A Generalized and Analytical Method to Solve Inverse Kinematics of Serial and Parallel Mechanisms Using Finite Screw Theory

Transcription

1 A Generalized Analyical Mehod o Solve Invere Kinemaic of Serial Parallel Mechanim Uing Finie Screw heory. Sun 1 S. F. Yang 1. Huang 1 J. S. Dai 3 1 Key Laboraory of Mechanim heory Equipmen Deign of Miniry of Educaion ianjin Univeriy ianjin China ao@ju.edu.cn School of Engineering he Univeriy of Warwick Covenry CV4 7AL UK 3 Cenre for Roboic Reearch King College London Univeriy of London London WCR LS UK Abrac. Invere kinemaic i a very imporan iue in he field of mechanim roboic which i he fundamenal problem in kinemaical analyi deign ynhei for boh erial mechanim (SM) parallel mechanim (PM). he objecive of invere kinemaic i o formulae compuable kinemaic equaion a he given poe of end-effecor of a SM or moving plaform of a PM hen olve all he join parameer (variable). Solving analyical oluion of invere kinemaic i he prerequiie for rajecory planning precie conrol manipulaion of mechanim. hi paper preen a generalized mehod o analyically do invere kinemaic of PM uing finie crew heory. Firly he kinemaic equaion of PM i algebraically formulaed hrough decribing finie moion generaed by he PM i limb join employing finie crew. hen he general procedure o analyically olve he finie crew baed kinemaic equaion are given. Finally a PM wih hree ranlaional one roaional Schoenflie moion i aken a an example o verify he validiy of he propoed mehod. Key word: Parallel mechanim Invere kinemaic Screw heory Finie crew. 1 Inroducion Invere kinemaic which i alo called invere poiion problem i aimed a formulaing kinemaic equaion of a mechanim a he given poe olving all he join parameer (variable). I i a fundamenal problem in kinemaical analyi deign ynhei for boh erial mechanim (SM) parallel mechanim (PM) [3 7]. Solving analyical oluion of invere kinemaic i he prerequiie for rajecory planning precie conrol manipulaion of mechanim. Becaue all he join parameer of a PM can be obained hrough olving he join parameer in each of i limb haring he ame moving plaform invere kinemaic of a Auhor' verion 1

2 . Sun S. F. Yang. Huang J. S. Dai PM can be decompoed ino everal invere kinemaic problem of SM. According o he mahemaical ool ha are ued o formulae kinemaic equaion he exiing mehod o deal wih invere kinemaic can be claified ino wo caegorie i.e. vecor chain baed mehod exponenial marix baed mehod. In vecor chain baed mehod hree-dimenional poiion vecor are ued o formulae he kinemaic equaion hrough building he mapping beween he poiion orienaion of he given poe. In he formulaed poiion equaion all join parameer are independen decoupled. Hence he equaion can be olved by mean of eliminaion. he vecor chain baed mehod can be raced back o he early reearch of heoreical kinemaic deailedly dicued concluded by Wampler [8] Craig [1]. Baed upon hi invere kinemaic of SM coniued by ix revolue (R) join are olved by Raghavan Roh [5] hrough engine value vecor analyi of everal univariae polynomial equaion wih high-order. Becaue lower mobiliy SM can be regarded a he ub-chain of ix degree-of-freedom (DoF) SM ix DoF SM can be regarded a he ubchain of SM wih higher DoF hi mehod can be exended o olve any SM. I hould be noed ha nonlinear equaion relaing he join parameer he given orienaion needed o be olved when he number of DoF of he SM i more han hree. hi bring huge difficulie o analyical oluion of invere kinemaic. hu numerical mehod are uually needed when olving higher DoF SM. Uing exponenial marix wih join parameer o decribe poe ranformaion beween adjacen link he kinemaic equaion can be obained by muliplying hee marice ogeher. In he formulaed kinemaic equaion all he join parameer are in he exponen he algebraic operaion can only be carried ou uing Baker-Campbell-Haudorff formula or aylor erie expanion. Becaue here are oo many erm in he exped marix polynomial he kinemaic equaion are hard o be analyically olved. hu oluion of invere kinemaic moly relie on numerical mehod []. For he kinemaic equaion formulaed by exponenial marix baed mehod invere kinemaic can alo be olved by geomerical mehod. Baed upon Kahan reearch work Paden [4] decompoed he invere kinemaic of SM ino everal ypical ub-problem hrough concluding he common rucure uni of SM. he analyical oluion of each ub-problem i given by geomerical algebraic derivaion. I hould be noed ha he Paden-Kahan ub-problem do no cover all he poible rucure uni of SM. Hence ome SM conno be olved applying hee ub-problem. From he above analyi i can be concluded ha he exiing mehod canno obain analyical oluion of invere kinemaic for arbirary SM becaue of he mahemaical ool ued. Boh vecor chain exponenial marix have ome limiaion in decribing finie moion formulaing kinemaic equaion of mechanim. Hence he clear algebraic mapping beween all he join parameer he given poe ha no been buil. A he concie non-redundan decripion of finie moion wih analyical compoiion crew riangle produc [6] finie crew ha he poenial o overcome he limiaion of vecor chain exponenial marix. A hown in he auhor previou work [6 9 10] he algebraic rucure of Auhor' verion

3 A Generalized Analyical Mehod o Solve Invere Kinemaic of Serial Parallel Mechanim Uing Finie Screw heory 3 finie crew were revealed he derivaive mapping beween finie inananeou crew wa buil reuling in a general conien mehod o unify ype ynhei kinemaic analyi under he umbrella of crew heory. In hi paper invere kinemaic will be carried ou employing finie crew which lead o a yemaic horough heoreical framework ha unifie opological poiion orienaion (poe) velociy modeling analyi ogeher. Baed upon he auhor previou work hi paper preen a generalized mehod o analyically do invere kinemaic of PM uing finie crew heory. he paper i organized a follow. Having a brief review of he ae-of-he-ar of he exiing mehod for invere kinemaic in Secion 1 Secion preen he new mehod o algebraically formulae kinemaic equaion of a PM i limb employing finie crew. In Secion 3 he general procedure o analyically olve he finie crew baed kinemaic equaion are given. A PM wih hree ranlaional one roaional Schoenflie moion i aken a an example o verify he validiy of he propoed mehod in Secion 4 before he concluion are drawn in Secion 5. Finie crew baed kinemaic equaion A finie moion of a rigid body from i iniial poe o arbirary poe can be preened a a roaion abou he Chale axi followed by a ranlaion along ha axi which can be decribed by a finie crew in quai-vecor [6] form a f r f S f S f an f 0 rf f f denoe he uni vecor poiion vecor of he finie moion axi are he angular linear diplacemen abou/along ha axi. A SM coniued by n one-dof join (R join primaic (P) join) i hown in Fig. 1. Uing finie crew o decribe he finie moion generaed by R P join he finie moion realized by he end-effecor can be expreed by crew riangle produc [6]. hu he kinemaic equaion of a SM a a given poe can be formulaed a Auhor' verion S f SM k S S S S () f SM n f SM n1 f SM1 f SM SM k SM k an R join rsm k SM k SM k 0 SM k P join k 1 n (1)

4 4. Sun S. F. Yang. Huang J. S. Dai S f SM denoe he given poe of he SM he denoaion of he ymbol in Eq. () can be referred o hoe in Eq. (1). Fig. 1 Finie moion of a SM For a PM compoed of l limb each limb i a SM haring he ame endeffecor i.e. he moving plaform of he PM. Hence all he join parameer can be obained hrough olving l kinemaic equaion relaing l limb in form of Eq. (). S f i k he ih limb S S S S i 1 l (3) f i ni f i ni1 f i1 ( k 1 n) denoe he finie crew generaed by he kh join in S i he given poe of he PM. Eq. (3) can be equivalenly rewrien uing crew riangle produc reuling in clear algebraic mapping beween he join parameer he given poe S Join Join 1 Bae End-effecor Join n-1 Join n S f SM S f SM1 x S f SM n S f SM n 1 ik. In hi way he join parameer can be olved by algebraic derivaion. 3 Generalized mehod o olve kinemaic equaion According o Reference [6] he reulan finie crew compoied by everal finie crew ha he quai-vecor form of Eq. (1). hu he lef ide of Eq. (3) can alway be rewrien ino he following form z O Auhor' verion y ik

5 A Generalized Analyical Mehod o Solve Invere Kinemaic of Serial Parallel Mechanim Uing Finie Screw heory 5 0 i f i f i n 1 1 an i f i ni f i i rf i f i f i S S S (4) ik ik f i r f i i of ha limb. If he poe of he PM i given a S i of he ih limb are funcion of he join parameer PM PM an f 0 PM r he following equaion can be derived baed upon Eq. (3)-(5) i an PM an f i i PM rf i f i f i r (7) i PM an an S Eq. (6) i he mapping beween he join parameer relaing roaional moion of he ih limb he orienaion of he moving plaform. Eq. (7) i he mapping beween he join parameer relaing ranlaional moion of he limb he poiion of he moving plaform. When i given all join parameer can be analyically olved uing Eq. (6) (7). he deailed ep of invere kinemaic for PM are lied a follow: Sep 1: Formulae kinemaic equaion of each limb a Eq. (3) (Eq. (6) (7)); Sep : Solve roaional parameer of each limb uing Eq. (6); Sep 3: Solve ranlaion parameer of each limb uing Eq. (7). 4 Example Auhor' verion A PM wih Schoenflie moion i for example hi PM i compoed of four limb in which every wo limb placed oppoiely have he ame rucure i.e. P 1P P 3R ar b P 1P R ar ar c. Given S we olved wo limb P 1P P 3R ar b one limb P 1P R ar ar b in hi Secion. Limb P1PP3RaRb: he kinemaic equaion can be formulaed by Eq. (3) (6) (7) (5) (6)

6 6. Sun S. F. Yang. Huang J. S. Dai b b a a an an S rb b ra a P3 P P 1 P3 P P1 (8) an ba an an an an a b a b a b a b ba PM an a b 1 an an ab a b a b an a an b an an a b a b a b an a an b an an a b ba PM pba r (10) ba PM an an a b a b an r an r an an r r p ba a b a b an a an b an an a b he wo roaional parameer a a b b a b b a a b a P P P3 P3 b can be olved from Eq. (9) a a b a b a arcan b arcan (11) b a b a a a b b he hree ranlaional parameer P1 P P3 P P3 P P1 P P 3 P P3 P3 can be olved from Eq. (10) a Auhor' verion P3 P P3 P (9) (1)

7 A Generalized Analyical Mehod o Solve Invere Kinemaic of Serial Parallel Mechanim Uing Finie Screw heory 7 E 3 1 E3 ba r p ba ba PM an an i a uni marix of order hree i he kew marix of. ba Limb P1PRaRaRc: he kinemaic equaion can be formulaed c c 0 0 an an an rc c he wo roaional parameer a a a1 a P P S 1 ra a ra 1 a P P1 a1 a can be olved in he imilar manner a Eq. (9) (11). he hree ranlaional parameer olved from he poiion par of Eq. (13) 1 b ba a1 P1 P can be E3 ca PM r p ca (14) ca PM an an p ca a 1 a c an ra an a rc c a 1 a c an an r r a c c a a c a1 a c a1 a c an a an c an an a c exp a a E r a r a 1 3 P he oluion of Eq. (14) i A A B C a arcan 1 B C Auhor' verion P1 exp a 1 a E3 r a r a P P P (13)

8 8. Sun S. F. Yang. Huang J. S. Dai P exp a 1 a E3 r a r a P P (15) a ra r a B a a C ra ra P P A 1 1 r r In hi way all he join parameer of hi PM can be analyically olved. 5 Concluion hi paper preen a generalized analyical mehod o olve invere kinemaic of SM PM uing finie crew heory. he main meri of hi mehod are: (1) he mehod can be applied o ge analyical oluion of invere kinemaic for arbirary SM PM. () he main advanage of hi mehod i accuracy he analyical oluion can be direcly ued in rajecory planning precie conrol of mechanim. (3) Unied wih he auhor previou work all opological poiion orienaion (poe) velociy modeling analyi can be unified ino he yemaic conien framework of crew heory. Acknowledgmen hi reearch work i parially uppored by ianjin Reearch Program of Applicaion Foundaion Advanced echnology under Gran No. 16JCYBJC Reference 1. Craig J. J.: Inroducion o Roboic: Mechanic Conrol nd ediion. Addion-Weley (1989). Milenkovic P.: Serie oluion for finie diplacemen of planar four-bar linkage. ASME Journal of Mechanim Roboic (011) 3. Murray R. Li Z. X. Sary S.: A Mahemaical Inroducion o Roboic Manipulaion. CRC Pre (1994) 4. Paden B.: Kinemaic Conrol Robo Manipulaor. PhD hei Deparmen of Elecrical Engineering Compuer Science Univeriy of California Berkeley (1986) 5. Raghavan M. Roh B.: Invere kinemaic of he general 6R manipulaor relaed linkage. ASME Journal of Mechanical Deign (1993) 6. Sun. Yang S. F. Huang. Dai J. S.: A way of relaing inananeou finie crew baed on he crew riangle produc. Mechanim Machine heory (017) Auhor' verion

9 A Generalized Analyical Mehod o Solve Invere Kinemaic of Serial Parallel Mechanim Uing Finie Screw heory 9 7. ai L. W.: Robo Analyi Deign: he Mechanic of Serial Parallel Manipulaor. John Wiley & Son Inc (1999). 8. Wampler C. W.: Manipulaor invere kinemaic oluion baed on vecor formulaion damped lea-quare mehod. IEEE ranacion on Syem Man Cyberneic (1986) 9. Yang S. F. Sun. Huang. Li Q. C. Gu D. B.: A finie crew approach o ype ynhei of hree-dof ranlaional parallel mechanim. Mechanim Machine heory (016) 10. Yang S. F. Sun. Huang.: ype ynhei of parallel mechanim having 31R moion wih variable roaional axi. Mechanim Machine heory (017) Auhor' verion