Trajectory Planning for a Welding Robot Based on the Bezier Curve

Transcription

1 Internatonal Journal of Control and Automaton Vol. 8 No. (5) pp. 5-6 ttp://dx.do.org/.457/jca Trajectory Plannng for a Weldng Robot Based on te Bezer Curve Cadong Wang Xnje Wang and Hu Zeng College of Mecancal and Electrcal Engneerng Zengzou Unversty of Lgt Instry Zengzou 45 Cna vwangcadong@6.com Abstract Accordng to te requrements for tower crane weldng robots precsely plannng te weldng trajectory s essental. Te optmal model for te least-tme trajectory plannng for te robot s establsed by Bezer curve wc as te caracterstc of sub-secton processng. Wt te consderaton of constrants suc as jont angular velocty and acceleraton te specfc operaton steps of genetc algortm (GA) optmzaton are presented. A weld seam trajectory example s ten proposed and te smulaton of a robot s trajectory plannng s carred out. Te result sows tat te curve s smoot and steady and can satsfy te requrements of tower crane weldng operatons. Ts metod sgnfcantly reces te moton tme and mproves workng effcency. Keywords: Weldng robot trajectory plannng Bezer curve genetc algortm optmzaton desgn. Introcton Tower cranes are essental equpment for te constructon nstry. Wt te rapd development of te world s nfrastructure te requrements for safety and relablty of tower cranes are ncreasng []. Most tower crane owners stll adopt manual weldng for ter procton. Te development of enterprses s restrcted by te problems of labor ntensty poor workng condtons low proctvty and a lack of qualty control []. Weldng robots ave many advantages suc as stable weld qualty ncreased proctvty and mproved workng condtons: as suc tey ave been wdely used n te: automotve aerospace sp-buldng macne processng electrcal and electroncs nstres []. Some large-scale constructon macnery companes wc ave ntroced weldng robots are plannng to explore te applcaton of flexble robot weldng tecnology on flat-ead and boom tower crane structural parts. However te weldng robots are expensve and tey are dffcult to use for tower crane weldng applcatons. Terefore ts work as sgnfcance for te enancement of te level of automaton of tower crane manufacture by evolvng te key tecnology of a tower crane weldng robot. It s dffcult to establs a precse system and model because of te g weldng qualty requred for a tower crane s standard secton as tey are welded: te weldng robot s complex workng envronment and ts load cange dramatcally. Durng moton control of te robot te angular acceleraton constrant angular velocty constrant and angle constrant of eac jont are te major constrants. In te general case of low-speed runnng as long as te jont angles are not overrun t as lttle nfluence on te trajectory plannng. But wen te robot runnng speed s g te jont angular acceleraton and angular velocty can easly exceed ter safe lmts wc wll force te drve current to become too large or ndeed cause an accdent. At ts pont te varous constrants sould be consdered on te trajectory plannng of te robot and te most commonly ISSN: IJCA Copyrgt c 5 SERSC

2 Internatonal Journal of Control and Automaton Vol. 8 No. (5) metod s to optmze te runnng tme. Many researcers ave proposed metods for calculatng te tme-mnmal trajectores for a robot e.g. te curvature constrants sould be consdered wen a moble car s turnng [4-6]. Cwa mposed jont velocty constrants and torque constrants on te trajectory of te manpulator as t ntercepted fast-flyng objects by optmzng te tme [7]. Saravanan used te NURBS curve to map te robot manpulators trajectory so tat t can satsfy varous constrants [8]. Algortms for map buldng and pat plannng n an outdoor envronment based on mult-sensor data fuson are avalable [9]. Based on te requrements of a tower crane standard secton weldng robot wc needs to plan ts weld trajectory precsely rng ts operaton te tme optmal model for robot trajectory plannng s establsed by te use of Bezer curves []. Addtonally te jont s angular velocty and acceleraton constrants are consdered and te smulaton of a robot s trajectory plannng undertaken.. Weldng Robot: Overall Program Te man task of a tower crane weldng robot s to complete te weldng operaton of te tower crane standard secton s man branc. Te weldng robot system comprses: a weldng manpulator a weldng locator weldng equpment bearngs control systems etc. Te weldng system for double locaton s demanded n te weldng process. Te manpulator s an essental part of te tower crane standard secton weldng system and drectly determnes te weld qualty. To ensure tat te weldng torc can complete ts weldng work n an arbtrary posture te 6-degree-of-freedom seral manpulator s adopted. Accordng to te desgn prncple of te manpulator tat: arm and wrst sould be separated [] te -D model of te manpulator s desgned as sown n Fgure. Te front tree jonts form te arm structure and determne te poston of te weldng torc. Te rear tree jonts axes are mutually perpendcular and consttute te wrst jonts wc determne te posture of te weldng torc. Te 6-degree of-freedom manpulator coordnate system s establsed by te D-H (Denavt-Hartenberg) metod. Te rear tree jont axes of te manpulator ntersect at one pont wc s te reference pont of te wrst. To make te knematc equatons smpler jonts 4 5 and 6 are set as a reference pont n te wrst. Te coordnate system s establsed as sown n Fgure and te D-H parameters are sown n Table. Fgure. Te D-H Coordnate System 5 Copyrgt c 5 SERSC

3 Internatonal Journal of Control and Automaton Vol. 8 No. (5) Table. D-H Parameters Rod number a d varable range mm ( ) ( ) mm ( ) () d to 4 a -9 (-9) to 7 a () to 9 4 a -9 4 () d to () to (). Trajectory Plannng Metod: te Bezer Curve.. Expresson of Fourt-order Cubc Bezer Curve 5-6 to 6 Te N t order vector equaton of te Bezer curve s: n n j nj j n n ( u) C ( u) P C ( u) up C ( u) u P C u P Were u P P n n n j n n n n j n j j Cn ( u) u Pj Bj n ( u) Pj j j Pn are te space vector C j n n! j!( n j)! j n j j Bj n Cn ( u) u. Based on te gven space vector ponts P P P and P te fourt-order cubc Bezer plane curve can be derved as: T ( u) u Mp u () Were u P 6 M u P u p u P P.. Bezer Curve Constrant Condtons To make te trajectory formed by te Bezer curve guarantee contnuty of robot jont angular velocty and angular acceleraton at te connectons we must derve condtons suc tat two Bezer curves sould be satsfed at te connectons. V V V and V are te control ponts of te frst Bezer curve. a a and a are te edge vector of te Bezer curve. W W W and W are te four control ponts of te second Bezer curve. b b and b are te edge vectors (were W and V concde W = V ). () Copyrgt c 5 SERSC 5

4 Internatonal Journal of Control and Automaton Vol. 8 No. (5) Fgure. Sttc Fgure of Two Bezer Curves We assume tat: b = a ( ) b a a ( ). Ten we obtan: W W b ( ) V V () W W b V ( ) V ( ) V. Te Bezer curve ( u) tat be constructed by four control ponts V V V and V can be derved as: ( u) ( u u u ) V (u 6u u) V ( u u ) V u V (4) d( u) ( u 6u ) V (9u u ) V ( 9u 6 u) V u V d ( u) ( 6u 6) V (8u ) V ( 8u 6) V 6uV If u we obtan: () V d( u) u V V ( V V ) a d () u 6V V 6V 6a 6a. Te Bezer curve ( u) tat be constructed by four control ponts W W W and W can be derved as: ( u) ( u u u ) W (u 6u u) W ( u u ) W u W (5) d u d ( u) In te same way we obtan te formula ( ) equaton 5. If u= we obtan () W V d( u) u W W ( W W ) b a d ( u) u 6W W 6W 6 a 6( ) a. from te dervatve of 54 Copyrgt c 5 SERSC

5 Internatonal Journal of Control and Automaton Vol. 8 No. (5) For te frst and second order dervatves of Bezer curves are contnuous at te d( u) d( u) d () d ( u) connectons u u u u.we obtan. So connect te two Bezer curves smootly te edge vector of te correspondng feature polygon sould satsfy te followng relatonsp: b a b a a (6) Te two sectons of te Bezer curves satsfy equaton 5 rng trajectory plannng for te manpulator and te values of frst and second dervatves at te control ponts are contnuous tat s to say te angular velocty and angular acceleraton of te jont are contnuous. By ndcatng te relatonsp of data ponts between two adjacent sub-secton curves equaton can be rewrtten as: W V V (7) W 4V 4V V V V V W Assumng tat te t secton secton Bezer curve connecton data ponts are P P and () () ts control ponts are V V V V by te nature of te Bezer curve V P V P. () () V V V (8) () () () () V V V V V So te trajectory of te secton Bezer curve of te manpulator s V () V T ( u) um () (9) V V ( ) Equaton 9 s a vector equaton: V V () V () and V are a set of control ponts. u s a parameter suc tat u wc gves te jont space trajectory of te manpulator. Vector control ponts ave two components: te tme component and jont angle component respectvely expressed by V and V.Te correspondng trajectory pont ( u) also as tme and jont angle components tey are respectvely expressed by t ( u) and q ( u ). t ( u) t t q () t T um ( () t t ) q () q T q ( u) um () q q () Eac Bezer trajectory s only decded by te four control ponts accordng to te Bezer trajectory formaton process wen a perod of trajectory runnng condtons or power does not meet te requrements some adjustment s needed. Te weldng robot s Copyrgt c 5 SERSC 55

6 Internatonal Journal of Control and Automaton Vol. 8 No. (5) workng envronment s constantly cangng so te trajectory plannng metod as to make a workng trajectory plan quckly and accurately. 4. Optmzaton of te Bezer Curve Trajectory Because te Bezer curve as te advantage of sub-secton processng and eac subsecton s only determned by four adjacent control ponts cangng one vertex of te caracterstc polygon wll only affect te adjacent four curves relate to te vertex te oter curves do not cange. Ts property of ts trajectory optmzaton means tat te entre trajectory of a manpulator can be optmzed step-by-step. As a result te wole runnng tme optmzaton of te manpulator along te Bezer curve can be converted nto te sub-secton runnng tme optmzaton for eac sub-secton. () Te Objectve Functon. Te sub-secton optmzaton s used for te robot runnng tme of eac sub-secton of Bezer curve ere. m T mn () Were T s te total runnng tme of te manpulator along te wole Bezer curve m are te runnng tme requred for te manpulator along eac sub-secton of te Bezer curve and m s te number of data ponts. () Desgn Varables. In equaton () m are ndependent varables so we set m as te desgn varables. () Constrant Condtons. ) Velocty constrants Settng te jont velocty of a certan trajectory of manpulator to ( u) te maxmum jont velocty sould occur at tme t or n te nterval ( t t ). Were max occurrence max max x () s te absolute value of te maxmum velocty accordng to ts actual s te absolute value of te velocty at tme t absolute value of te velocty n te nterval ( t t )( = m ). Ten solved by golden secton metod. dt( u) =( 6 ) (9 ) () ( 9 6 ) () u u t u u t u u t u ( t ) dq( u) =( 6 ) (9 ) () ( 9 6 ) () u u q u u q u u q u q Ten: x d ( u) dq( u) dq( u) dt ( ) ( )= A u u A u A dt dt B u B u B () () () () Were A = q 9q 9q q A 6q q 6q () () () () () B 9t 9t B 6t t 6t B t t. ) Acceleraton constrants d( u) d( u) qt qt ( u) dt dt t s te maxmum of te () A q q x can be (4) 56 Copyrgt c 5 SERSC

7 Internatonal Journal of Control and Automaton Vol. 8 No. (5) dt dq dt( u) dq( u) t q t q () () () () t ( t t t 6t 6 ) u 6t t 6t D D () () 6 6 D t t t t D 6t t 6t () () q ( q q q 6 q ) u 6q q 6q C u C () () () () C q q q 6 q C 6q q 6q () () () () ( u) B u B u B C u C D u D Au A u A / Au A u A Te maxmum angular acceleraton sould occur at tme t or n te nterval ( t t ). max max x (5) s te jont angular acceleraton of te manpulator at tme t x s te maxmum absolute angular acceleraton of te manpulator n te nterval ( t t ) ( = m ) and x can be solved by golden secton metod. (4) Genetc algortm (GA) optmzaton steps GA was used n ts work to solve te problem of robot trajectory plannng. Assume tat te robot trajectory as m nodes (ncludng te start and end ponts) [ t t] [ t t] [ tm tm] are taken as tme seres. Te lengt between two adjacent nodes s t t ( = m ). Were t s te tme at wc te weldng robot arrves at a node. Optmzaton steps of GA were as follows []: ) Codng Te tme range of eac Bezer curve for te weldng robot s encoded as cromosomes of te GA. Float code s adopted ere by wc a large area can be expressed. ) Intal populaton In te range of a number of ndvals are generated randomly. ) Te dstrbuton functon of ndval s ftness value n te populaton For every ndval n populaton ts ftness value s allocated by te followng formula: Satsfy te constrant condtons f (6) Not meetng te constrant condtons max Were max s te maxmum value of. Te ndval wc does not meet te constrant condtons as te mnmum value accordng to equaton 6. 4) Coce Roulette selecton s adopted ere. Te ftness value of an ndval j n te populaton s set to f ( j N). Under te roulette selecton metod te selectve probablty of ndval s: N p f / f (7) j j 5) Crossover Artmetc crossover operators. x and x are set as two parent ndvals. In te acton of artmetc crossover operators te progeny can be expressed as: Copyrgt c 5 SERSC 57

8 Internatonal Journal of Control and Automaton Vol. 8 No. (5) x cx ( c) x x ( c) x cx Were c s a random number between [ ]. 6) Mutatons Non-unform mutaton s adopted ere. Te ndval x k proced by non-unform mutaton based on x k s calculated by: U xk ( g xk xk ) x k (8) L xk ( g xk xk ) U L Were x k and x k are te upper and lower boundary g s te current number of teratons. ( g y) y ( / ) b m g G. Were m s a random number between [ ] G s te maxmum number of teratons and b s a sape coeffcent (set to ere). Te specfc parameters and steps are as follows. Step : Te populaton sze N = 5 genetc algebra G = 8 crossover probablty c =.6 mutaton rate m =. gven m data ponts P and (m- ) tme nterval t t. () Step : Te control ponts q (te jont angle component ofv ) can be found by equaton 8 based on te m data ponts P. Step : Wen g = te ntal populaton s generated randomly te correspondng t can be aceved based on. Step 4: Te Bezer curve can be obtaned based on q and t and t sould be cecked weter or not te curve meets te speed constrants. Step 5: Te adaptve value of ndvals n te populaton can be obtaned from equaton 7. Step 6: Take te operatons of roulette weel selecton artmetc crossover and nonunform mutaton on te current populaton. Step 7: g = g +. Step 8: If g < G ten swtc to step (). Oterwse termnate te algortm to yeld te optmal populaton. 5. Smulaton Analyss Takng te trajectory of weldng robot welds on te standard secton of a tower crane s man branc as an example we cose egt nodes on te weldng trajectory curve n Cartesan space (Table ). Addtonal nodes and 8 are obtaned based on te ntal condtons and boundary condtons wc sets te ntal pont velocty of te trajectory n eac sub-secton to zero. Puttng te nodal data nto te robot nverse knematc equatons we obtan te correspondng jont angle of te egt nodes as sown n Table. Table lsts te constrants of te robot wc are determned by te structure of te robot and te motor parameters. As sown n Table e to te tasks specfc requrements te trd fourt and sxt jont angles canged lttle terefore only tose curves for te frst second and fft jont angles are sown. To verfy te effect of GA optmzaton te optmzed runnng graps and ntal runnng graps of eac jont angle based on Bezer curves are plotted n Fgures to 5 respectvely. 58 Copyrgt c 5 SERSC

9 Internatonal Journal of Control and Automaton Vol. 8 No. (5) Table. Te Values of te Nodes Unt: Degree Nodes Jont Jont Jont Jont 4 Jont 5 Jont Table. Te Constrants of Robot s Jont Angle Velocty and Acceleraton Constrants Jont Jont Jont Jont 4 Jont 5 Jont 6 (rad/s) (rad/s^) Control pont Bezer Curve Control polygon 7 q /degree t / s Fgure. Te Angle Optmzaton Curve of te Frst Jont Fgures to 5 sow tat te angle varaton curve of te frst second and sxt jonts were smoot stable and tat tey met te basc requrements of manpulator operaton. Manpulator runnng tme parameters are sown n Table 4 were te runnng tme along a predetermned trajectory of te manpulator was seen to ave been sgnfcantly reced: te effect was perfect and t could meet te velocty constrants. Eac jont velocty curve can be obtaned by equaton (4). Te robots total runnng tme from te startng pont to te end as been reced from.45 s to 4. s. Te recton of te runnng tme mproved te robots workng effcency. Smulaton results sowed tat te robots runnng tme ad been reced compared to te ntal tme taken. Eac jont Copyrgt c 5 SERSC 59

10 Internatonal Journal of Control and Automaton Vol. 8 No. (5) dsplacement curve became smoot after GA optmzaton. Terefore te results realzed te tme optmal trajectory plannng of te robot and aceved te expected target Control pont Bezer Curve Control polygon 7 q /degree t / s Fgure 4. Te Angle Optmzaton Curve of te Second Jont 45 5 Control pont Bezer Curve Control polygon q /degree t / s Fgure 5. Te Angle Optmzaton Curve of te Fft Jont Table 4. Tme Optmzaton Results Unt: s Total tme Intal value Optmal value Conclusons Ts work studed te applcaton of te Bezer metod n trajectory plannng of a tower crane weldng robot. Accordng to te tme optmal metod a robot trajectory plannng model was establsed based on a Bezer curve metod. Te smulaton results sowed tat te algortm was smple easy to mplement and relable. Te metod could aceve te tme optmal trajectory of a robot wt knematc constrants. Ts work only 6 Copyrgt c 5 SERSC

11 Internatonal Journal of Control and Automaton Vol. 8 No. (5) took te velocty constrant nto consderaton rng trajectory plannng and dd not consder te drvng force constrants. Te robot trajectory optmzaton metod was undertaken by off-lne programmng and dd not consder real-tme plannng goals. Terefore te plannng of an optmal trajectory for suc a robot under actual operatonal condtons to realze on-lne real-tme trackng may be a useful objectve of future work. Acknowledgements Ts work s supported by Scentfc and tecnologcal project n Henan Provnce (45) Program for IRTSTHN (IRTSTHN) and Doctoral Researc Fund of Zengzou Unversty of Lgt Instry. References []. L. W. Wang and L. Wang Constructon Macnery vol. 9 no. (8) pp. -5. []. L. Hu S. Xn and T. We Journal of Harbn Unversty of Scence and Tecnology vol. 7 no. () pp []. Y. L. Xu T. Ln and S. B. Cen Metal processng (termal processng) vol. 9 no. 8 () pp. -6. [4]. M. Haddad W. Kall and H. E. Letet IEEE Transactons on Robotcs vol. 6 no. 5 () pp [5]. L. Hu S. Xn and T. Z. We Journal of Harbn Unversty of Scence and Tecnology vol. 7 no. () pp [6]. M. Jang E. Lee and S. Co Internatonal Journal of Control and Automaton vol. 5 no. 4 () pp [7]. D. Cwa K. Juno and Y. C. Jn IEEE Transactons on Systems Man And Cybernetcs vol. 5 no. 6 (5) pp [8]. R. Saravanan S. Ramabalab and C. Balamurugan Robotca no. 6 no. 6 (8) pp [9]. F. Yan Y. Zang and W. Wang Int. J. Computer Applcaton n Tecnology vol. 44 no. 4 () pp []. Y. M. Wang Mecancal Scence and Tecnology vol. no. () pp []. P. Wenger D. Cablat and M. Bal Journal of Mecancal Desgn vol. 7 no. (5) pp []. L. F. Tan and C. Curts Mecatroncs vol. 4 no. 5 (4) pp Autors Cadong Wang s an Assocate Professor at Zengzou Unversty of Lgt Instry Cna. He receved s PD from Te Scool of Mecatroncs Engneerng Harbn Engneerng Unversty Cna n. Hs man researc nterests are mecatroncs and robotc tecnology. Xnje Wang s a Professor at Zengzou Unversty of Lgt Instry Cna. Se receved er bacelor s master s and PD degrees from Huazong Unversty of Scence and Tecnology Cna n and 5 respectvely. Her man researc nterests nclude robotcs and automatc control. Copyrgt c 5 SERSC 6

12 Internatonal Journal of Control and Automaton Vol. 8 No. (5) Hu Zeng s presently a graate student majorng n mecancal desgn and teory at Zengzou Unversty of Lgt Instry Cna. He receved s bacelor s degree n from Henan Polytecnc Unversty Cna. Hs man researc nterest s parallel robot control tecnology. 6 Copyrgt c 5 SERSC