Optimization the Trajectories of Robot Manipulators Along Specified Task

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1 Receved: June 4, Optmzaton the Trajectores of Robot Manpulators Along Specfed Task Bendal Nadr 1,2 * Oual Mohammed 1 1 Structural Mechancs Research Laboratory, Department of Mechancal Engneerng, Unversty Blda-1, Algera 2 Faculty of Scence and Technology, Unversty of Khems Mlana, Road of Thenet El Had, Khems Mlana, 44225, Algera * Correspondng author s Emal: nadr_b1102@yahoo.com Abstract: Ths paper proposes a procedure for obtanng an optmal trajectory for robot manpulators along specfed tasks (spot weldng, drllng...). The purpose s to obtan smooth trajectores wth mnmum tme usng splne cubc functons under a varous knematc and/or dynamc constrants, takng nto account a full dynamc model of robot manpulator and wth a large number of va ponts. Ths procedure s based on the optmzaton by a genetc algorthm of a vector whch represents only a tme ntervals n order to fnd the best objectve functon. Two examples llustrated wth numercal detals for a two-lnk planar arm and sx-lnk manpulators, for the former, comparsons wth an alternatve optmzaton solver are exposed. Keywords: Splnes cubc, Imposed task, Trajectory plannng, Genetc algorthms. 1. Introducton The use of robot manpulators n the ndustry s often devoted to repettve tasks that execute at a sustaned pace. It s, therefore, natural that the productvty of ndustral processes closely dependent on the way to use these robots. In ths case, the rapdty of the robot can favour an effcency factor and consderable economc gan. The choce of the qualty of the robot's movements appears as a necessty, partcularly durng rapd transfer of objects, so for example, sortng or packng operatons on producton lnes. Moreover, there are several possbltes for the mplementaton of the same task; t would be wse to take advantage of the multtude of choces to opt for the least expensve soluton n terms of a predefned crteron. One way to acheve ths s to formulate the problem as an optmzaton problem. The work presented n ths paper s nterested n mposed task, n ths knd of trajectory the end of the chan tool nfluences on ts envronment wthout nterrupton and on a desred track. It s necessary n ths case to specfy the trajectory of the effector n Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 the operatonal space. Ths s the case, for example, cuttng work or the contnuous weldng. Several research projects were dedcated to the study the trajectory plannng problems of robot manpulators n the mposed tasks, we cte n ths context the papers of [1] where the authors proposed a fast and unfed approach based on partcle swarm optmzaton (PSO) wth K-means clusterng to solve the near optmal soluton of a mnmum-jerk jont trajectory constraned by a fxed traverse tme of robot manpulator, the cubc splnes were used to nterpolate between the nodes of the trajectory n an mposed tasks. In [2] used sequental unconstraned mnmzaton technques (SUMT) to do optmum trajectory plannng of a STANFORD manpulator, the cost functons used n optmal trajectory plannng are mnmum travelng tme, the mnmum mechancal energy of the actuators and mnmum penalty for obstacle avodance. The SQP method and ntal value fndng algorthm are used n the works of [3, 4]. The jerk mnmzaton s appled when the objectve functon of the optmzaton problem and the travelng tme are fxed, ther results show the man dsadvantage of SQP DOI: /jes

2 Receved: June 4, technque as t fnds a global mnmum only f the sutable ntal value s set. In [5] proposes a mnmum-tme trajectory of robot arm whch uses a clamped cubc splnes for modellng the trajectory and Harmony Search (HS) method for solvng the optmzaton problem, mnmum travelng tme trajectory plannng and consderng jont knematc constrants are essental n optmal trajectory plannng problems, ths s because these constrants are typcally concerned about ndustrals productvty. In [6, 7] the authors has been descrbed an expermental valdaton of the mnmum tmejerk trajectory plannng algorthm, the trajectores have been mplemented on Cartesan 3-axes manpulator equpped wth a pezoelectrc accelerometer, the obtaned expermental results have been dscussed by consderng the measure of the acceleraton (drectly related to the vbraton nduced on the mechansm) as the comparson parameter. In [8] a sem-nfnte optmzaton s proposed; the mnmum-tme trajectory s planned ncludng the knodynamc constrants where a hybrd technque usng genetc algorthms s put forward. In [9] the authors developed an approach based on fuzzy genetc algorthm usng real codng and eltsm approach to treat the problem of trajectory plannng of robotcs arm along specfed tasks to mnmze tme-jerk by consderng the knematc constrants, ths same problem s treated by the authors of [10] where an approach based on vector of tme ntervals s developed, the results of mnmum tme-jerk trajectory are satsfactory to solve the problem between hgh producton effcency and low structure vbraton of robotc arms. The am of ths study s to determne a smple and effcent approach to solve the problem of the trajectores plannng for robot manpulators along the mposed tasks the advantages of ths approaches are: Frst ts flexblty to deal wth the problems of dfferent robot manpulators wth a large number of degrees of freedom and complex mposed spots (many ntermedate ponts). The treatment of the constrants (geometrc, knematc and dynamc) s done by order of dffculty and wth a systematc way to reduce the proporton of the rejects durng the random selecton. By usng parametrc functons, t s possble to stablze the trajectory (stress mechancal stresses) and at the same tme to guarantee contnuty up to thrd order (Jerk). The dfferent problems studed have shown, on the one hand, the versatlty of ths approach and, on the other hand, ts effectveness n terms of the qualty of results by comparng t wth other technques proposed n [4, 8, 11]. Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 Ths paper s organzed as follows: secton 2 presents the problem of the trajectory plannng. Secton 3 presents a knematc and dynamc equaton of robot manpulators used n trajectory plannng. Secton 4 presents the parametrc functons used to model the trajectores. To make ths method complete and usable, we can see n secton 5 an approprate optmzaton technque. In Secton 6, two numercal examples.e. the two-lnk planar robot SCARA IBM7535 B04 and robot wth the sx degrees of freedom (6R) are presented to prove the effectveness of the proposed approach. Fnally, the work s concluded n Secton Problem statements The planar robot wth two degrees of freedom rotoïde and the robot manpulator wth sx degrees of freedom are consdered, the task s to move the robots n the workspace along a specfed path whle mnmzng an objectve functon. The dynamc model of the robots s derved usng Lagrange s energy functon and Euler-Lagrange s equatons. The equatons of moton nclude the nerta terms of the actuators and frcton forces. The objectve functon presents the tme nterval h between two va-knots of the trajectory and the optmzaton problem s defned as follows: Mnmze: F obj n1 h (1) 1 The subjects are: Jont veloctes: V j(t) V max j for j=1, n and =1, m-1 Jont acceleratons: W j(t) W max j for j=1, n and =1, m-1 Jont jerks: J j(t) max J j for j=1, n and =1, m-1 max Jont Cartesan velocty: cv j(t) cv j for j=1, n and =1, m-1 Jont angular velocty: av j(t) av max j for j=1, n and =1, m-1 Where V j max, W j max, J j max, cv j max and av j max denote the lmt knematcs performances of velocty, acceleraton, jerk, Cartesan velocty and angular velocty respectvely of the j-th jont deduced from technologcal and desgn data. DOI: /jes

3 Receved: June 4, Dynamc model The dynamc equatons for conservatve systems n terms of the Lagrangan functon are gven by: d L L dt q& q (2) The Lagrangan functon L s defned as the dfference between the knetc energy K and the potental energy P of the system: L K P (3) W ( t ) W ( t ) V t t t t t () 1 2h 2h q 1 ( 1) ( ) hw t q hw t h 6 h 6 And: W ( t ) W ( t ) Q t t t t t () 1 6h 6h q 1 ( 1) ( ) hw t q hw t ( t t) ( t 1 t) h 6 h 6 (6) (7) Where: q :generalzes coordnates of the system (Q for rotatonal jonts and d for prsmatc jonts). q : generalzed veloctes (angular velocty Ɵ for rotatonal jonts and lnear velocty d for prsmatc jonts). τ :generalzed forces. The generalzed forces τ are gven by [2] as: n n j j jk j k j1 1 k 1 j D q& C q& q& G (4) Where D j s the nertal system matrx, C jk s the Corols and centrpetal forces matrx, G s the gravty loadng vector and q s generalzed acceleratons. 4. Formulaton of the trajectory wth Cubc Splnes The cubc splne s a pecewse 3 rd degree polynomal functon. The veloctes and acceleratons of the ntal and termnal condtons (v 1, v m, a 1 and a m) are specfed to be zero. These condtons cause two equatons of the splne algorthm becomng zero and the path pattern cannot be solved. Therefore, two extra knots (poston values at tme t 2 and t m) are added and ther poston values are not specfed. Let Q (t) be the cubc polynomal for the -jont n the nterval tme [t, t +1]. The second dervatve of Q (t) s a lnear nterpolaton and can be wrtten as [2]: t t t t W t W t W t 1 ( ) ( ) ( 1) h h (5) Where: h =t +1-t Integratng Eq. (5) for the gven ponts Q (t )=q and Q (t +1)=q +1, the followng nterpolaton functons are obtaned: Usng the contnuty condtons on veloctes and acceleratons, a system of m-2 lnear equatons solvng for m-2 unknowns W (t ), =2,3,, m-1 s obtaned as: A W2 ( t2) W3 ( t3)... W 1( 1) T m tm B (8) In (8), the matrx A s non-sngular matrx and entres of the matrx B are changed for each jont. Then, the extra knots poston values can obtan from: 2 2 h1 h1 q2 q1 h1v 1 a1 W2 ( t2) (9) hm 1 hm 1 qm 1 qm hm 1vm am Wm 1( tm 1) (10) Optmzaton procedures 5.1 Intalzaton and route of generaton Let h be defned as the vector of desgn varables h = [h 1, h 2,, h m-1]. To ntalze the optmzaton process t s consdered that: ( 0 ) j2 j1 j2 j1 jm j,m1 h max,,..., j 1,...,m 1 max max max Vj Wj J j q q V V W W (11) As two extra knots are needed they are ntally taken as: q j2 ( q j1 q j3) / 2 and j, m 1 j, m 2 jm q ( q q ) / 2 (12) Durng the optmzaton process the ntermedate knots and the end pont wll generate only horzontally as seen n Fg.1, consequently the Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 DOI: /jes

4 Receved: June 4, Fgure. 1 Representaton of the Cubc Splne Trajectory wth the horzontal movement of the ntermedate knots and the end pont after generaton trajectory changes and moves also horzontally by mnmzng the objectve functon and obtanng the best vector h opt. 5.2 Genetc algorthms In order to make ths approach usable, t s requred to use an approprate optmzaton technque to solve the objectve functon presented prevously n Eq. (1), we propose to use a genetc algorthms GAs (Fg. 2). GAs are stochastc optmzaton algorthms based on the mechansms of natural selecton and genetcs. Ther operaton s extremely smple. We start wth a populaton of arbtrarly chosen potental solutons (chromosomes). Ther relatve ftness (ftness) s evaluated. On the bass of these performances, a new populaton of potental solutons s created usng smple evolutonary operators: Selecton rules select the ndvduals, called parents, whch contrbute to the populaton at the next generaton. Crossover rules the creaton of new ndvduals or chldren accordng to a very smple process for the next generaton. Mutaton rule s to randomly alter wth a certan probablty the value of a component of the ndvdual parent to form chldren. Fgure. 2 Flowchart of the genetc algorthms Ths cycle s repeated untl a satsfactory soluton s found. There are four man ponts that make the fundamental dfference between these algorthms and other methods: GAs Use a codng of the parameters, not the parameters themselves. GAs Work on a populaton of ponts, nstead of a sngle pont. GAs Use only the values of the objectve functon studed, not ts dervatve, or another auxlary knowledge. GAs select the next populaton Usng probablstc computatons, not determnstc ones. 6 Smulaton results and comparson The smulaton results by cubc splne compare our procedure wth two dfferent technques. For ths, two examples are presented n ths secton; the frst one concerns a planar 2R robot, t has been manly suggested to compare the results obtaned wth our procedure wth those obtaned wth the genetc /nterval algorthm used n [8]. The second example consders an actual manpulator 6R degrees of freedom, the results obtaned are compared wth those obtaned wth HHSA method used n [10]. The objectve functon taken n these smulatons s Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 DOI: /jes

5 Receved: June 4, presented by the Eq. (1). The purpose s to prove that the proposed procedure can be frutfully used wth non-trval problems. Notng that, durng the optmzaton process, the probablty of crossover s fxed to 65% and the probablty of mutaton to 4%. 6.1 Example 1: SCARA Two d.o.f planar robot The robot consdered n ths example s planar wth two degrees of freedom, the task s to plan under torque and veloctes constrants a trajectory n Cartesan path; the parameters nertal and dynamc constrants are summarzed n Table 1 and the lmt knematcs performance are the lnear and angular veloctes of the end-effector, whch are defned n (13) and (14) successvely. The robot s asked to move from a gven va ponts reported n Table 2. Moreover, we consder the arm at rest n the ntal and fnal postons ( q& nt q& fn 0; q& nt q& fn 0). (a) cv l q& l ( q& q& ) 2 l l q& ( q& q& )cos( q ) 0.7 m / s (13) av & 1 & z q q2 1.5 rd / s (14) (b) Table 1. Parameters nertal and dynamc constrants of 2R robot. Lnk 1 Lnk 2 M (kg) L (m) Torques (N. m) Table2. Values of the va-pont of the trajectory of 2R robot Va-ponts (Rad) Jont Extra knots Extra knots (c) (d) Fgure. 3 Results of SCARA 2R robot: (a) jonts angles, (b) jonts veloctes, (c) jonts acceleratons and (d) torques jonts Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 DOI: /jes

6 Receved: June 4, (a) volate one of the knodynamc constrants wll be automatcally rejected. The Fg. 4-b shows the optmal trajectory n Cartesan space (XY) of Scara 2R robot whch performs an mposed task. The mnmzaton evaluated by our algorthm gves a vector of tme ntervals h = [0.0031, , , , , , , , , , ] and the estmated global mnmum travelng tme s 11 1 h =2.0356sec, the soluton found by genetc/nterval algorthm used n [8] gves 11 1 h = sec. 6.2 Example 2: a sx d.o.f robot (b) Fgure. 4 Cartesan and Angular veloctes of Endeffector (a) and optmal trajectory n Cartesan space (b) of SCARA 2R robot The profles results, of each jont mnmum tme of smooth trajectory modelled wth the cubc splne functon and ther dervatves, jonts veloctes, jonts acceleratons and torques are shown n Fg. 3, the trajectory was gven by ten knots postons and two extra knots, to represent the trajectory and solve these extra knots we use both Eq. (9) and Eq. (10), the crcles and the crosses presented n jonts angle ndcate the knot postons and the two dummy knots respectvely, the dynamc constrants have been respected and the jont 1 torque presents a segment of broad tme saturaton. In addton Fg. 4-a shows the Cartesan and Angular veloctes of End-effector, the knematc constrants appled n the end-effector were respected. It should be mentoned that ths approach treats all the constrants n a sequental way and by order of dffculty; the knematcs and dynamcs constrants by the ncluson of a full manpulator dynamc model and any trajectory whch would Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 In ths example, we wll treat a robot manpulator wth sx degrees of freedom ts structure contans sx artculatons pvot as studed n [11]. The knot postons (va-ponts) and knematcs constrants of the 6R robot manpulator jonts system are shown n Table 3 and Table 4 respectvely. Each smooth jont of cubc splne trajectores ncludng ther dervatves of veloctes, acceleratons and jerk are llustrated n Fg. 5. The optmal trajectory wth mnmum tme was gven by four knots postons and fve tme ntervals for all the sx jonts, by applyng our approach the knematcs lmtaton constrants consdered n the optmzaton process are satsfed for the 6 DOF and n a sequental way, the veloctes constrants, the acceleratons constrants and jerk constrants Fg. 6, shows the smulaton of robot manpulator 6R n 3D Cartesan coordnates (XYZ), where the optmal trajectory found n ths mposed tasks s descrbed between the va-ponts. Notng that, durng the optmzaton process we have fxed the generaton number to 80, the results hstores of the generaton are shown n the bottom of Fg. 7, these results approve that our approach converges rapdly toward the optmal soluton. Table3. Knot postons of each robot manpulator jont Jont 1-10 Knot (degrees) Extra Extra knot knot DOI: /jes

7 Receved: June 4, (a) (b) (c) (d) (e) (f) Fgure. 5 Results of robot manpulator 6R wth (a) to (f) represent respectvely jonts 1 to 6 under knematcs constrants Where: represent respectvely poston, velocty, acceleraton and jerk. The crcles ndcate the knot postons and the crosses ndcate the two dummy knots Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 DOI: /jes

8 Receved: June 4, Table4. Knematcs constrants of each of robot jont Jont Veloctes (deg/s) Acceleraton (deg/s 2 ) Jerk (deg/s 3 ) (a) (b) The mnmzaton of objectve functon Eq. (1) evaluated by our algorthm gves a vector of tme ntervals h = [0.7289, , , , ] and equal to executon tme 5 1 h=8.5298sec. The mnmum-tme s the most sgnfcant data to compare the technques used to fnd the optmal smooth trajectory plannng of robots manpulator along specfed tasks. The results show that the approach descrbed n ths paper can obtan the bes soluton compared wth the prevous related lteratures ncludng [4] whch used a Sequental Quadratc Programmng (SQP) technques whch obtaned the objectve functon equal to 5 1 h =85726 sec, and [11] whch used a Hybrd Harmony Search Algorthm (HHSA) whch obtaned the objectve functon equal to h = sec. 5 1 (c) Fgure. 6 Trajectory results of robot manpulator 6R: (a) among knot 1 and 3, (b) among knot 1 and 4 and (c) among knot 1 and 6 wth knematcs constrants Fgure. 7 Hstores generaton results of robot manpulator 6R 7. Concluson Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 In ths paper, a procedure of study the trajectory plannng problem s suggested for robot manpulators along specfed paths, the trajectores uses cubc splnes functons whch guarantee the smoothng of the trajectory and at the same tme guarantee the contnuty of veloctes, acceleratons and jerk wth an nterpolaton of the gven va ponts (plus two dummy knots). Ths procedure s based on the optmzaton a vector of tme ntervals and t makes possble to treat all the constrants of ths problem n a sequental way and by order of dffculty (the geometrcal, knematcs and torque constrants) by the ncluson of a full manpulator dynamc model. Moreover, the tme effcency guaranteed by ths approach permts to solve the problems wth a relatvely large number of va ponts and degrees of freedom of robot manpulators; as well, our proposed method converges quckly and has a faster calculaton speed. In consequence, the proposed optmzaton strategy of the trajectory along mposed tasks s feasble. DOI: /jes

9 Receved: June 4, Ths work opens the door for further nvestgatons such as usng the B-Splne functons or Non- Unform Ratonal B-Splne (NURBS) functons, takng nto account obstacles n a workspace and a mnmzaton of mult-objectve functons for robots arms n mposed tasks, so as to evaluate the applcablty the proposed method and ts results n other stuatons. References [1] H. Ln, A fast and unfed method to fnd a mnmum-jerk robot jont trajectory usng partcle swarm optmzaton, Journal of Intellgent & Robotc Systems, Vol.75, pp , [2] S.F.P. Saramago and V. Steffen Jr, Trajectory modelng of robot manpulators n the presence of obstacles, Journal of Optmzaton Theory and Applcatons, Vol.110, No.1, pp.17-34, [3] A. Gasparetto and V. Zanotto, A new method for smooth trajectory plannng of robot manpulators, Mechansm and Machne Theory Vol.42, pp , [4] A. Gasparetto and V. Zanotto, A technque for tme-jerk optmal plannng of robot trajectores. Robotcs and Computer-Integrated Manufacturng, Vol. 24, pp , [5] P. Tangpattanakul and P. Artrt, Mnmum- Tme Trajectory of Robot Manpulator Usng Harmony Search Algorthm, In: 6th Internatonal Conference on Electrcal Engneerng/Electroncs, Computer, Telecommuncatons and Informaton Technology, Chonbur, Thaland, [6] V. Zanotto, A. Gasparetto, A. Lanzutt, P. Boscarol, and R. Vdon, Expermental valdaton of mnmum tme-jerk algorthms for ndustral robots, Journal of Intellgent & Robotc Systems, Vol.64, pp , [7] A. Gasparetto, A. Lanzutt, R. Vdon, and V. Zanotto, Expermental valdaton and comparatve analysts f tme-jerk algorthms for trajectory plannnh, Robotcs and Computer- Integrated Manufacturng, Vol.28, pp , [8] C. Guarno Lo Banco, and A. Pazz, A semnfnte optmzaton approach to optmal splne trajectory plannng of mechancal manpulators. n: M.A. Goberna, M.A. Lopez (Eds.), Semnfnte Programmng: Recent Advances, Sprnger-Verlag, Berln, pp , [9] M. Cong, X. Xu, and P. Xu, Tme-jerk synthetc optmal trajectory plannng of robot based on fuzzy genetc algorthm, Int J Intellegent Systems Technologes and Applcatons, Vol.8, pp1-4, [10] N. Bendal and M. Oual, On-lne trajectory plannng of tme-jerk optmal for robotc arms, Medterranean Journal of Modelng and Smulaton, Vol.6, pp.34-44, [11] P. Tangpattanakul, A. Meesomboon, and P. Artrt, Optmal Trajectory of Robot Manpulator Usng Harmony Search Algorthm, Recent Advances n Harmony Search Algorthm, the seres Studes n Computatonal Intellgence, Sprger-Verlag,Berln,Vol.270, pp.23-36, Internatonal Journal of Intellgent Engneerng and Systems, Vol.11, No.1, 2018 DOI: /jes