Journal of System Design and Dynamics

Transcription

1 Vbraton Control of Safe Robot Arm wth MR-Based Passve Complant Jont * Seung-kook YUN**,, Seong-Sk YOON**,, Sungchul KANG**, and Munsang KIM** **Intellgent Robotcs Research Center, Korea Insttute of Scence and echnology, Hawolgok-dong1 39-1, Sungbuk-ku, Seoul , Korea E-mal: yunsk@mt.edu, yoon@cello.korea.ac.kr, {kasch, munsang}@kst.re.kr Mr. Seung-kook Yun s now wth Computer Scence and Artfcal Intellgence Laboratory, Massachusetts Insttute of echnology, Cambrdge, USA. Dr. Seong-sk Yoon s currently wth ACI Devson, Samsung Electro-Mechancs Co., 581, Dong-Myon, Yeong-gun, Chungcheongnam-Do , Korea. Abstract In ths paper, vbraton control of a safe arm wth passve complant jonts and vsco-elastc coverng for a human-frendly servce robot s presented. he passve complant jont (PCJ) s desgned to passvely attenuate the appled force. he rotary sprng gves the arm complant property, and yet t can be a source of vbraton. We use an nput-preshapng method whch s motvated by the nput shapng technque (IS) based on mpulse responses. Experments prove that both of fast moton and force attenuaton of the safe arm can be acheved. Key words: Passve Complant Jont, Magneto-Rheologcal Damper, Vsco-Elastc Coverng, Safety, Input Shapng echnque 1. Introducton *Receved 21 Dec., 27 (No ) [DOI: /jsdd.2.475] Buldng a safe robot manpulator and performng hgh-speed manpulaton s a trade-off when speed and precson are consdered (1)(2)(3). he less stff complance we use n a robot arm, the more dffcult the arm has fast moton. In our prevous work (4), a safe arm of a servce robot was developed. In order to have a safe nteracton n a human-robot envronment, the arm has passve complant jonts composed of a magneto-rheologcal rotary damper and rotary sprngs. Despte of the guaranteed safety, unwanted vbraton can occur because of sprngs ncluded n the PCJ durng fast jont moton. Although the damper of the PCJ partly suppresses vbraton, quck moton cannot be performed wthout vbraton. hs vbraton can cause unstable moton or collson to the envronment, whch degrades safety. In ths paper, we propose a new method to enhance performance of our safe manpulator by vbraton control based on nput preshapng. he Input Shapng echnque(is) (5) based on mpulse responses s consdered to solve a problem of vbraton, because of ts smple structure and hgh effcency. he IS s an excellent soluton to reduce vbraton for a lnear tme nvarant(li) system, however t s not so effectve for a non-lnear or a tme varyng system and the safe arm s a tme varyng system n whch ts nertal propertes change accordng to the arm confguraton. Although t s almost mpossble to fnd the exact tmng of the second mpulse for a tme varyng system, there have been approaches to fnd the sequence. Park proved that a proper IS flter wth two mpulses can elmnate a resdual vbraton of a tme nvarant system (6). Rappole (7) appled a tme varyng nput shapng technque(vis) to a two lnk flexble manpulator, and Park (8) proposed a practcal method of the VIS to reduce vbraton of an ndustral robot. he former used a look-up table whch contaned nformaton on confguraton-dependent frequences, whle the latter had a smple equaton relatng a perod 475

2 of vbraton to a length from the swng axs to the wrst axs. Although these methods reduce vbraton of a tme varyng system, an estmaton of tmng of the second mpulse tme s always ncorrect because of dynamcs of a manpulator. Motvated from the IS based on mpulse responses, we desgn a new algorthm to reduce vbraton at both the start and the end poston. Wth the developed safe arm and the proposed control method, experments for vbraton reducton are performed to verfy that vbraton reducton s successfully acheved. hs paper s composed of fve sectons. In Secton 2, we ntroduce how to desgn the 6-dof safe arm ncludng three PCJs. In Secton 3, the nput preshapng method for vbraton reducton of the PCJ s proposed as a controller for the safe arm. Experments on vbraton control by the proposed nput preshapng method are shown n Secton 4. Fnally, we conclude ths work n Secton Safe Arm Desgn We ntroduce a passve complant jont wth a magneto-rheologcal damper and a sprng. Wth the PCJs and soft cover, we show a 6DOF arm for a servce robot. he detals of the arm are n Ref. (4). (a) (b) Fg.1 (a) Desgn of passve complant jont mechansm. (b) Developed PCJ 2.1 Passve Complant Jont Desgn A schematc desgn and components of a passve complant jont(pcj) s shown n Fg. 1. It conssts of a magneto-rheologcal (MR) rotary damper and a rotary sprng. In the PCJ, a MR rotary damper s ntroduced for dampng effect. he PCJ has a resolver sensor wth 16-bts hgh resoluton to read the relatve poston between the housng and the next lnk due to the sprng dsplacement. he relatve poston s converted nto a relatve velocty sgnal by a numercal dfferentaton and a flterng. hen the velocty s sent to a dampng tuner whch converts Coulomb frcton property of the MR flud nto vscous one whch makes the dynamc system lnear. Note that the damper and sprng are located n parallel between the housng and next lnk Safe Arm Desgn By ntegratng the MR-based passve complant jonts and the vsco-elastc coverng, a new safe arm s developed for servce robot applcatons. A sx-dof safe arm wth 3Kg-payload, 3kg weght and 1m reach has been developed as depcted n Fg. 2. hs arm s used for the PSR-2 (Publc Servce Robot) developed at the KIS (Korea Insttute of Scence and echnology). akng account of the weght ncrease of the arm due to the weght of MR dampers and sprngs, PCJs are adopted only to three jonts n the lower part of the 6-dof servce robot arm. he three PCJs gve the arm complance nto the three 476

3 drectons. Parameters for the PCJs are shown n Fg. 2(c) and able.1. θ, x, y complant y x θ (a) (b) m 4=1.7 m5= m 1=4.7 m 2=8. m 3= payload 13. (c) Fg.2 (a) KIS Safe Arm (b) schematc dagram of the arm (c) lengths of the lnks and estmated weghts of pont masses able 1 Specfcatons of the PCJs Axs number Axs 1 Axs 2 Axs 3 Sprng Constant (Nm/rad) Maxmum dampngorque (Nm) Vscous dampng coeffcent (Nm sec/rad) Controller Desgn: Input Preshapng Method for the Safe Arm Although the PCJ enables passve complance, t also can be a source of unwanted vbraton and deflecton by gravty. hey can cause unstable moton or naccuracy of the jont angle. In partcular, bg amount of resdual vbraton can occur durng fast rotaton of the PCJ. o cope wth ths problem, we desgn a vbraton controller based on the nput shapng technque. 3.1 Estmaton of me Varyng Perod of Vbraton from PCJs As an effectve soluton to reduce ths vbraton, an nput shapng technque(is) based on mpulse response was proposed (5). he IS uses two mpulses to elmnate resdual vbraton. Fgure 3 shows how the IS works wth two mpulse responses. If the second mpulse tme s properly selected, the superposed response wll have no vbraton after the second mpulse. he IS s a smple flter snce two mpulses are convoluted wth a desred trajectory or an open loop control nput. 477

4 Fg.3 Vbraton suppresson va IS For ths tme varyng nput shapng technque(vis), we need to know perods of vbraton n all confguratons that the arm can have n ts workspace. Note that t s proved that a proper IS flter wth two mpulses can elmnate a resdual vbraton of a tme nvarant system (6). A dynamc equaton of the -th PCJ of the safe arm excludng gravty s as follows: J ( θ ) θ + C( θ, θ ) + K ( θ θ ) = τ (1) m where J s a nerta term, C s a Corols and centrfugal term, K s a sprng constant, θ m s a motor angle of -th axs and τ s a control torque. A perod of vbraton for the -th PCJ can be expressed as follows: J = 2 π ( = 1,2,3) (2) K Because the safe arm s a seral manpulator, a perod of vbraton for each jont s functon of the next jont angles assumng a sprng constant s tme-nvarant. We consder the perod of vbraton for the thrd jont as a constant because change of nerta along the wrst confguraton s comparably small. herefore, tme varyng ISs for the frst and second jont are needed. In ths paper, vbraton control of the second jont s only mplemented snce the same method can be appled to the frst jont. perod of vbraton (s) expermental value ftted cosne curve θ 3 (deg) Fg.4 Perod of vbraton at second jont along jont confguratons o determne parameters for VIS, the perods of vbraton at dfferent confguratons 478

5 are expermentally measured. In Fg.4, the perod of vbraton for the second jont accordng to the thrd jont angle s shown. We can expect the perod of vbraton s proportonal to square root of cosθ 3 because nerta s proportonal to the square root of cosθ 3. he curve of perod of vbraton s a functon of cosθ 3 as follows: = α + β cos( θ θ ) (3) perod 3 o where α =.2, β =.22, θ ο = 3. We can make a table of the perod of vbraton along each jont confguraton usng obtaned data. When controllng the PCJ, we can use the table or the equaton. Although ths paper deals wth only the case of no-payload, we can extend the method for any payload condtons by measurng the vbraton perods wth varous payloads and nterpolatng values from the curve equatons. 3.2 Modfed VIS In prevous VISs, a controller estmates the second mpulse tmng by comparng a jont confguraton wth measured perods of vbraton. Even f the estmaton s correct and the measured data are exact, however, there wll be stll a lttle vbraton because of tmng error due to dynamcs of the system. When nerta ncreases, the second mpulse occurs later than the correct tmng as shown n Fg.5. Fg.5 Estmaton error of the second mpulse tmng o compensate ths error, we propose to estmate the second mpulse tme as a mean of the measured parameters at the start and the end. he proposed method gves better result unless dynamcs s faster than a changng rate of the mpulse tmng. We have found the dynamcs of the developed manpulator s not fast enough that our estmaton holds. Furthermore, t wll gve a perfect result f the square root of nerta changes lnearly. For ths, the nerta change of our manpulator shows approxmately lnearly snce tme between the mpulses s very short compared to the jont trajectory. hen the second mpulse tme guessed at the moton-start as follows: half + half _ start half _ VIS = (4) 2 where 479

6 half _ start = π J( θ( t)) K θ = θ ( t= ) start (5) half _ VIS = π J( θ( t)) K θ = θ( t= estmated tme) (6) 35 1 motor angle [deg] lnk angle motor angle PCJ dsplacement [deg] tme [s] tme [s] (a) (b) Fg.6 (a) Motor angle and lnk angle of the second PCJ (b) two modes of vbraton n PCJ dsplacement PCJ dsplacement [deg] 1.5 modfed VIS for 1st fltered orgnal PCJ dsplacement [deg] 1.5 conventonal IS for 2nd tme [s] tme [s] (a) (b) Fg.7 Smulaton results: (a) IS for the frst mode (b) IS for the second mode 3.3 Input Preshapng Method for the safe arm Generally, there are two phases of vbraton durng the moton of a flexble jont. he frst one s excted by start moton, and the second s by end moton. In Fg.6, a smulaton result s shown when the second axs moves by 3 degree durng one second. Lookng at the rght part of the drawng carefully, we can fnd that there are two knd of vbraton; the frst one s started at zero second, and the second s started at one second whch s the begnnng of the steady state. wo vbratons have dfferent frequences snce the safe arm s a tme varyng system. herefore, we need two dfferent IS flters to elmnate both of vbraton. We desgn a proposed VIS flter for the frst vbraton, and a conventonal IS flter for the 2nd vbraton because the system s almost tme nvarant when the jont moton reaches the steady state. Smulaton results are shown In Fg.7. Fgure 7(a) results from a controller wth a 48

7 modfed VIS, and Fgure 7(b) comes from a conventonal IS for the second vbraton. We can see the modfed VIS elmnates the frst vbraton but second vbraton, and the conventonal IS can elmnate only the second vbraton. In case of a lnear tme nvarant system, we can easly elmnate both of vbraton by usng a merged IS flter by convolutng the frst IS flter to second one. However, ths prncple does not work n case of a lnear tme varyng system. Now we propose a new nput-preshapng method for the safe arm. We already have IS parameters of each vbraton that are composed of each magntude of two mpulses and the second mpulse tmng. All parameters we need to know are lsted as follows: A 1 : magntude of 1st mpulse for the modfed VIS A 2 : magntude of 2nd mpulse for the modfed VIS 2A : half perod of 1st vbraton B 1 : magntude of 1st mpulse for the conventonal IS B 2 : magntude of 2nd mpulse for the conventonal IS 2B : half perod of 2nd vbraton Where A + A = B + B = (7) to guarantee the same steady-state value. We set a new trajectory whch s convoluted by the mpulses as follows: θ () t = R [ Aθ () t + Aθ ( t ) u( t )] < t< * d 1 d 2 d 2A 2A f θ () t = R [ Bθ () t + Bθ ( t ) u( t ) + α] < t * d 1 d 2 d 2B 2B f (8) where f s the tme of steady state pont and θ d s a orgnal desred trajectory and R s set to satsfy the followng: θ dt = θ dt (9) + f f 2 B * d d, therefore the total dsplacement of the reference trajectory s the same as that of the shaped one. Note that the fnal dsplacement of the jont s mantaned by R. α = Aθ ( ) + Aθ ( ) Bθ ( ) Bθ ( ) (1) 1 d f 2 d f 2A 1 d f 2 d f 2B, and for t < ut ( ) = 1 for t > (11) whch s a step functon to enable the effect of the mpulses. All calculatons are executed off-lne, not a real-tme estmaton. If a reference trajectory s gven, we can estmate all parameters from the frequency and jont confguraton table. Fgure 8 shows a smulaton result usng proposed nput preshapng method. You can see that smple change of the reference nput make clear response wthout vbraton. he proposed method gves better result than that of both tme nvarant and tme varyng nput shapng flters because ths reflects not only tme varyng property of the system but also the 481

8 dynamc effect durng the control w/ preshaped w/ unshaped 1 w/ preshaped w/ unshaped desred velocty [deg/s] PCJ dsplacement [deg] tme [s] tme [s] Fg.8 Smulaton: proposed nput preshapng method 4. Expermental Results We prove the proposed nput preshapng by experments wth the safe arm. Vbratons of the second and thrd jont are successfully suppressed wth the IS and the proposed controller. As for the set-up, a PC-based UMAC moton controller (9) s used for controllng the sx-dof safe arm, and MR dampers are ndependently controlled. Encoder sgnals are sent to PC va USB and RS-232C port. motor angle3 (deg) command velocty (deg/s) t m e [s] ( a ) t m e [ s] ( b ) w / I S w / o I S w / I S w / o I S PCJ dsplacement (deg) w/ IS w/o IS tme [s] (c) Fg.9 Experment: vbraton reducton for thrd axs: (a) motor angle of the 3rd jont. (b) command velocty of the thrd motor, (c) PCJ dsplacement of 3rd PCJ defned n equaton(5) An experment of rotatng the thrd jont of the safe arm wth the IS s presented n Fg. 9. A desred trajectory s 6 deg/sec durng a second. We have found that the frequency of the vbraton s around 3Hz. he mplemented IS reduces vbraton from the 3rd axs dramatcally. Note that the dfference between the PCJ dsplacement of a startng pont and steady state s caused by gravty of the lnk. As shown n Fg. 9, IS works well even n gravty. A lttle amount of overshoot s stll shown, but ths s due to the nerta of the thrd 482

9 lnk not to the vbraton mode. After the second mpulse tmng, lttle vbraton s shown. 3 Fg.1 Change of jont confguraton durng the experment Wth the proposed VIS controller, an experment of rotatng the 2nd axs wth varyng effectve nerta s performed. he jont moves from 3 degree to zero degree, whle the thrd jont rotates from 3 degree to +3 degree as shown n Fg.1. Wth a gven trajectory and the table of perods of vbraton versus jont confguratons, we can expect that the frequency of vbraton vares from 1.85Hz to 1.55Hz. A trajectory by the proposed algorthm s generated and tested. Fgure 11 shows the preshaped nput and the vbraton of the second PCJ. he controller wth the preshaped nput clearly suppresses both of vbraton ncludng vbraton of startng moton and endng moton. θ 2 (deg) velocty (deg/s) tme [s] (a) w/o preshaped w/ preshaped PCJ dsplacement (deg) w/ unshaped nput w/ preshaped nput tme [s] (b) tme [s] Fg.11 Experment: vbraton reducton for 2nd axs: (a) motor angle of the 2nd jont, (b) command velocty of the 2nd motor, (c) dsplacement of the 2nd PCJ 5. Concluson In ths paper, we propose a new nput preshapng controller for a safe arm that has passve complant jonts wth sprngs and dampers and a soft cover. wo modes of vbraton are caused by startng moton and by endng moton, and a reference trajectory s shaped to reduce the vbraton by the proposed controller. Parameters of mpulse sequences by the 483

10 modfed VIS and a conventonal IS are used for nput preshapng. Expermental results show that proposed method mplemented successfully to the safe arm. In future, we wll explore strateges for contact moton by the safe arm as well as devse lghtweght passve jonts. References (1) M. Znn, O. Khatb, B. Roth, and J. Salsbury, Playng t safe [humanfrendly robots], Robotcs and Automaton Magazne, IEEE, vol. 11(24), pp (2). Morta and S. Sugano, Development and evaluaton of seven dof ma arm, n IEEE Internatonal Conference on Robotcs and Automaton, 1997, pp (3) A. Bcch and G. onett. Fast and Soft Arm actcs: Dealng wth the Safety-Performance rade-off n Robot Arms Control IEEE Robotcs and Automaton Magazne, Vol. 11(24), pp (4) S.-S. Yoon, S. Kang, S.-J. Km, Y.-H. Km, M. Km, and C.-w. Lee, "Safe Arm wth MR-based Passve Complant jonts and Vsco-elastc Coverng for Servce Robot Applcaton, In Proceedngs of the IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems(23), pp (5) N.C. Snger, and W.P. Seerng, Preshapng Command Inputs to Reduce System Vbraton, ASME Journal of Dynamc Systems, Measurement and Control, Vol.112(199), pp (6) J. Park, P.H. Chang, and E. Lee, Can a me Invarant Input Shapng echnque Elmnate Resdual Vbratons of LV Systems, Proc. of the Amercan Control Conference(22), pp (7) B.W. Rappole, Mnmzng Resdual Vbratons n Flexble Systems, Master thess, Department of Mechancal Engneerng (1992), MI. (8) H.Y. Park, P.H. Chang, and J.S. Hur, me-varyng Input Shapng echnque Appled to Vbraton Reducton of An Industral Robot, Proc. of the IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems(1999), pp (9) he Deltaau Corporaton ( 484