A Comparative Study of Three Inverse Kinematic Methods of Serial Industrial Robot Manipulators in the Screw Theory Framework

Transcription

1 University f Wllngng Research Online Faculty f Engineering Infrmatin Sciences - Papers: Part B Faculty f Engineering Infrmatin Sciences A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk Emre Sariyildiz University f Wllngng, emre@uw.edu.au Eray Cakiray Istanbul Technical University Hakan Temeltas Istanbul Technical University Publicatin Details Sariyildiz, E., Cakiray, E. & Temeltas, H. (). A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk. Internatinal Jurnal f Advanced Rbtic Systems, 8 (5), 9-4. Research Online is the pen access institutinal repsitry fr the University f Wllngng. Fr further infrmatin cntact the UOW Library: research-pubs@uw.edu.au

2 A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk Abstract In this paper, we cmpare three inverse kinematic frmulatin methds fr the serial industrial rbt manipulatrs. All frmulatin methds are based n screw thery. Screw thery is an effective way t establish a glbal descriptin f rigid bdy avids singularities due t the use f the lcal crdinates. In these three frmulatin methds, the first ne is based n quaternin algebra, the secnd ne is based n dualquaternins, the last ne that is called expnential mapping methd is based n matrix algebra. Cmpared with the matrix algebra, quaternin algebra based slutins are mre cmputatinally efficient they need less strage area. The methd which is based n dual-quaternin gives the mst cmpact cmputatinally efficient slutin. Paden-Kahan sub-prblems are used t derive inverse kinematic slutins. 6-DOF industrial rbt manipulatr's frward inverse kinematic equatins are derived using these frmulatin methds. Simulatin experimental results are given. Keywrds screw, manipulatrs, rbt, industrial, serial, methds, kinematic, inverse, three, study, framewrk, cmparative, thery Disciplines Engineering Science Technlgy Studies Publicatin Details Sariyildiz, E., Cakiray, E. & Temeltas, H. (). A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk. Internatinal Jurnal f Advanced Rbtic Systems, 8 (5), 9-4. This jurnal article is available at Research Online:

3 Internatinal Jurnal f Advanced Rbtic Systems ARTICLE A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk Regular Paper Emre Sariyildiz, Eray Cakiray Hakan Temeltas Department f Cntrl Engineering, Istanbul Technical University, Turkey Crrespnding authr e-sariyildiz@htmail.cm Received 6 Apr ; Accepted 4 Aug Sariyildiz et al.; licensee InTech. This is an pen access article distributed under the terms f the Creative Cmmns Attributin License ( which permits unrestricted use, distributin, reprductin in any medium, prvided the riginal wrk is prperly cited. Abstract In this paper, we cmpare three inverse kinematic frmulatin methds fr the serial industrial rbt manipulatrs. All frmulatin methds are based n screw thery. Screw thery is an effective way t establish a glbal descriptin f rigid bdy avids singularities due t the use f the lcal crdinates. In these three frmulatin methds, the first ne is based n quaternin algebra, the secnd ne is based n dualquaternins, the last ne that is called expnential mapping methd is based n matrix algebra. Cmpared with the matrix algebra, quaternin algebra based slutins are mre cmputatinally efficient they need less strage area. The methd which is based n dual quaternin gives the mst cmpact cmputatinally efficient slutin. Paden Kahan subprblems are used t derive inverse kinematic slutins. 6 DOF industrial rbt manipulatr s frward inverse kinematic equatins are derived using these frmulatin methds. Simulatin experimental results are given. Keywrds Dual Quaternin, Industrial Rbt Manipulatr, Paden Kahan sub prblems, Quaternin, Screw Thery, Singularity Free Inverse Kinematic.. Intrductin In industrial applicatins f rbtic autmatin systems, it is demed that the rbt manipulatrs track a desired trajectry precisely. This gal can be achieved by finding a map which transfrms the desired trajectry int the mtin f jints f the rbt manipulatrs. It can als be described as a mapping frm Cartesian crdinate space t the jint space. Kinematic gives us this mapping withut cnsidering the frces r trques which cause the mtin. Since the kinematic based slutins are easy t btain requires less number f cmputatins cmpared with dynamical equatins, they are frequently used in the industrial rbt applicatins. Several methds are used in rbt kinematics screw thery is ne f the mst imprtant methds amng them. Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Int Cmparative J Adv Rbtic Study Sy, f, Three Vl. Inverse 8, N. Kinematic 5, 9-4 Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 9

4 Danialidis et al. cmpared screw thery with the mst cmmn methd in rbt kinematics called hmgenus transfrmatin methd they fund that screw thery based slutin ffers mre cmpact cnsistent way fr the rbt kinematics than hmgeneus transfrmatin ne []. Althugh screw thery based slutin methds have been widely used in many rbtic applicatins fr the last few decades, the elements f screw thery can be traced t the wrk f Chasles Pinst in the early 8s. Using the therems f Chasles Pinst as a starting pint, Rbert S. Ball develped a cmplete thery f screws which he published in 9 []. In screw thery every transfrmatin f a rigid bdy r a crdinate system with respect t a reference crdinate system can be expressed by a screw displacement, which is a translatin by alng a λ axis with a rtatin by a θ angle abut the same axis [3]. This descriptin f transfrmatin is the basis f the screw thery. There are tw main advantages f using screw thery fr describing the rigid bdy kinematics. The first ne is that it allws a glbal descriptin f the rigid bdy mtin that des nt suffer frm singularities due t the use f lcal crdinates. The secnd ne is that the screw thery prvides a gemetric descriptin f the rigid mtin which greatly simplifies the analysis f mechanisms [4]. Several applicatins f screw thery have been intrduced in the kinematic prblem. Yang Freudenstein were the first t apply line transfrmatins t spatial mechanism by using quaternin algebra [5]. Yang als investigated the kinematic f special five bar linkages using dual 33 rthgnal matrices [6]. Pennck Yang extended this methd t rbt kinematics [7]. J.M. McCarthy presented a detailed research n dual 3 3 rthgnal matrices its applicatin t velcity analysis using screw thery [8]. Defining screw mtin using dual 3 3 rthgnal matrices is nt a cmpact cmputatinally efficient slutin methd. Since, it needs 8 parameters t define screw mtin while just 6 parameters are needed it requires dual matrix calculatins. B. Paden investigated several subprblems that are called as Paden Kahan sub prblems [9 ]. The Paden Kahan sub prblems can be used t slve the general kinematic prblem by transfrming it int the Paden Kahan sub prblems. R.M. Murray et al. slved 3 DOF 6 DOF rbt manipulatr kinematics by using screw thery with 4 4 hmgeneus transfrmatin matrix peratrs Paden Kahan sub prblems []. Then J.Xie et al applied this methd t the 6 DOF space manipulatr []. Als, this methd was applied t the redundant rbt manipulatrs using a hybrid algrithm by W. Chen et al. [3]. They used expnential mapping methd t btain the general screw mtin f the rbt manipulatrs. Hwever, expnential mapping methd defines the general screw mtin by using less parameter than dual 33 rthgnal matrices; it requires 6 parameters t define the screw mtin while just 6 parameters are needed suffers frm cmputatinal lad. J. Funda analyzed transfrmatin peratrs f screw mtin he fund that dual peratrs are the best way t describe screw mtin als the dual quaternin is the mst cmpact efficient dual peratr t express screw displacement [4 5]. A. Perez J.M. McCarthy analyzed dual quaternin fr 4 DOF cnstrained rbtic systems [6]. R. Campa et al. prpsed kinematic mdel cntrl f rbt manipulatrs by using unit quaternins [7]. Finally E. Sariyildiz H. Temeltas investigated the kinematics f 6 DOF serial industrial rbt manipulatrs by using quaternins in the screw thery framewrk they shwed its superir perfrmance ver D H methd [8]. Mrever authrs als develped the methdlgy by emplying dual quaternin peratrs in rder t increase cmputatinal perfrmance [9]. In this paper, we present a cmparisn study fr the three inverse kinematic frmulatin methds which are all based n screw thery. In these methds, the first ne uses quaternins as a screw mtin peratr which cmbines a unit quaternin plus a pure quaternin, the secnd ne uses the dual quaternin, which is the mst cmpact efficient dual peratr t express the screw displacement, the last ne uses matrix algebra t express the screw mtin. The first tw methds given in [8] [9] are extensively develped in mathematical frmulatins all f these methds are analyzed in details. Additinally, the methds are implemented int the 6 DOF industrial rbt manipulatr namely Stäubli RX 6L in rder t shw real time perfrmance results. Cmparisn results f these frmulatin methds are given in sectin VI. This paper is als included the mathematical preliminary in sectin II, screw thery by using matrix quaternin algebras in sectin III, the kinematic scheme f n DOF serial rbt manipulatr in sectin IV, frward inverse kinematic slutins f the 6 DOF serial rbt manipulatr in sectin V experimental results in sectin VII. Simulatins are made by using expnential mapping methd prpsed methds. The methds are cmpared with respect t cmpactness f transfrmatin peratrs cmputatinal efficiency. Cnclusins future wrks are drawn in the final sectin.. Mathematical Preliminary Quaternins In mathematics, the quaternins are hyper cmplex numbers f rank 4, cnstituting a fur dimensinal vectr space ver the field f real numbers [, ]. The quaternin can be represented in the frm, q ( q, q ) () where q is a scalar qv ( q, q, q3 ) is a vectr. If q then, we get pure quaternin. The sum the prduct f tw quaternins are then, V Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

5 qa qb ( qa qb ),( qav qbv ) () q q ( q q q q ),( q q q q q q ) (3) a b a b av bv a bv b av av bv where,, dente quaternin prduct, dt prduct crss prduct respectively. Cnjugate, nrm inverse f the quaternin can be expressed in the frms q ( q, ) ( q, q, q, q ) q v 3 3 q q q q q q q that satisfies the relatin (4) (5) q q q q (6) q q q q. When, we get unit quaternin that satisfies the relatin q q. q Definitin : Let qa qb be tw pure quaternins the prduct f these tw quaternins be q q ( q q q q ), a b a b av bv ( q q q q q q ) a bv b av av bv ( q q ),( q q ) av bv av bv Then, let s define tw new functins by using the prduct f tw pure quaternins given by V q q a b av bv (7) q q be the vectr part f the quaternin multiplicatin S qa qb qav qbv be the scalar part f Dual Quaternin the quaternin multiplicatin The dual quaternin can be represented in the frm q ( q S, q V ) r q q q (8) where qs q S q S is a dual scalar, q V qv qv is a dual vectr, q q are bth quaternins is the dual factr [, 3]. The sum the prduct f tw dualquaternins is then a b a b a b q q ( q q ) ( q q ) (9) a b a b a b a b q q ( q q ) ( q q q q ) () where, dente the quaternin dualquaternin prducts respectively. Cnjugate, nrm inverse f the dual quaternin are similar t the quaternin s cnjugate, nrm inverse respectively. q q ( q ) () q q q () q q q q (3) When q, we get a unit dual quaternin that satisfies the relatin q q q q q. Definitin : Let qa qb be tw dual quaternins the prduct f tw dual quaternins be q ab q aq b qab qab ( qabs, qabv ) ( qabs, qabv ) (4) Then, let s define fur new functins by using the prduct f tw dual quaternins definitin given by, S R q q q be scalar part f the real a b abs part f multiplicatin S D q aq b qabs be scalar part f the dual part f multiplicatin V R q aq b q ab V be vectr part f the real part f multiplicatin V D q aq b q abv be vectr part f the dual part f multiplicatin Pint Line Transfrmatins Using Quaternins Unit quaternin can be defined as a rtatin peratr [4 6]. Rtatin abut an axis f n by an angle can be expressed by using the unit quaternin given by q cs,sin n (5) Let s define a directin vectr in pure quaternin frm given by l (, v ). Here, the vectr v is a unit directin vectr. Rtatin f l (, v) can be defined given by R l q l q 6) A general rigid bdy transfrmatin cannt be expressed by using nly a unit quaternin. Since as stated abve, we need at least six parameters t define rigid bdy transfrmatin. Therefre, we shuld use at least tw quaternins t define rigid bdy transfrmatin. The unit dual quaternins can als be used as a rigid bdy transfrmatin peratr [7]. Althugh it has eight parameters it is nt minimal, it is the mst cmpact efficient dual peratr [4, 5]. This transfrmatin is very similar with pure rtatin; hwever, nt fr a pint but fr a line. A line in Plücker crdinates L( m,d ) ( l l m in dual quaternin frm, see Appendix) can Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk

6 LR R R be transfrmed t ( m, d ) by using unit dualquaternins as fllws, [8] 3. Screw Thery l q l q (7) R All prper rigid bdy mtins in 3 dimensinal space, with the exceptin f pure translatin, are equivalent t a screw mtin, that is, a rtatin abut a line tgether with a translatin alng the line [, 7]. If the axis f the screw mtin des nt pass thrugh rigin as shwn in figure then, the screw mtin is expressed given by I3x3 p (, ) k T R d d I3x3 p R(, d) kd ( I3x3 R(, d)) p (8) Rtatin: q cs, sin d Translatin: t q p q p q () where q is the amunt f pure translatin is the psitin vectr f sme pint n the line in the pure quaternin frm. Screw Thery Using Dual Quaternin A screw mtin can be expressed by using dualquaternin as fllws: q S qv t q q V q S t t q V k cs sin k sin sin cs d m d k cs k sin ( ) d m () Figure. General screw mtin Screw Thery Using Quaternin In equatin (8), screw mtin is expressed by using 4x4 hmgenus transfrmatin matrices. It uses sixteen parameters while just 6 parameters are needed. We can express screw mtin in a mre cmpact frm than hmgenus transfrmatin matrices by using quaternin algebra. If we separate general screw mtin as a rtatin translatin then, we have Rtatin: R(, d ) Translatin: ( (, )) k d I3x3 R d p (9) Then, we can express these equatins by using quaternins as fllws This representatin is very cmpact als it uses algebraically separates the angle pitch k infrmatin. If we write k d d m then, the equatin () becmes mre cmpact as the fllwing () Further details f general screw mtin frmulatin by using dual quaternin can be fund in [9]. 4. Manipulatr Kinematic A. Frward Kinematic T find frward kinematic f serial rbt manipulatr we fllwed these steps: Ntatin:. Label the jints the links: Jints are numbered frm number t n, starting at the base, the links are numbered frm number t n. The i th jint cnnects link i t link i.. Cnfiguratin f jint spaces: Fr revlute jint we describe rtatinal mtin abut an axis we measure all jint angles by using a right hed crdinate system. Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

7 3. Attaching crdinate frames (Base Tl Frames): Tw crdinate frames are needed fr n degrees f freedm pen chain rbt manipulatr. The base frame can be attached arbitrarily but in general it is attached directly t link the tl frame is attached t the end effectr f rbt manipulatr. Frmulizatin: Quaternin based methd. Determining the jint axis vectrs: First we attach an axis vectr which describes the mtin f the jint.. Obtaining transfrmatin peratrs: Transfrmatin peratrs f all jints are btained by using quaternins as fllws i i Rtatin: q i cs, sin d i i i i i i Translatin: q p q p q (3) where is an arbitrary pint n the i th axis i,, n 3.Frmulizatin f rigid mtin: Using (3) () rigid transfrmatin f serial rbt manipulatr can be expressed as q q q q n n n n n n n n n q q p q q p q q (4) where q n q n indicate rtatin translatin respectively. The rientatin the psitin f the end effectr can be expressed as fllws n n Orientatin: l q l q Psitin: ep n ep n n p q p q q (5) where p, l indicate the psitin the rientatin f ep the end effectr befre the transfrmatin pep, l indicate the psitin the rientatin f the end effectr after the transfrmatin. Dual Quaternin based methd. Determining jint axis vectrs mment vectrs: First we attach an axis vectr which describes the mtin f the jint. Then the mment vectr f this axis is btained fr revlute jints by using the equatin (A.).. Obtaining transfrmatin peratr: Transfrmatin peratrs f all jints are btained by using dual quaternins as fllws n q i ( q is, q iv ) q r i qi qi i i where q i cs, sin d i (6) qi pi qi pi qi qi r q i sin m i (7) 3. Frmulizatin f rigid mtin: Using the equatin () rigid transfrmatin f serial rbt manipulatr can be expressed as q q q q n n (8) The rientatin the psitin f the end effectr can be expressed by using the equatin (A.4) as fllws Orientatin: l qn l qn Psitin: pep qvn qvn qvn qvn qvn q Vn (9) B. Inverse Kinematic Paden Kahan sub prblems are used t btain the inverse kinematic slutin f rbt manipulatr. There are sme main Paden Kahan sub prblems als new extended sub prblems [9,, 3]. Three f them which ccur frequently in inverse slutins fr cmmn manipulatr design are used in the inverse kinematic prblem. These subprblems can be seen in Appendix. T slve the inverse kinematic prblem, we reduce the full inverse kinematic prblem int the apprpriate sub prblems. Slutin f the inverse kinematic prblem f 6 DOF serial rbt arm is given in the next sectin. 5. Kinematic Analysis f 6 Df Industrial Rbt Arm In this sectin, the kinematic prblem f serial rbt arm which is shwn in figure is slved by using the new frmulatin methds. Quaternin Based Methd Frward Kinematics Step: Firstly, the axes f all jints shuld be determined. d [,,] d [,, ] d3 [,, ] d4 [,,] d5 [,, ] d 6 [,,] (3) Any pint n these axes can be written as p [,, l ] p [,, l ] p 3 l l [,, ] p4 [ l l,, l ] p5 [ l l,, l ] 6 l l l p [,, ] (3) Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 3

8 pb intersectin f the wrist axes the secnd ne is which is at the intersectin f the first tw axes. The last three jints d nt affect the psitin f the pint pw the first tw jints d nt affect the psitin f the pint p. Then we can easily write the fllwing equatins, b 3 w w 6 6 q p q q q p q q (34) b q p q q p (35) The psitin f the pint pb is free frm the angles f first tw jints. If we subtract these tw equatins frm each ther then, we get b 3 w 3 3 b q p q q q p q q b q pw q q q p q q (36) If we take the end effectr psitin qin ( q, q, q, q3 ) at the intersectin f the wrist axes, we have pw qin. Hence we can write Figure. 6 DOF serial arm rbt manipulatr in its reference cnfiguratin Step : The transfrmatin peratrs which are in quaternin frm can be written by using the equatin (3). Step3: Finally, the frward kinematic equatin f serial rbt manipulatr can be btained as fllws, q Orientatin: 6 l6 q6 Psitin: q6 p6 q6 q6 (3) where q6 q q q3 q4 q5 q6 i i i i i i i i q q p q q p q q Inverse kinematics q p q p q. (33) In the inverse kinematic prblem f serial rbt manipulatr, we have rientatin psitin knwledge f the end effectr. These are tw quaternins we will calculate all jint angles by using these quaternins. The first ne gives us the rientatin knwledge f the rbt manipulatr q in the secnd ne gives us the psitin knwledge f the end effectr q in. T find all jint angles cmplete inverse kinematic prblem must be cnverted int the apprpriate Paden Kahan sub prblems (see Appendix). This slutin can be btained as the fllwing, Step : Firstly, we put tw pints at the intersectin f the rtatin axes. The first ne is p which is at the w 3 w 3 6 w 6 6 in q p q pb q p q q pb q pb (37) Using the prperty that distance between pints is preserved by rigid mtins; take the magnitude f bth sides f equatin (37), we get sub prblem 3. The parameters f sub prblem 3 are p = p w qin q = p b. 3 can be fund by using sub prblem 3. Step : If we translate pw by using knwn 3 we btain a new pint p. We get sub prblem by using the pint p as the initial psitin the pint q as the final psitin f the sub prblem mtin. The pints p q can be frmulized as fllws ( 3 w 3) 3 q q p q q q pb q pb (38) in ( 3 w ) q q p q p q p q q q (39) Hence, p q3 pw q3 p3 q3 p3 q3 q q in. can be fund by using sub prblem. Step 3: T find wrist angles we put a pint pc which is n the d 6 axis it des nt intersect with d 4 d 5 axes. Thus the pint pc is nt affected frm the last jint angle. Furth fifth jints angles determine the psitin f the pint p c. Fr knwn, 3 angles, we can write the fllwing equatin 45 c in q p q q q q q q q q q q (4) The equatin (4) gives us the sub prblem. T btain the parameters f sub prblem we shuld find the pints p q. The pint p is equal t p c. We shuld just find q. The pint q can be fund given by m c m m m m m t t q q p q q p q q p q q q (4) in 4 Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

9 Where qm ( q q q3) qin t q q p q p t q q p q ( q q ) p ( q q ) 3 3 ( q q ) p ( q q ) (4) l l l q l q q l l q ( 6 6 ) 6 l l l q l q q l l q (46) ( 5 5 ) 5 q q q q q q q Where q q q q q q Step 4: The first five jint angles are btained. Only the last jint angle is unknwn. Of curse, we can find the last jint angle by using the given sub prblems hwever; it can be als easily slved by using algebraic equatins fr the knwn jint angles. In this step we will find the last jint angle by using algebraic equatins. Paden Kahan based slutin will be given in dual quaternin based slutin. The last jint angle can be fund frm the rientatin part as the fllwing, k ( q q q q q ) qin 6 arctan k(), ( k()) (43) Dual Quaternin Based Methd Frward Kinematics Step: Firstly, the axes f all jints shuld be determined. d [,,] d [,, ] [,, ] d [,,] d [,, ] d [,,] 44) 4 5 Then, the mment vectrs f all axes must be calculated. m = p d m = p d m 3 = p 3 d3 m 4 = p 4 d4 m 5 = p 5 d5 m 6 = p 6 d 6 (45) where p,p,p 3,p 4,p5 p6 are any pint n the crrespnding directin vectrs d,d,d 3,d 4,d5 d 6. Step : The transfrmatin peratr that is in dualquaternin frm can be written by using the axis mment vectrs the equatin (7). Step3: Finally, the frward kinematic equatin f serial rbt manipulatr can be btained as fllws, d 3 6 Then, the rientatin f the end effectr is the real part f the dual quaternin l 6 the psitin f the end effectr is V R q 5l 5q 5 V D q 5l5q 5 V Rq 6l6q 6 p6 V Rq 6 l 6 q 6 V Dq 6 l 6 q 6 V Rq 6l6q 6 Inverse kinematic (47) In the inverse kinematic prblem f serial rbt manipulatr, we have the psitin rientatin infrmatins f the end effectr such that q in ( qin, qin ) where qin ( q, q, q, q3 ), that is the rientatin f the end effectr, is the real part f the dual quaternin q in qin ( q, q, q, q3 ), that is the psitin f the end effectr, is the dual part f the f the dual quaternin q in. The general inverse kinematic prblem shuld be cnverted int the apprpriate Paden Kahan subprblems (see Appendix) t btain the inverse kinematic slutin. This slutin can be btained as fllws, Step : Firstly, we put tw pints at the intersectin f the rtatin axes. The first ne is p which is at the intersectin f the wrist axes the secnd ne is w pb which is at the intersectin f the first tw axes. The last three jints d nt affect the psitin f the pint p the first tw jints d nt affect the psitin f the pint p w b. If we use the prperty that distance between the pints is preserved by rigid mtins we get the fllwing equatin q V Rq lq ( V R q l q V D q l q ) ( V R q l q V D q l q ) V R q l q V R q l q ( V R q l q V D q l q ) qin p ( V R q l q V D q l q ) V R q l Using the prperty that distance between the pints is preserved by rigid mtins; take the magnitude f bth sides f equatin (48), we get b (48) Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 5

10 ( ) ( ) V R q l q V D q l q V R q l q V D q l q V R q l q V R q l q V R l V D l V R l V D l V R l V R l qin p b (49) The equatin (49) gives us the sub prblem 3. The parameters f the sub prblem 3 are a V Rl 6 V Dl b = V Rl V Dl V R l V D l V R l V R l, V R l V D l V R l V R l l is the axis f the jint 3 that is d3 δ = q p. 3 can be fund by using the sub prblem 3. Step : If we use knwn 3 in the equatin (48) then, we btain V R q l q V D q l q V R q l5 q V D q l5 q V Rq l6 q q in V R q l q where l6 q 3l6q 3 l 5 q 3l5 q 3. The equatin (5) gives us the sub prblem. The parameters f subprblem are a V Rl 6 V Dl 6 in V R l5 V D l5 V R l6 V R l6 l is axis f the jint that is d, l is the axis f the jint that is d b q in. can be fund by using the sub prblem. Step3: T find the wrist angles, let s cnsider a pint pi p6 d 6 (initial pint) that is n the axis d 6, it is nt cincident with the d 4 d 5 axes. Tw imaginer axes are used t find p e (end pint) that is the psitin f the pint pi after the rtatin by 4 5 angles. The pint pi is the intersectin pint f the tw imaginer axes. Let s define the tw imaginer axes which are n the d6 axis intersect n the pint pi given by d 7 [,,], [,,] d 8, (5), b p p p [ d, ly ly d, lz lz lz d ]. m p d. i 7 8 6x 6y 6z The mment vectrs are m7 pi d 7 8 i 8 Then we can easily write, V Rq 45l 8 q 45 V Dq 45l 8 q 45 V Rq 45 l7 q 45 V Dq 45 l7 q 45 V Rq 45l8 q q in d 6 V R q l q The equatin (5) gives us the sub prblem. The parameters f sub prblem are a V Rl 8 V Dl 8 V R l7 V D l7 V R l8 V R l8 l is the imaginer axis d 7, l is the imaginer axis d8 b qin d can be fund by using the subprblem. Step4: T find the last jint angle, we need a pint which is nt n the last jint axis. We call it pd p5 d 5. Tw imaginer axes are used t find p d that is the psitin f the pint pd after the rtatin by 6. The pint pd is the intersectin pint f the tw imaginer axes. Let s define the tw imaginer axes which are n the d5 axis intersect n the pint pd given by d 9 [,, ], d [,,] p d = p 9 = p [ d5x, ly ly d5y, lz lz lz d5z ]. The mment vectrs are m9 p9 d 9, m p d. Then we can easily write, V R q l q V D q l q V R q 6l9 q 6 V D q 6l9 q 6 V Rq 6l q qin d5 V R q l q The equatin (5) gives us the sub prblem. The parameters f sub prblem are a V Rl V Dl V R l V D l V R l V R l 9 9 (5),, (5) 6 Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

11 l is axis f the jint 6 that is d 6, b qin d 5. 6 can be fund by using the sub prblem. 6. Simulatin Results Stäubli RX6L industrial rbt arm is used fr the simulatin studies. Stäubli RX6L series rbt arms feature an articulated arm with 6 degrees f freedm fr high flexibility. It spreads a wide range area in the industrial rbt applicatins. The kinematic simulatin studies are made by using Matlab virtual reality tlbx (VRML) f Matlab. Stäubli RX6L iges file which can be freely btained frm the Stäubli s web page, is als used fr the animatin applicatin which is shwn in Figure 3. In this animatin a single rbt arm carries a bx frm its initial psitin t the target psitin as shwn in figure 3. T implement this case, first a path is determined fr the bx. Then, the inverse kinematic prblem f serial arm is slved by using this path. Here, a cmparative study f the presented methds the expnential mapping methd is wrked ut. 6 DOF rbt arm s frward inverse kinematic slutins using expnential mapping methd can be fund in []. Expnential mapping methd uses hmgeneus transfrmatin matrices as a screw mtin peratr. Hmgeneus transfrmatin matrices require 6 memry lcatins fr the definitin f rigid bdy mtin while the quaternins require eight memry lcatins. The strage requirement affects the cmputatinal time because the cst f fetching an per frm memry exceeds the cst f perfrming a basic arithmetic peratin. The perfrmance cmparisn f hmgeneus transfrmatin matrix quaternin peratrs can be seen in table table. Methd Strage Multiplies Add & Ttal Subtracts Rtatin Matrix Quaternin Table. Perfrmance cmparisn f rtatin peratins Methd Strage Multiplies Add & Ttal Subtracts Hmgenus Trans. Matrix Dual Quaternin Table. Perfrmance cmparisn f rigid transfrmatin peratins In rder t btain the rigid bdy transfrmatin fr a n link serial rbt manipulatr 64( n ) multiplicatins 48( n ) additins must be dne if expnential mapping methd is used. 57( n ) multiplicatins 4( n ) additins must be dne if quaternin based methd is used. 48( n ) multiplicatins 4( n ) additins must be dne if dual quaternin based methd is used. Fig. 4 shws that as the degrees f freedm increase the methd which uses dual quaternin as a rigid bdy transfrmatin peratr becmes mre advantageus. (I) (II) Numbers f Ttal Calculatins Hm. Trans. Matrix Quaternin Dual-Quaternin 5 5 Degrees f Freedm Figure 4. Perfrmance cmparisn f the rigid bdy transfrmatin chaining peratins Cmputatinal efficiency f these three frmulatin methds are given in figures 5 6. (III) (IV) Figure 3. Stäubli RX6 industrial rbt arm carries a bx frm its initial psitin t the target psitin Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 7

12 7. Experimental Study Figure 5. Simulatin times f the frward kinematic slutins (secnd) In the experimental study, we used Stäubli RX series serial rbt arm CS8 cntrller which includes a lw level prgramming package t cntrl the rbt under VxWrks real time perating system. The kinematic algrithms are applied t Stäubli RX 6L rbt arm, that is shwn in figure 7, by using Stäubli Rbtics LLI Prgramming Interface S6.4, which is a C prgramming interface fr lw level rbt cntrl. LLI sts fr Lw Level Interface it is a sftware package which includes minimum functins required t cnstruct a rbt cntrl mechanism via C/C++ API [3]. The kinematic algrithms are written in C++ language by using the library functins f LLI sftware package embedded t the cntrller. Figure 6. Simulatin times f the inverse kinematic slutins (secnd) As it can be seen frm figures 4, 5 6, the methds which use quaternin dual quaternin as a screw mtin peratr are mre cmputatinally efficient than expnential mapping methd. Since, quaternin dual quaternin methds describe screw mtin using less parameter have less cmputatinal lad. Hwever bth quaternin dual quaternin describe screw mtin using eight parameters, dual quaternin gives us better results than quaternin. Since, screw thery with dual quaternin methd is mre cmpact than quaternin based methd. And als as the degree f freedm f the system increases the methd which uses quaternin as a screw mtin peratr gets mre cmplicated. Therefre, dual quaternin gives us better results than quaternin in kinematics. The cmputatin time is evaluated using Matlab s tic tc cmms n a Cre Du. GHz PC with GB RAM. In rder t verify the simulatin results we made tw different path tracking applicatins by using the prpsed methds. The path tracking applicatins is implemented by using the fllwing blck diagram which shwn in figure 8. In this blck diagram a desired path, which is an elliptic path fr the first applicatin, is generated in the desired path blck it is transferred t the inverse kinematic blck. In the inverse kinematic blck we btain the jint angles by using the prpsed inverse kinematic algrithms. Then the angles which are btained in the inverse kinematic blck are directly applied t the rbt manipulatr. Rbt psitin rientatin are directly measured als they are calculated in the frward kinematic blck. Desired path is cmpared with the measured calculated results the psitin rientatin errrs are btained. Figure 7. Stäubli RX 6L serial rbt arms psitin psitin psitin Psitin Errr Clck time DP rientatin Desired Path IK angles rientatin Inverse Kinematics angles FK rientatin Frward Kinematics Orientatin Errr Figure 8. Blck Diagram f Path Tracking Algrithm 8 Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

13 In the first applicatin, an elliptical path is generated n the x z crdinates plane fr the rbt arm. Then, the inverse kinematic is slved by using this path. Figure 9 shws the desired measured paths fr the elliptical path tracking applicatin. Path tracking results are shwn in figures 9,,. Distance n the crdinate z (mm) Distance n the cridante x (mm) Figure 9. Path tracking n the crdinates x z Errr n the crdinate x (mm).5 x Errr n the crdinate z (mm) x -3 desired path measured path - 5 Figure. Path tracking errrs n the crdinates x z crdinates Jint (rad) Jint (rad) Jint 3 (rad).5 5 Jint (rad/sec) Jint 4 (rad/sec) x -3-5 x -3-5 Jint (rad/sec) Jint 5 (rad/sec) Figure. Jint velcities f rbt arm during the elliptical path tracking applicatin Figure shw the jint psitin velcities during the elliptical path tracking applicatin. Bth f the jint psitin velcities are smth. Figure shws the path tracking errr. As it can be directly seen frm the figures 9,, a satisfactry path tracking applicatin is implemented fr an elliptical path by using the prpsed methds. We als defined a cubic path which passes thrugh the singular cnfiguratins f rbt arm. Then, we implemented the prpsed methds t the path tracking applicatin. The path tracking results fr the psitin rientatin f the serial rbt arm are shwn in figures 3, 4 5. Figure 3 shws the psitin f the rbt arm s end effectr n the x, y z crdinates. Figure 4 shws the rientatin angles f the rbt arm using the rll, pitch yaw angles. Ellipses, which are shwn in figures 3 4, shw that rbt arm passes thrugh singular cnfiguratins at these pints. Figure 5 shws path tracking errrs fr psitin rientatin. As it can be directly seen frm the figures 3, a satisfactry singularity free trajectry tracking applicatin is implemented by using the prpsed methds. Jint 3 (rad/sec) Jint 6 (rad/sec) x -3-5 Jint 4 (rad).6.4. Jint 5 (rad).6.4. Jint 6 (rad) Figure. Jint psitin f rbt arm during the elliptical path tracking applicatin Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 9

14 Distance n the crdinate x (mm) Lcus f rbt arm singular cnfiguratins desired path measured path Distance n the crdinate y (mm) desired path measured path Lcus f rbt arm singular cnfiguratins Distance n the crdinate z (mm) Lcus f rbt arm singular cnfiguratins desired path measured path Figure 3. Path tracking fr the x, y z crdinates Rll Angle (Degree) Lcus f rbt arm singular cnfiguratins desired path measured path Pitch Angle (Degree) Lcus f rbt arm singular cnfiguratins desired path measured path Yaw Angle (Degree) desired path measured path Lcus f rbt arm singular cnfiguratins Figure 4: Path tracking fr the Rll, Pitch Yaw rientatin angles - 3 Errr n the crdinate x (mm) Errr in the Rll Angle (rad) x x Errr n the crdinate y (mm) Errr in the Pitch Angle (rad) x -4 Figure 5: Path tracking errrs fr psitin rientatin x Errr n the crdinate z (mm) Errr in the Yaw Angle (rad) x x Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

15 8. Cnclusin In this paper, three frmulatin methds f the kinematic prblem f serial rbt manipulatrs have been cmpared in terms f cmputatinal efficiency cmpactness. These frmulatin methds are based n screw thery. The main advantage f screw thery in rbt kinematics lies in its gemetrical representatin f link jint axes f a manipulatr, giving better understing f its cnfiguratin in the wrkspace aviding singularities due t the use f the lcal crdinates. Thus, the prpsed methds are singularity free their gemetrical descriptin is quite simple. In these frmulatin methds, the first ne uses quaternins as a screw mtin peratr which cmbines a unit quaternin plus pure quaternin, the secnd ne uses dual quaternins the last ne that is called expnential mapping methd uses 4x4 hmgeneus matrices as a screw mtin peratr. Screw theries with quaternin dual quaternin methds are mre cmputatinally efficient than expnential mapping methd. And als screw thery with dual quaternin is the best methd. On these accunts, the wider use f the screw thery based methds with dual quaternin in rbt kinematic studies has t be cnsidered by rbtics cmmunity. The use f quaternins dual quaternins in trajectry planning mtin cntrl prblems fr rbtic manipulatrs are nt discussed in this study. Hwever thse are leaved fr future studies. Als, velcity dynamic analysis based n screw thery with quaternin algebra shuld be studied in the future wrks. 9. Appendix A. Line Gemetry Dual Numbers A line can be cmpletely defined by the rdered set f tw vectrs. First ne is a pint vectr p which indicates the psitin f an arbitrary pint n line, the ther vectr is a free directin vectr d which gives the line directin. A line can be expressed as: L( p, d ) (A.) The representatin L( p, d) is nt minimal, because it uses six parameters fr nly fur degrees f freedm. Plücker Crdinates An alternative line representatin was intrduced by A. Cayley J. Plücker. Finally this representatin named after Plücker [3]. Plücker crdinates can be expressed as: Intersectin f Lines Lp ( m, d) where m p d (A.) Let s define tw lines in Plücker crdinates given by L p a a a ( m, d ) ( m, d ) (A.3) L p b b b The intersectin pint f tw intersectin lines can be expressed as the fllwing, [33] r d m (( d m ) d ) d d m (( d m ) d ) d (A.4) a a b b a a b b a a b b where r indicates the intersectin pint. Dual Numbers The dual number was riginally intrduced by Cliffrd in 873 [34]. In analgy with a cmplex number a dual number can be defined as: u u u (A.5) where u u are real number. Dual numbers can be used t express Plücker crdinates given by. where Lp ( u, u ) u u (A.6) u ( x x, y y, z z ) u pd A. Paden Kahan sub prblems by using quaternin algebra Sub prblem : Rtatin abut a single axis A pint a rtates abut the axis f l until the pint a is cincident with the pint b. This rtatin is shwn in Figure A.. Figure A.: A line in Cartesian crdinate system Figure A.: Rtate a abut the axis f l until it is cincident with b Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk

16 Let r be a pint n the axis f l, x = a -r y = b-r be tw vectrs. The rtatin angle abut the axis f l can be fund as fllws, arctan S l x y, S x y (A.7) where S l l S l l S l x S l y where x x S l xl y q x q S l q x q l q [cs,sin ] l. Sub prblem : Rtatin abut tw subsequent axes Firstly, a pint a rtates abut the axis f l by then abut the axis f l by, hence the final lcatin f a is cincident with the pint b. This rtatin is shwn in Figure A.3. Let r be the intersectin pint f the tw axes, x = a -r y = b-r be tw vectrs. Let c be the intersectin pint f the rtatins that is shwn in Fig. (A.3) let z cr, be the vectr that is defined between the pints c r, z [, z ] pure quaternin frm f the vectr z. We can als define tw rtatins given by q x q z q y q where, q [cs,sin ] l q [cs,sin ] l x = [, x] y = [, y ]. (A.8) Since l, l l l are linearly independent, we can write z l l, V l l S l l S l l S l y S l x x S l l V l l (A.9) Sub prblem 3: Rtatin t a given distance A pint a rtates abut the axis f l until the pint is a distance frm b as shwn in Fig. A.4. Let r be a pint n the axis f l x [, a r] y [, b r] be pure quaternin frms f the vectrs x y respectively. The rtatin angle abut the axis f l can be fund as fllws, x y cs x y where S l x y S x y arctan, x [, x ] x S l x l y [, y ] q x q S l q x q l S l a b (A.) q [cs,sin ] l. (A.) Figure A.3: Rtate a abut the axis f l fllwed by a rtatin arund the axis f l until it cincident with the pint b. Figure A.4: Rtate a abut the axis f l until it is a distance frm b Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4

17 . References [] N.A. Aspragaths, J.K. Dimitrs, A cmparative study f three methds fr rbt kinematics, IEEE Transactins n Systems, Man, Cybernetics, Part B: Cybernetics, vl. 8, pp.35 45, Apr. 998 [] R. S. Ball, The Thery f Screws, Cambridge, U.K., Cambridge Univ.Press, 9 [3] J.D. Adams, D.E. Whitney, Applicatin f Screw Thery t Cnstraint Analysis f Mechanical Assemblies Jined by Features, Jurnal f Mechanical Design, vl. 3, Issue, pp. 6 33, March [4] Z. Huang Y.L. Ya, Extensin f Usable Wrkspace f Rtatinal Axes in Rbt Planning, Rbtica, vl. 7, pp.93 3, 999 [5] A.T. Yang F. Freudenstein, Applicatin f dualnumber quaternins t the t the analysis f spatial mechanism, ASME Transactins Jurnal f Applied Mechanics, vl. 86, pp. 3 38, June 964 [6] A.T. Yang, Displacement analysis f spatial five link mechanism using (3 x 3) matrices with dual number elements, ASME Jurnal f Engineering fr Industry, vl. 9, n., pp. 5 57, Feb.969 [7] G.R. Pennck A.T Yang, Applicatin f dualnumber matrices t the inverse kinematic prblem f rbt manipulatrs, ASME Jurnal f Mechanisms, Transmissins Autmatin in Design, vl. 7, pp. 8, 985 [8] J. M. McCarthy, Dual rthgnal Matrices in manipulatr kinematics, Internatinal Jurnal f Rbtics Researches, vl. 5, pp. 45 5, 986 [9] B. Paden, Kinematics Cntrl Rbt Manipulatrs, PhD Thesis Department f Electrical Engineering Cmputer Science, University f Califrnia, Berkeley,986 []W. Kahan, Lectures n cmputatinal aspects f gemetry, Department f Electrical Engineering Cmputer Science, University f Califrnia, Berkeley,986 [] M. Murray, Z. Li, S.S. Sastry, A mathematical intrductin t rbtic manipulatin, Bca Ratn FL: CRC Press, 994. [] J.Xie, W. Qiang, B. Liang C. Li, Inverse Kinematics Prblem fr 6 DOF Space Manipulatr Based On The Thery f Screw, Internatinal Cnference n Rbtics Bimimetics, Dec. 7 [3] W. Chen, M. Yang, S. Yu, T. Wang, A hybrid algrithm fr the kinematic cntrl f redundant rbts, IEEE Internatinal Cnference n Systems, Man Cybernetics, vl.5, pp , 3 Oct. 4 [4] J. Funda R.P. Paul, A cmputatinal analysis f screw transfrmatins in rbtics, IEEE Transactins Rbtics Autmatin, vl.6, pp , June 99 [5] J. Funda, R.H. Taylr R.P. Paul, On hmgeneus transfrms, quaternins, cmputatinal efficiency, IEEE Transactins Rbtics Autmatin, vl.6, pp , June 99. [6] A. Perez J. M. McCarthy, Dual Quaternin Synthesis f Cnstrained Rbtic Systems, Jurnal f Mechanical Design, vl.6, n.3, pp , 4. [7] R. Campa, K. Camarill, L. Arias, Kinematic Mdeling Cntrl f Rbt Manipulatrs via Unit Quaternins: Applicatin t a pherical Wrist, Prceedings f the 45th IEEE Cnference n Decisin & Cntrl, San Dieg, USA, 3 5 Dec 6 [8] E. Sariyildiz H. Temeltas, Slutin f Inverse Kinematic Prblem fr Serial Rbt Using Quaterninns, Internatinal Cnference n Mechatrnics Autmatin, 9 [9] E. Sariyildiz H. Temeltas, Slutin f Inverse Kinematic Prblem fr Serial Rbt Using Dual Quaterninns Plücker Crdinates, Advanced intelligent mechatrnics, 9 [] W.R. Hamiltn, Elements f Quaternins, Vl. I, Vl. II, New Yrk, Chelsea, 869. [] J.C.K JChu, Quaternin Kinematic Dynamic Differential Equatins, IEEE Transactins Rbtics Autmatin, vl.8, pp , Feb 99 [] D. Han, Q. Wei Z. Li, Kinematic Cntrl f Free Rigid Bdies Using Dual Quaternins, Internatinal Jurnal f Autmatin Cmputing, pp.39 34, July 8 [3] K. Dngmin, Dual Quaternin Applicatin t Kinematic Calibratin f Wrist Munted Camera, Jurnal f Rbtic Sysyems, 3(3), 53 6, 996 [4] R. Mukundan, Quaternins: Frm Classical Mechanics t Cmputer Graphics Beynd, Prceedings f the 7 th Asian Technlgy Cnference in Mathematics, [5] J. C. Hart, G. K. Fracis, L.H. Kaauffman, Visualizing Quaternin Rtatin, ACM Transactins n Graphics, vl.3, n. 3, pp , July 994 [6] Q. Tan J.G. Balchen, General Quaternin Transfrmatin Representatin Fr Rbt Applicatin, IEEE Transactins n Systems, Man Cybernetics, Systems Engineering in the Service f Humans, vl.3, pp , Oct. 993 [7] J.M. Selig, Gemetric Fundamentals Of Rbtics, nd Editin: Springer, Nvember 5 [8] D. Gan, Q. Lia, S. Wei, J.S. Dai, S. Qia, Dual quaternin based inverse kinematics f the general spatial 7R Mechanism, Prceedings f the Institutin f Mechanical Engineers, Part C: Jurnal f Mechanical Engineering Science, vl., pp , 8 [9] K. Daniilidis, H Eye Calibratin Using Dual Quaternins, The Internatinal Jurnal f Rbtics Research, vl.8, n.3, pp , March 999 [3] T. Yue sheng X. Ai ping, Extensin f the Secnd Paden Kahan Sub prblem its Applicatin in the Inverse Kinematics f a Manipulatr, IEEE Cnference n Rbtics, Autmatin Mechatrnics, Chengdu, pp , Sept 8 Emre Sariyildiz, Eray Cakiray Hakan Temeltas: A Cmparative Study f Three Inverse Kinematic Methds f Serial Industrial Rbt Manipulatrs in the Screw Thery Framewrk 3

18 [3] F. Pertin, J. M. B. des Tuves, Real Time Rbt Cntrller Abstractin Layer, In Prc. Int. Sympsium n Rbts (ISR), Paris, France, March 4. [3] H. Bruyninckx J. D. Schutter, Intrductin t Intelligent Rbtics 7th editin, Oct. [33] J.H Kim, V. R. Kumar, Kinematics f rbt manipulatrs via line transfrmatins, Jurnal f Rbtics Systems, vl.7, n.4, pp , 99. [34] Y.L. Gu J.Y.S. Luh, Dual Number Transfrmatin Its Applicatin t Rbtics, IEEE Jurnal f Rbtics Autmatin, vl.ra 3, n.6, Dec Int J Adv Rbtic Sy,, Vl. 8, N. 5, 9-4