Inverse Displacement Analysis of a General 6R Manipulator Based on the Hyper-chaotic Least Square Method

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1 Internatonal ournal of Advanced Robotc Systems ARICLE Inverse Dsplacement Analyss of a General 6R Manpulator Based on the Hyper-chaotc Least Square Method Regular Paper Youxn Luo * We Y and Qyuan Lu 3 College of Mechancal Engneerng Hunan Unversty of Arts and Scence Changde R.R.Chna Sany Heavy Industry Co. Ltd Changde P.R.Chna 3 College of Mechancal Engneerng Hunan Unversty of Arts and Scence Changde R.R.Chna *Correspondng author E-mal: LLYX3@6.com Receved Sept ; Accepted 5 an Luo et al.; lcensee Inech. hs s an open access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal wor s properly cted. Abstract he hyper chaotc least square method for fndng all real solutons of nonlnear equatons was proposed and the nverse dsplacement analyss of a general 6R manpulator was completed. Applyng the D H method a matrx transform was obtaned and the frst type twelve constraned equatons were establshed. Analysng the characterstcs of the matrx the second type twelve constraned equatons were establshed by addng varables and restrcton. Combnng the least square method wth hyper chaotc sequences the hyper chaotc least square method based on utlzng a hyper chaotc dscrete system to obtan and locate ntal ponts to fnd all the real solutons of the nonlnear questons was proposed. he numercal example was gven for two type constraned equatons. he results show that all the real solutons have been obtaned and t proves the correctness and valdty of the proposed method. Keywords 6R manpulator nverse dsplacement hyperchaotc sequences least square method. Introducton he nverse dsplacement analyss of a robot plays an mportant role n robotcs whch s drectly related to the problems of the off lne programmng trajectory plannng and real tme control. It s most dffcult n relaton to nverse dsplacement analyss on a general space 6R manpulator. hs ssue wth the 6R mechansm belongs to the same problem whch has been called the Mount Everest of the poston analyss on a space mechansm []. In 973. Rastegar and V. Schenman [] ponted out that the number of the solutons was up to 3 n the nverse dsplacement analyss of a general 6R seral robot. In 985 L. W. sa and A. P. Morgan [3] smplfed the above problem n eght quadratc equatons usng the hgh dmensonal approxmaton analyss on the nverse nematcs problem and carred out the numercal soluton adoptng a polynomal contnuous method and then speculated as to the concluson that the soluton number s up to 6 on a varety of dfferent structures of 6R robot. In 989 M. Raghavan and B. Roth [] made use of the nature of the ratonale generated by usng the Youxn Luo We Y and Qyuan Lu: Inverse Dsplacement Int Adv Robotc Analyss Sy of a General Vol R Manpulator Based on the Hyper-chaotc Least Square Method

2 separaton elmnaton method and mult varable equatons and calculated a 6th degree polynomal from the half angle tangent formula of the jont varables. In 99 C. Wampler and A. P. Morgan[5] proposed that all the cases nvolvng the nverse dsplacement problem were only solved usng the extenson method but the algorthm based on the extenson method ran very slowly. From 99 to 99 M. Raghavan and B. Roth [6 8] frst proposed a method solvng the characterstc polynomal of the general 6 DOF manpulator ncludng all the specal crcumstances of 6R 5RP RP 3R3P and wth the am of solvng all of the 6 DOF robot mechansm. In 99 C. W. Wampler and A. P. Morgan [9] completed the 6R nverse poston problem usng a generc case soluton methodology wth polynomal equatons. In 993 D. Kohl and M. Osvatc[] used the product of the powers and the elmnaton method to obtan a set of equatons contanng 6 6 matrx coeffcents and to drectly have 6th degree polynomals of no extraneous rootssnce the soluton to the problem can be smplfed to the egenvector problem. In the same year D. Manocha and. F. Canny [] made use of the symbol processng and matrx computaton to transform ths problem nto the decomposton problem of the matrx to be calculated and solved the matrx egenvalues and egenvectors to obtan the robot jont varables. However the algorthm fnally computed the th order square matrx to solve solutons ncludng extraneous roots rather than drectly the 6 solutons. In 999 H.. Su et al. [] proposed an algorthm based on algebrac elmnaton to solve the robot nverse nematcs. he method can get a one varable equaton wth one jont varable frst of all whch greatly reduces the computaton procedure of the real tme teratve. However the algorthm needs to expand the determnant polynomal to solve 6th degree polynomals. A slower computng speed and may occur and cause problems when calculatng the polynomal. Y. Q. Yu[3] solved the nverse nematcs problem of a general 6R seral robot and smulated the robot moton by usng VC++ wth OpenGL n 3 but there was a poor consequence upon the real tme reverse problem. How to qucly fnd all the real solutons s mportant to the research of the mechansm but t s also one of the basc problems for the general 6R and 7R robot mechansms. he research nto chaos s one of the mportant achevements of the st century. How to utlze chaotc characterstcs s one of the mportant wors of the modern mechansm of research. I have solved the forward dsplacement problem of 6 RPS n the real area usng the chaotc method n []. Wth ths method t advances that the ponts of a ula set whch are generated by the Newton teraton method wll appear n the neghbourhood where the acoban matrx of the solvng equatons s equal to zero. However ths supposton has not been proven and the process for solvng the multvarable acoban matrx s very complex. he chaotc sequence method s new n whch the ntal pont of the Newton teraton s generated usng the chaotc and hyper chaotc system and all the real solutons on the mechansm synthess can be effectvely solved [5 7]. When the solutons are not n convergence usng Newton method or a quas Newton method the mathematcal programmng method can be adopted [8]. However ts computatonal effcency s very low for the process of solvng the nverse nematcs problem of a 6R mechansm by usng the mathematcs quadratc programmng method based on hyper chaos. Wth the Newton teraton method the requrement for the ntal value s hgh. However the least square method can expand the selected scope of the ntal value and t can sometmes mae one ntal value convergence whle Newton method cannot. For these reasons combnng the least square method wth hyper chaotc sequences and utlzng the ntal value that was terated n the least square method based on hyper chaotc Hénon mappng the nverse nematcs problem of a general 6R mechansm was solved n ths paper. he example provded shows that the method proposed n ths paper s verfed as correct and effectve.. Hénon hyper chaotc system he Lyapunov exponent s one of the effectve methods depctng the chaos specfc property of nonlnear systems. If one of the Lyapunov exponents s postve the system s chaotc and f a system has two or more postve Lyapunov exponents the system s hyperchaotc. he greater the number of the postve Lyapunov exponents the hgher the degree of nstablty n the system [9]. In general f the systematc state varable number s hgher t probably appears that the unsteady level s also hgher. A general Hénon mappng was desgned [] as follows: x x a xn x bx n where 3 n expresses the dmenson of the system s the dscrete tme and a and b are adjustable parameters. When = the above mappng s the famous Hénon mappng. When fxed parameters a=.76 and b=. and the dmensons vary from to after computng t was found that by ncreasng n the smple relaton of the number n of the postve Lyapunov exponents wth the system dmenson n s n n n [] namely when a system dmenson s the larger of two the system s hyperchaotc. For n> a smulaton study was conducted[8] whch also obtaned the same result. ( Int Adv Robotc Sy Vol

3 3. he least squares method n nonlnear equatons he nonlnear equaton s expressed as: f f f n ( x [ ( x ( x] ( Its soluton s wrtten n x [ x x x n ]. Assumng f ( x the least squares teratve method can be x descrbed as follows roughly: ( o select the ntal value x ( o tae the teraton accordng to Eq. (3 ( By Eq. ( to construct the chaos set x ( j ( n n s the varable number of hyper chaotc system n s the number of the postve Lyapunov exponents and s also the number of varables. j N where N s the length of the chaos set and obtan x ( j ; ( o tae the jth chaotc sequence x (: j as the ntal value of the least square teratve method Eq. (3 has * been terated j tmes to fnd all of the solutons x n Eq.(. 5. he mathematcal model x x x ( ( f ( (3 Accordng to the structural parameters of the robot.e.the D H parameters and the ln coordnate system the general 6R robot nematcs can be descrbed as: where f ( x s the functon value of f ( x at the pont x and s the acoban matrx of f ( x at the pont x. Although s always symmetrc postve and semdefnte t seems that usually there exsts the nverse matrx (. But the value of ts determnant det( s so small that there exsts a serous ll condtoned problem. Many scholars have put forward mproved algorthms and the most well nown of these s the damped least square method also nown as the L M method. Recently a Chnese scholar C. X. Zhan proposed a new method called the C. X. Zhan method n a paper. Wth the new method computatonal effcency s better than wth the L M method and the convergence rate s faster. So far ths method s the best for solvng the least squares problems and t s adopted n ths paper. Its basc steps are as follows: Frstly t s assumed that A A s decomposed to A LDL (where L s a lower trangular matrx and D s a dagonal matrx. Secondly Eq. (3 s rewrtten as LDL κ( x x f ( x. hrdly after ncreasng the dampng to D the teratve equaton based on the C. X. Zhan method s expressed as: L ( D I L ( x x f ( x ( κ where I s the n order unt matrx and s the dampng factor where. he selecton of the dampng factor and the convergence crtera sseen n [].. he hyper chaotc least square method n nonlnear equatons Usng the least square method based on a hyper chaotc system the process for fndng all of the solutons of nonlnear equatons s as follows: A A (5 A A 3A A 5A 6 where A s a homogeneous transformaton matrx from the th ln coordnate system to the ( th coordnate system; A s a homogeneous transformaton matrx from the coordnate system of the robot end to one of the base. Q P A Q ZX cos sn X cos sn Z sn cos sn cos P a cos a sn b where a b and are the structural parameters of the robot and they are determned by the specfc robot structure; ( 6 s the ont varable. he nverse nematcs problem of the general 6R robot s that the jont varable s found under the condtons of gvng a b and A. Eq. (5 s expanded as A( A( A(3 A( A( A( A(3 A( A(3 A(3 A(33 A(3 A6 ( A6 ( A6 (3 A6 ( A6 ( A6 ( A6 (3 A6 ( A6 (3 A6 (3 A6 (33 A6 (3 In Eq. (6 the correspondng parameters are equal and there are twelve scalar equatons and four denttes. o the mechansm sx of these scalar equatons only are ndependent. Nowwth Eq. (7 substtuted nto twelve (6 Youxn Luo We Y and Qyuan Lu: Inverse Dsplacement Analyss of a General 6R Manpulator Based on the Hyper-chaotc Least Square Method 3

4 scalar equatons we can obtan Eq. (8: x sn x cos x3 sn x cos 5 sn 3 x6 cos 3 x7 sn x8 cos x x 9 sn5 x cos5 x sn 6 x cos where: 6 (7 F [ f ( x f ( x] (8 x [ x x x] f A( A6( f A( A6( f3 A(3 A6(3 f A( A6( f5 A( A6( f6 A( A6( f7 A(3 A6( f8 A( A6( f9 A(3 A6(3 f A(3 A6(3 f A(33 A6(33 f A(3 A6(3. Wthout consderng sn cos twelve scalar equatons of Eq. (8 are ndependent. After the soluton the results should be tested as to whether they meet the condton equaton sn cos. Removng those extraneous roots whch are not satsfed condtons the solutons n Eq.(8 are found. Eq.(5 s transformed nto Eq.(9: A A A A A AA ( After expanson the equaton s as follows: l( 3 5 l( 3 5 l3( 5 r( 6 r( 6 r3( 6 l( 3 5 l3( 3 5 l( 3 5 l ( 3 5 l3( 3 5 l( 3 5 l ( ( ( 3 5 l33 5 l3 5 ( r( 6 r3( r( r ( ( ( 6 r3 r r ( ( ( 3 6 r33 r3 where l j and r j are the non trval components of the left and rght n Eq.(9 respectvely. Snce the values of the thrd and fourth columns n Eq.( are not assocated wth 6 utlzng the non trval components of the thrd and fourth columns obtans sx equatons: F [ f ( x f ( x] ( 6 where: f l3 r3 f l r f3 l3 r3 f l r f l r f l r Mang use of the equaton equatons are added as: where 7 sn cos sx F [ f ( x f ( x] ( f x : x f f x5 x6 f f x x f 7 8 x3 x 9 x7 x8 9 x x he calculaton based on Eq. ( and Eq. ( yelds the soluton x [ x x x ]. Because there s no 6 n Eq. ( [ x x x ] as the nown varables are substtuted nto Eq.(3 and then x and x are found. l r x x (3 where the equaton l r can be replaced by the other scalar equaton correspondng to the frst or second column. After the soluton we can chec whether the frst column n Eq.( s equal to the second one. 6. Numercal example Gven the mechansm parameters shown n ab. and the pose of the end as follows: Q P where the unt of P s m the nverse dsplacement of the mechansm should be found. o a ( m b ( m ( able. he parameters of a 6R manpulator Int Adv Robotc Sy Vol

5 No No able. he computng results of the varables After a random number s generated an ntal pont of the hyper chaotc set and the number of hyper chaotc varables n s selected as 3 accordng to the generalzed Hénon hyper chaotc system n Eq. ( and the hyperchaotc sequence ncludng postve Lyapunov exponents was bult (the selected chaotc sequence length N was. If N s selected as bgger the computng tme would be longer and f N s too small t may be that not every real soluton would be found. herefore n relaton to these problems of the mechansm synthess N should frst be taen as smaller when counted approxmately and then be selected as larger. In general N s selected at (8 tmes the varable number. After an teratve calculaton for 98s Eq.(8 was solved to fnd x. hen ( 6 was convertedas shown n able. hese results were the same as e.g of []. If solvng Eq.( Eq.( and Eq.(3 from Eq. (9 6 groups of solutons were found after runnng for 6.5s. hese results were the same as e.g of [] and the computatonal effcency ncreased greatly. Snce the hyper chaotc Newton teraton method n [5] and the hyper chaotc sequence from Eq.( made the matrx sngular the result was not found. In the mathematcal programmng method based on hyper chaos n [8] the fmncon functon only obtaned two groups of real solutons after runnng many tmes whle the fmnmax functon was only obtaned for one or two groups. For the frst nd of constrant equatons sn and cos ( 6 were not restrcted.e. sn cos was not used n the soluton process and the varables x x x were dealt wth drectly. hus extraneous roots were ncreased.e. there exsted x (. he second type of constrant equaton has no extraneous roots by the use of sn cos and so the effcency of ts soluton s relatvely hgh. 7. Conclusons Applyng the D H method a transform matrx was obtaned and the frst type twelve constraned equatons were establshed. hen usng the characterstcs of the matrx after transformng sx constrant equatons were establshed. Sx supplement equatons were also establshed by ncreasng sx varables and the relaton of the sne and cosne functons and the second type of twelve constraned equatons for the poston analyss were establshed. Combnng the least square method wth hyper chaotc sequences a hyper chaotc least square method based on utlzng a hyper chaotc dscrete system to obtan and locate ntal ponts to fnd all the real solutons of the nonlnear questons was proposed and the calculaton steps solvng the nverse dsplacement analyss of 6R manpulator contanng two types of constraned equatons were gven. wo types of constrant equatons can be solved but the second type of constrant equaton has a hgh effcency. hs method solved the problem that t s dvergent when utlzng a Newton teratve method based on chaos and hyperchaos the quas Newton method and the hyper chaotc mathematcal programmng method. he numercal example shows that ths new method s correct and effectve. Usng ths method n the real area t s more effcent and the result s the same as the homology analyss method. he new method provded n ths paper s also sutable for solvng the nverse dsplacement analyss of 7R manpulator. 8. Acnowledgments hs research s supported by the Natonal Natural Scence Foundaton of P.R. Chna (No:575 the Youxn Luo We Y and Qyuan Lu: Inverse Dsplacement Analyss of a General 6R Manpulator Based on the Hyper-chaotc Least Square Method 5

6 grant of the th Fve Year Plan for the constructon program of the ey dscplne (Mechancal Desgn and heory n Hunan provnce. 9. References []. Duffy Analyss of Mechansms and Robot Manpulators Edward Amold London 98. [] B. Roth and. Rastegar V. Schenman On the Desgn of Computer Controlled Manpulators In On the heory and Practce of Robotcs and Manpulators Frst CISM IFoMM Symposum pp [3] L. W. sa and A. P. Morgan Solvng the Knematcs of the Most General Sx and Fve Degree of Freedom Manpulators by Contnuaton Methods ransactons of the ASME ournal of Mechansms ransmssons and Automaton n Desgn 7( pp [] M. Raghavan and B. Roth Knematc Analyss of the 6R Manpulator of General Geometry In Internatonal Symposum on Robotcs Research pp [5] C. Wampler and A. P. Morgan Solvng the 6R Inverse Poston Problem Usng a Generc Case Soluton Methodology Mechansms and Machne heory 6( pp [6] M. Raghavan and B. Roth Knematc Analyss of the 6R Manpulator of General Geometry. Proceedngs of the 5th Internatonal Symposum on Robotcs Research MI Press Cambrdge pp [7] M. Raghavan and B. Roth A General Soluton for the Inverse Knematcs of all Seres Chans Proceedngs of the 8th CISM IFoMM Symposum on Robotcs and Manpulators Cracow Poland pp [8] M. Raghavan and B. Roth Inverse Knematcs of the General 6R Manpulator and Related Lnages Proceedngs of the ASME Desgn echncal Conference Chcago Illnos DE5 pp [9] C. W. Wampler and A. P. Morgan Solvng the 6R nverse poston problem usng a generc case soluton methodology Mech. Mach. heory 6( pp [] D. Kohl and M. Osvatc Inverse Knematcs of General 6R and 5RP Seral Manpulators ournal of Mechancal Desgn ransactons Of the ASME 5( pp [] D. Manocha and. F. Canny Real me Inverse Knematcs for General 6R Manpulators Proceedng of the 99 IEEE Internatonal Conference on Robotcs and Automaton Nce France 99. [] H.. Su Q. Z. Lao C. G. Lang et al. Real tme algorthm of nverse nematcs of the general 6R manpulator based on the algebrac elmnaton Robot (7 pp [3] Y. Q. Yu P. Wang and Q. Z. Lao Poston Inverse nematcs of general 6R manpulator Chna Mechancal Engneerng ( pp [] Y.X. Luo and D.X. L Fndng All Solutons to Forward Dsplacement Analyss Problem of 6 SPS Parallel Robot Mechansm wth Chaos Iteraton Method ournal of Chnese Engneerng Desgn ( pp [5] Y.X. Luo D.X. L and X. F. Fan et al. Hyper chaotc Mappng Newton Iteratve Method to Mechansm Synthess ournal of Mechancal engneerng 5(5 pp [6] Y.X. Luo and D.G. Lao Couplng Chaos Mappng Newton Iteratve Method and Its Applcaton to Mechansm Accurate Ponts Movement Synthess ournal of Chnese Mechancal ransmsson 3( pp [7] Y.X. Luo X.F. L L.L.Luo et al. he Research of Newton Iteratve Method Based on Chaos Mappng and ts Applcaton to Forward Solutons of the 3 RPR Planar Parallel Mechansm Chnese Machne Desgn and Research 3( pp [8] Y.X. Luo Hyper chaotc Mathematcal Programmng Method and ts Applcaton to Mechansm Synthess of Parallel Robot ransactons of the Chnese Socety for Agrcultural Machnery 39(5 pp [9] Wolf. B. Swft H. L. Swnney et al. Determnng Lyapunov exponents from a tme seres Physcal D NonlnearPhenomena6(3 pp [] M. Zhao D. Y. Huang. H. Peng Generalzed synchronzaton of generalzed Hénon maps ournal of Shenzhen Unversty (Scence & Engneerng ( pp [].Y. Zhang Mathematcal methods of mechancs. Shangha aotong Unversty Press Shangha 3. [] W.G. Song he prncple theory methods and algorthms of the robot mechancal system Mechancal Industry Press Bejng. 6 Int Adv Robotc Sy Vol