Optimal Motion Planning For a Robot Arm by Using Artificial Bee Colony (ABC) Algorithm

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1 Internatonal Journal o Modern Engneerng Research (IJMER) wwwjmercom Vol, Issue 5, Sep-Oct 01 pp--8 ISSN: Optmal Moton Plannng For a Robot Arm by Usng Artcal Bee Colony (ABC) Algorthm PV Savsan, 1 R L Jhala, 1 (Research Scholar PAHER Unversty, Udapur, Rajasthan, Inda) (Proessor, Marwad Educaton Foundaton, Dept o Mech Engg,Gujarat, Inda) Abstract: Ths work s concentrated to optmze the trajectory or planer two-lnk robot The whole travel o the trajectory s dvded nto two parts, whch conssts o ourth order polynomal trajectory or the one part and th order trajectory or the second part There are many optmzaton algorthms, whch can be mplemented to solve such problems Evolutonary algorthms are also tred by many researchers to solve such trajectory optmzaton problems Evolutonary algorthms are capable o gvng global optmum soluton and tradtonal optmzaton technques converge to local optma and so they are not sutable or the trajectory problems In ths work artcal bee colony (ABC) algorthm s mplemented to solve trajectory optmzaton problem The objectve uncton or the proposed ABC s to mnmze travelng tme and space, whle not exceedng a maxmum pre-dened torque The objectve uncton consst o our parameter, excessve drvng torque, total jont travelng dstance, total jont Cartesan length and total consumed tme or robot moton Keywords: Artcal bee colony algorthm, R robotc arm, Trajectory I INTRODUCTION Many researches on optmal trajectory plannng have been reported n ree workspace as well as n obstacle exstence work space The evolutonary algorthm s extensvely used as practcal optmzaton computaton methods, because o the advantage that ths algorthm can avod local optmum value Ths work ocused on the trajectory optmzaton o R robotc arm n ree work space as well as crcular obstacle exstence robot work space usng ABC Forth order and th order polynomals are used to descrbe the segments that connect ntal, ntermedate and nal pont at jont space In last year s, evolutonary algorthms have been appled n large number o elds An optmal gallopng trajectory proposed by Gju Chae and Jong Hyeon Park[] whch cost low energy and guarantees the stablty o the quadruped robot They optmzed trajectory based on energy and stablty usng GA, whch provdes a robust and global soluton to a mult-body, hghly nonlnear dynamc system For generatng smooth trajectory plannng or speced path, Zoller and Zentan [] ocused on the problem o the trajectory plannng and dealt wth constant knetc energy moton plannng Ths method produced trajectory characterstcs smoother and better than whch dd obtaned rom tme optmal method Zhe Tang et al [6] proposed a thrd order splne nterpolaton based trajectory-plannng method to plan a smooth bped swng leg trajectory by reducng the nstant velocty change Chwa et al [7] proposed Mssle Gudance Algorthm to generate on lne trajectory plannng o robot wwwjmercom arms o the ntercepton o a ast maneuverng object The authors employed the gudance law throughout the trackng phase, and dynamc constrants such as torque and velocty constrants and satsed the matchng condton o the poston and velocty at the tme o the ntercepton altogether Garg and M Kumar [8] use GA technques or robot arm to denty the optmal trajectory based on mnmum jont torque requrements The authors use polynomal o th degree n tme or trajectory representaton to jont space varables Pres and Machado [10] propose a path plannng method based on a GA whle adoptng the drect knematcs and the nverse dynamcs The optmal trajectory s the one that mnmze the path length, the rpple n the tme evoluton and the energy requrements, wthout any collson wth the obstacle n the workspace S G Yue et al [] ocused on the problem o pont-to-pont trajectory plannng o lexble redundant robot manpulator (FRM) n jont space The proposed trajectory to mnmze vbraton o FRMs s based on GA Pres et al [5] use genetc algorthm to optmze a planar robot manpulator trajectory The man purpose o ths paper s to present the optmum robot trajectory n ree workspace and obstacle exstence workspace by usng ABC In the rst part o the work mathematcal model s ormed In the second part optmzaton o the developed robot trajectory s done by usng ABC II ARTIFICIAL BEE COLONY (ABC) TECHNIQUE Artcal Bee Colony (ABC) Algorthm s an optmzaton algorthm based on the ntellgent oragng behavour o honey bee swarm The colony o artcal bees conssts o three groups o bees: employed bees, onlookers and scouts [9,11,1] An employed bee searches the destnaton where ood s avalable They collect the ood and returns back to ts orgn where they perorm waggle dance dependng on the amount o ood avalable at the destnaton The onlooker bee watches the dance and ollows employed bee dependng on the probablty o the avalable ood means more onlooker bee wll ollow the employed bee assocated wth the destnaton havng more amount o ood The employed bee whose ood source becomes abandoned convert nto a scout bee and t searches or the new ood source For solvng optmzaton problems the populaton s dvded nto two parts consstng o employed bees and onlooker bees An employed bee searches the soluton n the search space and the value o objectve uncton assocated wth the soluton s the amount o ood assocated wth that soluton Employed bee updates ts poston usng Equaton (1) and t updates new poston t s better than the prevous poston, e t ollows greedy selecton Page

2 Internatonal Journal o Modern Engneerng Research (IJMER) wwwjmercom Vol, Issue 5, Sep-Oct 01 pp--8 ISSN: Where v j s the new poston o employes bee, x j s the current poston o employed bee, k s a random number between (1, N(populaton sze)/) and j =1,,,Number o desgn varables R j s a random number between (-1, 1) An onlooker bees chooses a ood source dependng on the probablty value assocated wth that ood source, p, calculated usng Equaton () p v j x F j N / n1 j ( xj xkj ) Where F s the tness value o the soluton and N/ s the number o ood sources whch s equal to the number o employed bees The Employed bee whose poston o the ood source cannot be mproved or some predetermned number o cycles than that ood source s called abandoned ood source That employed bee becomes scout and searches or the new soluton randomly usng Equaton () x x j F j mn R n j j rand( 0,1)( xmax xmn ) The value o predetermned number o cycles s an mportant control parameter o the ABC algorthm, whch s called lmt or abandonment The value o lmt s generally taken as Number o employed bees Step by step procedure or the mplementaton o ABC s gven as ollows (1) () () Step 5 Calculate probablty assocated wth the derent solutons usng Equaton () Onlooker bee ollows a soluton dependng on the probablty o that soluton So more the probablty o the soluton more wll be the onlooker bee ollowng that soluton Step 6 Update the poston o onlooker bees usng Equaton (1) I the value o objectve uncton o the new soluton s better than the exstng soluton, replace the exstng soluton wth the new one Step 7 Identy abandon soluton and replace t wth the newly generated soluton usng Equaton () Step 8 Contnue all the steps rom step untl the speced number o generatons are reached III MATHEMATICAL MODELLING For the present work two degree o reedom planar robotc arm s consdered as shown n gure 1, where the endeector s requred to move rom startng pont to goal pont n ree work space as well as wthout colldng wth the obstacle n work space For the moton plannng, pontto-pont trajectory s taken whch s connected by several segments wth contnuous acceleraton at the ntermedate va pont ABC s used as optmzaton tool Step 1 Intalze ABC parameters whch are necessary or the algorthm to proceed These parameters ncludes populaton sze whch ndcates the number o employed bees and onlooker bees, number o generatons necessary or the termnaton crtera, value o lmt, number o desgn varables and respectve range or the desgn varables Step Generate random populaton equal to the populaton sze speced Each populaton member contans the value o all the desgn varables Ths value o desgn varable s randomly generated n between the desgn varable range speced Frst hal o the populaton wll consst o employed bees Each populaton member assocated wth employed bees ndcates each ood source Step Obtan the value o objectve uncton or employed bees The value o objectve uncton so obtaned ndcates the amount o nectar (ood) assocated wth that destnaton (ood source) Step Update the poston o employed bees usng Equaton (1) I the value o objectve uncton o the new soluton s better than the exstng soluton, replace the exstng soluton wth the new one wwwjmercom Fgure 1 Two degree o reedom planar robotc arm For the moton plannng, pont-to-pont trajectory s taken whch s connected by several segments wth contnuous acceleraton at the ntermedate va pont as shown n gure For a robot, the number o degrees o reedom o a manpulator s n and the number o endeectors degree o reedom s m I one wshes to be able to specy the poston, velocty, and acceleraton at the begnnng and the end o a path segment, a ourth order and a th order polynomal are used Let us assume that there s mp ntermedate va ponts between the ntal and nal ponts[1 ], 1 t a 0 a 0,, mp 1 1 t a t a t a t, () 5 Page

3 Internatonal Journal o Modern Engneerng Research (IJMER) wwwjmercom Vol, Issue 5, Sep-Oct 01 pp--8 ISSN: Where (a 0,,a ) are constants, and the constrant are gven as: o the nal conguraton (n-m) Thereore, or -lnk robot case, t used m p = 1, n = and one degree o reedom o redundancy or the nal pont, there are sx parameters to be determned IV FITNESS FUNCTION Present robot trajectory use the our parameters to meet the crtera o the robotc manpulator n ree work space All parameters are translated nto penalty unctons to be mnmzed Each parameter s computed ndvdually and s ntegrated n the tness uncton evaluaton The tness uncton adopted or evaluatng the asprant trajectores s dened as: Fgure Intermedate pont on planed trajectory 1 ot q c tt (18) a 0 1 a0 a 1T at at at a 1 1 a a 1 at at at Where T s the executon tme rom pont to pont +1 The ve unknowns can be solved as[1] The ntermedate pont (+1)'s acceleraton can be obtaned as: 1 a 6aT 1aT (10) The segment between the number mp o ntermedate ponts and the nal pont can be descrbed by th order polynomal as:, 1 ( t) b ( mp) 0 b t b t 1 b t b t Where the constrants are gven as: b 0 b t b 0 b 1T b T b T b T b 5T b 1 5, (5) (6) (8) (9) (11) (1) (7) (1) (1) 1 b 1 b T b T b T (15) 1 5b T b b 6b T 1b T 0b 5T (17) (16) In addton, these constrants specy a lnear set o sx equatons wth sx unknowns whose soluton s explaned n [1] As ormulated above, the total parameters to be determned are the jont angles o each ntermedate va pont (n m p parameters), the jont angular veloctes o each ntermedate pont (n m p parameters), the executon tme or each segment (m p +1 parameters), and the posture The optmzaton goal conssts n ndng a set o desgn parameters that mnmze accordng to the prortes gven by the weghtng actors β ( = 1,, ), where each derent set o weghtng actors must results n a derent soluton For ths work the weght actors are, 1,,, 1,,,1 The ot ndex represents the amount o excessve drvng, n relaton to the maxmum torque τ max, that s demanded or the th jont motor or the trajectory The ndex q represents the total jont travelng dstance o the manpulator, The ndex c represents total Cartesan trajectory length and The ndex t T represents the total consumed tme or robot moton All our ndex are calculated as gven n [1] For obstacle exstence workspace, obstacle avodance objectve uncton ob has been combned wth ree space tness uncton to orm over all tness uncton, as shown below: / ob (0) By ob, the robot manpulator has the ablty to avod the obstacle collson durng ts movement rom pont to pont n sde the workspace and t s calculated as [1] For the present work sx parameter are optmzed : q1, q, q, q1, q, q t1, t Where q and q are ntermedate jont angle and velocty or th jont respectvely, t 1 s executon tme rom ntal to ntermedate va pont, and t s executon tme rom ntermedate to nal pont Lmts o all the varables are as ollows: q / q / 01 t 8 q g () () (1) () wwwjmercom 6 Page

4 Internatonal Journal o Modern Engneerng Research (IJMER) wwwjmercom Vol, Issue 5, Sep-Oct 01 pp--8 ISSN: Implementaton o ABC For the case study, robotc arm movement s consdered as moton o human hand, whch start at (x=0 m, y= -076 m, ) and get nal poston (x= 076 m, y= 0) Varous parameters or the llustraton are taken as: Length o lnk 1 =008 m,length o lnk =008 m, Mass o lnk 1 and = 175 gms For ABC maxmum generaton =000 and populaton= 10 The man target here s to mnmzng travelng tme and space, whle not exceedng the predened torque(jont-1=5 Nm, jont-=0nm),n ree workspace and wthout collson wth any obstacle Fnally results o ABC are dscussed V RESULTS AND DISCUSSION Trajectory o R robot s optmzed usng ABC Table-1 shows total travelng tme, total jont travelng dstance and total cartensan trajectory length or both ree work space and obstacle avodance work space Fgure jont angle varaton wth respect to tme Table 1 Comparson o optmum results obtaned n ree workspace and obstacle exstence workspace Result value Free workspace Obstacle exstence Total travelng 0106 workspace 1078 tme(sec) Total jont travelng dstance(rad) Total Cartesan 10 6 trajectory length(m) Ftness value As noted rom the table 1 the values o all the parameters are more n obstacle exstence workspace The amount o ncrement allows the robot to manpulate n the work space wthout colldng wth obstacle For the optmum result uncton evaluaton s taken as 0000 or both ree and obstacle exstence workspace Fgure shows the varaton o jont angle wth respect to tme n ree robot workspace Dotted lne represents the varaton o jont angle n ree workspace whle ull lne shows the angle varaton n obstacle exstence workspace Jont 1 angle vares rom 0 to ~ -16 rad n sec when the arm moves n ree workspace, whle t takes more than sec n obstacle exstence workspace to cover the same range o angle Jont angle s almost zero when the robot arm moves n ree workspace and varaton o 0 to ~15 rad n s obtaned when t moves n obstacle exstence workspace Fgure torque varaton wth respect to tme Fgure 5 tness uncton wth uncton evaluton wwwjmercom 7 Page

5 Internatonal Journal o Modern Engneerng Research (IJMER) wwwjmercom Vol, Issue 5, Sep-Oct 01 pp--8 ISSN: Fgure depcts that or both the workspace (ree and obstacle exstence) torque or jont 1 and jont are wthn the predened lmt Torque o jont 1 vares rom 0 to ~ Nm and jont vares rom 0 to ~ 1 Nm when robot arm manpulate n ree workspace over sec Manpulaton o robot arm n obstacle exstence workspace gves the torque varaton or jont 1( 0 to ~0 Nm) and jont ( ~ -6 Nm to ~1Nm) over 1 sec Ftness value varaton wth ree and obstacle exstence workspace s shown n gure 5 Ftness value n obstacle exstence workspace s more than the tness value n ree workspace VI CONCLUSIONS Trajectory plannng method based on ABC wth specc objectve unctons s presented Case study o human moton s taken or the R planner robotc arm and trajectory s optmzed n ree and obstacle exstence workspace Comparson shows that tness value or the ree workspace s lesser than value n obstacle exstence workspace The jont torque o the robot dd not exceed ts maxmum pre-dened torque Snce ABC uses the drect knematcs, the sngulartes do not consttute a problem REFERENCES [1] Kazem, I B,Mahd, A I,Oudha, A T (008) Moton Plannng or a Robot Arm by Usng Genetc Algorthm Jordan Journal o Mech and Ind Engg, pp [] Chae, G, Park, J, H (007) Gallopng Trajectory Optmzaton and Control or Quadruped Robot Usng Genetc Algorthm In: IEEE Internatonal Conerence on Robotcs and Bommetcs, pp [] Zoller, Z, Zentan, P (1999) Constant Knetc Energy Robot Trajectory Plannng perdca polytechncaser Mech Eng, (), pp 1 8 [] Yue, S,G, Henrch, D, Xu, X, L, Toss, SK (00) Pont-to-Pont Trajectory Plannng o Flexble Redundant Robot Manpulators Usng Genetc Algorthms J Robotca, 0, pp [5] Solterro, P, Tenrero, M, Moura O, 00, Fractonal Order Dynamc n a Genetc Algorthm In:11th Internatonal Conerence On Advanced Robotcs, pp 6-69 Colomba, Portugal [6] Tang, Z, Zhou, C, Sun Z (00) Trajectory Plannng For Smooth Transton o A Bped Robot In:IEEE Internatonal Conerence on Robotc & Automaton,pp [7] Chwa, D, Kang, J, Im, K,Cho, J (00) Robot Arm Trajectory Plannng Usng Mssle Gudance Algorthm In: SICE Annual Conerence, pp [8] Grag, P, Kumar, M (00) Opmzaton Technques Appled To Multple Manpulators or Path Plannng and Torque Mnmzaton Engneerng Applcatons o Artcal Intellgence,15(), pp 1-5 [9] Savsan, V, Rao, R, Vakhara, D P (010) Mult objectve optmzaton o mechancal elements usng artcal bee colony optmzaton technque ASME Early Career Techncal Journal [10] Soltero, P, Machado, J (000) A GA Perspectve O The Energy Requrement For Manpulators Maneuverng In A Workspace Wth Obstacles Cec 000-Congress On Evolutonary Computaton, pp , Santago, Calorna, USA [11] Crag, J (1989) Introducton to Robotcs: Mechancs and ControlWesley [1] Karaboga, D (005)An Idea Based On Honey Bee Swarm For Numercal Optmzaton Techncal Report-TR06, Ercyes Unversty, Engneerng Faculty, Computer Engneerng Department [1] Basturk, B Karaboga, D (006) An Artcal Bee Colony (ABC) Algorthm or Numerc uncton Optmzaton IEEE Swarm Intellgence Symposum, May 1-1, Indanapols, Indana, USA wwwjmercom 8 Page

NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS

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