1997, Time-Optimal Negative Iput Shapers, J. Dy. Syst., Meas. Cotrol, 119, pp Obergfell, K., 1998, Ed-Poit Positio Sesig ad Cotrol of Flexible Multi-Lik Maipulators, Atlata: Georgia Istitute of Techology. 214 Õ Vol. 126, MARCH 2004 Trasactios of the ASME