1. Albano LANZUTTI SMOOTH TRAJECTORY PLANNING ALGORITHMS FOR INDUSTRIAL ROBOTS: AN EXPERIMENTAL EVALUATION 1. DIPARTIMENTO DI INGEGNERIA ELETTRICA, GESTIONALE E MECCANICA UNIVERSITA' DI UDINE, UDINE ITALY ABSTRACT: An analyss of te expermental results of a new metod for smoot trajectory plannng for robot manpulators s presented n ts paper. Te tecnque s based on te mnmzaton of an objectve functon tat s composed of two terms: te frst one s proportonal to te trajectory executon tme, te second one s proportonal to te ntegral of te squared jerk. Te need for a smoot trajectory and te need for a fast executon can be adjusted by cangng te values of two constants tat weg te two terms. Te trajectory executon tme s not set a pror and te knematc constrants on te robot moton are taken nto account. Cubc splnes and fft-order B-splnes are used to compose te overall trajectory. Two dfferent trajectory plannng tecnques (te frst one mnmzes te maxmum absolute value of te jerk along te wole trajectory, wle te second one ensures only te contnuty of te poston, velocty and acceleraton values) ave been mplemented wt te am to compare te outcomes of te tests. Te descrbed metods are appled to a 3-d.o.f. Cartesan robot and te expermental tests are carred out by usng an accelerometer mounted on te manpulator end-effector. KEYWORDS: Trajectory plannng, Smootness, Tme-jerk optmzaton, Expermental valdaton INTRODUCTION One of te most mportant current robotc ndustral requrements s te estmaton and te reducton of te manpulators vbratonal penomena. Indeed, te demand for ncreasng productvty troug fast and g precson moton s growng, tus te desgners are forced to reduce te masses of te robot structures, resultng n a loss of structural rgdty and an ncrease of flexblty affectng also te dynamc response of te system. A proper calbraton of te manpulator control system and a dedcated acton on te trajectory plannng pase [1] can be consdered as a soluton of te problem. Te trajectory plannng s a fundamental ssue for robotcs applcatons and automaton n general. At g operatng speeds, requred n many current tasks, te possblty to generate trajectores tat satsfy specfc targets and requrements s a basc step to ensure optmal results. Robotc movements and trajectores tat ave smootness propertes are becomng more wdely used n modern applcatons. Indeed, te plannng of trajectores wt a bounded value of te jerk s an mportant target, snce ts allows to reac ger task executon speeds, reduce te exctaton of te resonant frequences of te manpulator structure and mprove te trackng accuracy. Te analyss of te scentfc lterature sows tat te trajectory plannng problem s based on te optmzaton of some objectve functon or of some parameters. Crtera tat are based on mnmum executon tme, mnmum energy or actuator effort, mnmum jerk or ybrd optmalty approaces can be found. Wt te am to ncrease te productvty n te ndustral sector, te frst trajectory plannng tecnques proposed were te mnmum-tme algortms. Startng from unconstraned problems [2], ts type of optmzaton s recently evolved n mnmum tme algortms under knematc constrants (.e. maxmum values for velocty, acceleraton and jerk) [3]. A second mportant crteron for trajectory plannng s focused on te mnmzaton of te actuator effort,.e. te mnmzaton of te energy requred to te manpulator actuators [4]. If te energy consumpton s mnmzed nstead of te executon tme, te effort of te actuators and te stresses of te manpulator are reduced, moreover te resultng trajectory s easer to track. Ts type of optmzaton s ten preferable n applcatons wt lmted capacty of energy source. Anoter type of trajectory plannng algortms s based on te optmzaton of te jerk along te wole lengt of te pat [5-6]. If ts tecnque s used, te exctaton of te resonant frequences of copyrgt FACULTY of ENGINEERING - HUNEDOARA, ROMANIA 127
te mecancal system s reduced. Tus, te stresses to te actuators and to te robot structure are ntrnscally reduced, and te trackng errors decrease. As mentoned n te foregong, startng from te fundamental optmzaton tecnques above descrbed, ybrd optmalty approaces are mplemented. For nstance, ybrd tme-energy-optmal trajectory plannng algortms can be found n [7]. Wt te am to reac te advantages of te jerk reducton n fast trajectores, ybrd tme-jerk optmal tecnques are proposed [8-12]. Tese algortms are based on dfferent prmtves tat are used to nterpolate te pat (e.g. trgonometrc splnes n [8], polynomals of fourt and fft order n [10]) and dfferent optmzaton procedures (e.g. genetc algortms are used n [9], SQP algortm n [11-12]. One of te most popular algortms for plannng smoot trajectores s descrbed n [5-6]. Based on nterval analyss, ts tecnque mnmzes te absolute maxmum value of te jerk along a trajectory wose executon tme s known and set a pror. Cubc splnes are used to nterpolate te va-ponts of te pat and te output of te algortm s a set of tme ntervals tat produces te lowest jerk peak. A mnmum tme-jerk trajectory plannng tecnque s presented n [11-12]. Two algortms based upon a mnmzaton of an objectve functon tat takes nto account te speed and te smoot of te trajectory are presented. More n detal, te objectve functon s composed of a term tat s proportonal to te total executon tme and of a term tat s proportonal to te ntegral of te squared jerk along te pat, bot wegted by two parameters. A metod based on te objectve functon defned n [11-12] and extended by consderng also te power consumpton of te actuatng motors and te jonts pyscal lmts (so tat te tecnque s a tme-jerk-energy plannng algortm) s presented n [13]. In ts paper, te two trajectory plannng algortms presented n [11-12] are consdered. Unlke most jerk-mnmzaton tecnques, ts metod does not requre a trajectory executon tme set a pror, and takes nto account te robot moton constrants. Tus, one can defne te upper bounds on te absolute values of velocty, acceleraton and jerk for eac robot jont. In order to demonstrate te benefts of te used algortms (.e. reduced mecancal stresses and reduced vbratonal penomena), te trajectores so planned are nput to a 3-d.o.f. Cartesan robot and te vbratons of ts arms durng ter movements are evaluated by usng an accelerometer. Wt te am of evaluatng te results obtaned wt te mnmum tme-jerk tecnque, bot te metod descrbed n [5-6] and a classcal splne based plannng algortm ave been mplemented and expermentally tested on te Cartesan manpulator. Te paper s organzed as follows: n secton 2 te plannng algortms [11-12] and [5-6] and te man caracterstcs of te plannng tecnques under test are explaned; te smulatons and expermental results of te used tecnques, wt a bref descrpton of te expermental set-up, are analyzed n secton 3. THE TRAJECTORY PLANNING ALGORITHMS. MINIMUM TIME-JERK TRAJECTORY PLANNING ALGORITHM Te mnmum tme-jerk algortm (descrbed wt many detals n [11-12]) concerns trajectores off-lne geometrcally defned. Accordngly, a pat planner at te top level generates te geometrc pats (obstacle avodance problems are solved at ts level) as a sequence of nodes n te operatve space wc represent successve postons and orentatons of te end-effector of te manpulator. Te executon tme of te planned trajectory s not set a pror (t s a result of te algortm), and te upper bounds on velocty, acceleraton and jerk (te knematc constrants) are taken nto account. Te generated trajectory s optmzed n te sense of te best compromse between executon tme and value of te jerk. In order to aceve ts task, a ybrd objectve functon made of two terms avng opposte effects s consdered. Te frst term s proportonal to te trajectory executon tme, wereas te second term s proportonal to te ntegral of te squared jerk. Te two effects are wegted by te coeffcents k T and k J respectvely. In order to represent te trajectores, two specfc prmtves are cosen. Te frst prmtve s a cubc splne (te algortm s so called SPL3J) and te objectve functon s gven by: FOBJ = k n 1 T = 1 +k N n 1 2 ( α j,+ 1 α j, ) J j= 1 = 1 were α j, s te acceleraton of te j-t jont at te -t va-pont, n s te number of te va-ponts of te pat, N s te number of robot jonts and s te tme nterval between two va-ponts (for more detals [12]). Te second prmtve s a fft-order B-splne, degree p = 5 and order k = 2r = 6, (te algortm s so called BSPL5J) and te objectve functon s gven by: FOBJ = k vp+ 1 T = 1 + k N tf n 2 J j= 1 0 k= 1 CPJ j,k N p 3, k () t 2 dt (1) (2) 128 Tome IX (Year 2011). Fasccule 1. ISSN 1584 2665
were vp s te number of va-ponts, n+1 s te number of control ponts (n = vp + 2(r-1)), N,p (t) s te base functon, CPJ j,k s te control pont of te jerk and t f s te total executon tme of te trajectory (for more detals [11]). Te soluton of te optmzaton problem s a vector of tme ntervals between two subsequent va-ponts tat mnmze te objectve functons (1) or (2). Wt a sutable coosng of te value of te two wegts k T and k J, n bot solver metods above descrbed, a balance between speed and smootness of te trajectory can be reaced. Te lmt condtons are te mnmum executon tme (.e. k J = 0) and te mnmum jerk value (.e. k T = 0). A crteron to make te coce of te two wegts s reported n [14]. GLOBAL MINIMUM JERK TRAJECTORY PLANNING ALGORITHM For a comparatve analyss of te expermental results, te global mnmum jerk trajectory plannng algortm (so called GMJ) presented n [5-6] as been mplemented. In ts tecnque, te executon tme of te trajectory s set a pror and te manpulator knematc constrants are not taken nto account. Moreover, cubc splnes are used to nterpolate te sequence of ponts of te geometrc pat tat s planned n off-lne mode. Te algortm can be summarzed as follows. If s te vector of te tme ntervals between two consecutve va-ponts, and defned j k, () as te value of jerk of te -t splne at te k-t jont, te optmzaton problem of te GMJ algortm s : subject to: mn + n R max { j ( ) : = 1,, n; k = 1,, m} k, n = 1 were n s te number of va-ponts, m s te number of robot jonts and T s te trajectory executon tme. Te output of te algortm s a set of tme ntervals tat mnmzes te absolute maxmum value of te jerk along te wole pat. COMPARATIVE ANALYSIS OF THE PLANNING TECHNIQUES PROPERTIES Wt te am of evaluatng te trajectores obtaned by runnng te tree tecnques above descrbed (SPL3J, BSPL5J and GMJ), a fourt algortm as been mplemented. It s based on cubc splnes (so as to ensure te contnuty of poston, velocty and acceleraton values). Te duraton of te tme ntervals between two va-ponts s proportonal to te trajectory executon tme, tat s set a pror, and te number of va-ponts (accordngly te algortm s called PROP). Te algortm so = T Table I : Man propertes of te SPL3J, BSPL5J, GMJ and PROP algortms Algortm Prmtve Trajectory tme Optmzaton Knematc Constrants SPL3J cubc splne calculated jerk-tme Yes BSPL5J quntc B-splne calculated jerk-tme Yes GMJ cubc splne mposed max jerk No PROP cubc splne mposed No (3) (4) mplemented does not take nto account te knematc constrants of te manpulator. In Table I te man propertes of te four algortms are reported. An mportant remark on te convergence tme of te four tecnques tat ave been used must be done: for all te tested trajectores, GMJ algortm gves te soluton after several ours, wereas SPL3J, BSPL5J and PROP algortms take less tan a mnute to produce te solutons. Ts drawback s very mportant f, for example, te tecnques wll be used to plan trajectores for ndustral applcatons were sort tmes of soluton are necessary. EXPERIMENTAL EVALUATION OF VIBRATIONAL PHENOMENA. IMPLEMENTED TRAJECTORIES Tree dfferent trajectores ave been mplemented n Matlab TM and ten nput to te Cartesan manpulator wt te am to test and valdate te benefts of usng smoot trajectory plannng algortms. Te target of te expermental tests s to compare te vbratonal penomena on te robot end-effector tat are nduced by te movements of ts arms after applyng te four tecnques above descrbed on te same geometrc pat. Ts means tat te trajectores va-ponts and te executon tme assocated to eac pat are te same for eac algortm. In order to reac te second target, te tree trajectores are frst smulated wt te SPL3J and BSPL5J (te values of k T and k J are set wt te am to get te same executon tme) and te executon tme so obtaned s ten nput n te GMJ and PROP algortms. In ts way, wt te same test startng condtons for te four tecnques, a more strct evaluaton of te vbratonal penomena can be conducted. Te tree trajectores are below descrbed: copyrgt FACULTY of ENGINEERING - HUNEDOARA, ROMANIA 129
Trajectory #1: te frst pat mplements a pck-and-place movement. Te k T and k J values are respectvely 860 and 0.005 for SPL3J tecnque, 10 and 1 for BSPL5J and te executon tme s 7.4 s. Te va-ponts of te trajectory are reported n Table II and n Table III te smulated mean and maxmum jerk values for eac algortm are ncluded. ANNALS OF FACULTY ENGINEERING HUNEDOARA Internatonal Journal Of Engneerng Table II : Trajectory #1 va-ponts X poston Y poston Z poston [mm] [mm] [mm] 1 0 0 0 2 0 0-170 3 10 12.9-190 4 30 38.6-200 5 175 225-200 6 320 411.4-200 7 340 437.1-190 8 350 450-170 9 350 450 0 Va-ponts Table III : Maxmum and mean jerk values for te four algortms max [mm/s 3 ] mean [mm/s 3 ] x y z x y z SPL3J 183.54 230.43 322.49 90.04 115.68 182.84 BSPL5J 178.90 231.13 376.55 74.90 96.16 178.77 GMJ 169.52 212.19 305.21 88.90 114.18 193.02 PROP 588.30 756.70 1105.20 264.32 340.06 533.65 Trajectory #2: te second example mplements a L-saped pat. Te k T and k J values are respectvely 845 and 0.005 for SPL3J tecnque, 139 and 1 for BSPL5J and te executon tme s 5 s. In Table IV te trajectory va-ponts are reported. For eac tecnque, te smulated maxmum and mean jerk values are sown n Table V. Table IV : Trajectory #2 va-ponts Va-ponts X poston Y poston Z poston [mm] [mm] [mm] 1 0 0 0 2 270 0 20 3 290 0 40 4 290 20 60 5 290 290 80 Table V : Maxmum and mean jerk values for te four algortms max [mm/s 3 ] mean [mm/s 3 ] x y z x y z SPL3J 531.75 532.01 195.93 278.86 278.22 84.96 BSPL5J 615.39 696.89 189.58 244.31 215.50 64.04 GMJ 511.99 524.35 184.15 273.33 293.09 56.49 PROP 1259.40 1259.20 34.60 474.02 473.92 23.04 Trajectory #3 : te last trajectory s a square wt fve va-ponts, wose sequence s reported n Table VI. Te k T and k J values are respectvely 1280 and 0.5 for SPL3J tecnque, 10 and 1 for BSPL5J and te executon tme s 14.5 s. Te smulated maxmum and mean jerk values for te four algortms are reported n Table VI : Trajectory #3 va-ponts Va-ponts X poston Y poston Z poston [mm] [mm] [mm] 1 10 10 0 2 330 10 0 3 330 330-170 4 10 330-170 5 10 10 0 Table VII. As mentoned before, te SPL3J and BSPL5J algortms optmze te trajectores n te sense of best trade-off between te executon tme and te ntegral of te squared jerk, wereas te GMJ tecnque mnmzes te absolute maxmum value of te jerk along te pat. Startng from tese consderatons, te lowest maxmum values of te jerk and te lowest mean jerk values are expected f te GMJ and SPL3J/BSPL5J are used respectvely. Table VII : Mean jerk values for te four algortms max [mm/s 3 ] mean [mm/s 3 ] x y z x y z SPL3J 68.96 61.95 32.91 35.39 32.38 17.26 BSPL5J 86.44 69.30 36.82 30.25 27.66 14.70 GMJ 49.42 57.28 30.43 32.35 34.11 18.12 PROP 82.59 82.59 43.88 38.87 38.87 20.65 If Tables III, V and VII are consdered, t s possble to fnd a confrmaton to te above antcpatons: te GMJ algortm provdes te lowest maxmum jerk values f compared wt te oter tree tecnques, wle SPL3J and BSPL5J feature te lowest mean jerk values. Ten, t s possble to verfy tat te PROP metod s te worst n terms of bot te mean and te maxmum jerk values. EXPERIMENTAL SET-UP Te expermental tests, amed to evaluate te vbraton penomena durng te executon of te tree trajectores planned wt te four algortms, are made on a Cartesan manpulator (Fgure 1), controlled usng a real tme external controller. Te 3-d.o.f. manpulator as tree prsmatc jonts, wose knematc bounds are sown n Table VIII, a workspace of 500x600x500 mm (X, Y and Z) and an accuracy of 0.1 mm. Te jonts are actuated by means of brusless servo-motors, coupled wt te robot arms by usng a cogged belt and equpped wt resolver poston sensors. Eac motor s lnked to te transmsson belt by a reducton gear ead. 130 Tome IX (Year 2011). Fasccule 1. ISSN 1584 2665
An embedded multfuncton board, te Sensoray S626, s used n order to realze a lnk between te manpulator and te external control loop. Te poston real tme controller s set up on an AMD Atlon(tm) XP 2400 (1.99 GHz wt 480 MB of RAM memory) by means of te xpc Target TM toolbox of Matlab TM. In order to evaluate te vbraton penomena of te robot durng te movements of ts arms nduced by te planned trajectores, a un-axal accelerometer s mounted on te end-effector. Te devce as a maxmum value of acceleraton of ±5g and an accuracy of 1036 mv/g. It s mportant to empasze te fact tat te evaluaton of te vbraton penomena s only focused on te performance of te four trajectory plannng algortms, snce te performances of te real tme controller are not consdered as fundamentals for te expermental tests. Startng from ts assumpton, te only nputs tat can be canged n a smulated ndustral task are te trajectores parameters, n good accordance wt te condtons found n ndustral envronments, were generally a user s not allowed to cange te parameters of te macne controller. Jont Table VIII : Knematc bounds of te Cartesan manpulator Knematc Bounds Velocty Acceleraton Jerk [mm/s 3 ] [mm/s] [mm/s 2 ] 1 225 700 2400 2 225 700 2400 3 225 700 2400 Fgure 1: Cartesan manpulator used for testng te trajectory plannng algortms EVALUATION OF THE TRAJECTORIES SMOOTHNESS Te smootness of te tree trajectores planned wt te four algortms s expermental tested by means an accelerometer mounted on te robot end-effector. Te drecton used to measure te vbraton of te manpulator as been cosen by takng nto account te mean values of te smulated jerk along te pat. By consderng ts assumpton and te Tables III, V and VII, te X cartesan drecton as been cosen for trajectores #2 and #3, wereas te Z cartesan drecton as been cosen for trajectory #1. Table IX : Measured acceleratons mean value Acceleratons mean value [m/s 2 ] SPL3J BSPL5J GMJ Trajectory #1 0.12 0.12 0.13 Trajectory #2 0.40 0.43 0.37 Trajectory #3 0.48 0.55 0.47 Fgure 2 : Smulated vs. measured acceleraton (Trajectory #1 - SPL3J) Fgure 3 : Smulated vs. measured acceleraton (Trajectory #1 - BSPL5J) Fgure 4 : Smulated vs. measured acceleraton Fgure 5 : Smulated vs. measured acceleraton (Trajectory #1 - GMJ) (Trajectory #1 - PROP) In Table IX te mean values of te measured acceleratons are reported. If te PROP values are consdered as reference, a mean mprovement of 36% s obtaned f SPL3J and GMJ tecnques are consdered, a mean mprovement of 31% s obtaned f BSPL5J algortm s used. copyrgt FACULTY of ENGINEERING - HUNEDOARA, ROMANIA 131
Te comparson between te four trajectory plannng algortms for all te pats mplemented, confrms te effectveness of te SPL3J, BSPL5J and GMJ tecnques n reducng te vbratons f compared to te PROP metod. All te expermental tests demonstrate tat te real beavor of te Cartesan manpulator s effectvely represented by te smulatons, snce te smulated acceleratons obtaned by runnng te algortms and nput to te manpulator ave a tme course comparable wt te acceleratons measured by te accelerometer mounted on te end-effector. To confrm ts, n Fgures 2-5 a comparson between te smulated and te measured acceleratons (for Trajectory #1) s reported. CONCLUSION In te present paper a mnmum tme-jerk trajectory plannng tecnque as been expermental evaluated and valdated. Ts metod tat takes nto account bot te ntegral of te squared jerk along te trajectory and ts executon tme, s mplemented by usng two types of prmtves: cubc splnes (SPL3J) and fft-order B-splnes (BSPL5J). Te knematc constrants are consdered n te optmzaton problem, and te executon tme s not set a pror. An accelerometer mounted on te robot end-effector as been used wt te am to measure te acceleratons of te manpulator jonts, n order to evaluate te vbraton penomena of te Cartesan robot. Tree test-trajectores ave been mplemented on a Cartesan manpulator and te expermental results ave been compared wt te results obtaned wt a global mnmum jerk (GMJ) metod, one of te most popular for plannng smoot trajectores, and wt a classc splne algortm. Te outcomes of te tests demonstrate te effectveness of te smoot trajectory plannng tecnques, snce te results prove te reductons of te vbraton penomena of te robot arms durng te trajectory executon. REFERENCES [1.] T. BROGARDH, Present and future robot control development An ndustral perspectve, Annual Revews n Control, 31 (1) (2007), 69-79. [2.] A. PIAZZI and A. VISIOLI, Global mnmum-tme trajectory plannng of mecancal manpulators usng nterval analyss, Internatonal Journal of Control, 71 (4) (1998), 631 652. [3.] K. JOONYOUNG, K. SUNG-RAK, K. SOO-JONG, K. DONG-HYEOK, A practcal approac for mnmum-tme trajectory plannng for ndustral robots, Industral Robots : An Internatonal Journal, 37 (1) (2010), 51 61. [4.] G. FIELD and Y. STEPANENKO, Iteratve dynamc programmng: an approac to mnmum energy trajectory plannng for robotc manpulators, Proc. of te IEEE Internatonal Conference on Robotcs and Automaton, 3 (1996), 2755 2760. [5.] A. PIAZZI, A. VISIOLI, An nterval algortm for mnmum-jerk trajectory plannng of robot manpulators, Proceedngs of te 36t Conference on Decson and Control, (1997), 1924 1927. [6.] A. PIAZZI, A. VISIOLI, Global mnmum-jerk trajectory plannng of robot manpulators, IEEE Transactons on Industral Electroncs, 47 (1) (2000), 140 149. [7.] H.XU, J.ZHUANG, S.WANG and Z.ZHU, Global Tme-Energy Optmal Plannng of Robot Trajectores, Proc. of te Internatonal Conference on Mecatroncs and Automaton, (2009), 4034 4039. [8.] D. SIMON and C. ISIK, A trgonometrc trajectory generator for robotc arms, Internatonal Journal of Control, 57 (3) (1993), 505 517. [9.] P. HUANG, Y. XU and B. LIANG, Global mnmum-jerk trajectory plannng of space manpulator, Internatonal Journal of Control, Automaton and Systems, 4 (4) (2006), 405 413. [10.] K. PETRINEC and Z. KOVACIC, Trajectory plannng algortm based on te contnuty of jerk, Proc. of te 15t Medterranean Conference on Control & Automaton, (2007). [11.] A. GASPARETTO and V. ZANOTTO, A new metod for smoot trajectory plannng of robot manpulators, Mecansm and Macne Teory, 42 (4) (2007), 455 471. [12.] A. GASPARETTO and V. ZANOTTO, A tecnque for tme-jerk optmal plannng of robot trajectores, Robotcs and Computer-Integrated Manufacturng, 24 (3) (2008), 415 426. [13.] F. LOMBAI and G. SZEDERKENYI, Trowng moton generaton usng nonlnear optmzaton on a 6-degree-offreedom robot manpulator, Proc. of IEEE Internatonal Conference on Mecatroncs, (2009). [14.] A. GASPARETTO, A. LANZUTTI, R. VIDONI and V. ZANOTTO, Trajectory plannng for manufacturng robots: algortm defnton and expermental results, Proc. of ASME 2010 10t Bennal Conference on Engneerng Systems Desgn and Analyss ESDA2010, (2010). 132 Tome IX (Year 2011). Fasccule 1. ISSN 1584 2665