A Comparative Study of Two Methods for Forward Kinematics and Jacobian Matrix Determination

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1 17 Internatinal Cnference n Mechanical, Syste and Cntrl Engineering A Cparative Study f Tw Methds fr Frward Kineatics and Jacbian Matrix Deterinatin Abdel-Nasser Sharkawy1,, Niks Aspragaths 1 Mechanical Engineering Departent Faculty f Engineering, Suth Valley University Qena 8353, Egypt Departent f Mechanical Engineering and Aernautics University f Patras Ri 654, Greece e-ail: eng.abdelnassersharkawy@gail.c, asprag@ech.upatras. transfratin ne. Sariyildiz et al [9] cpared three inverse kineatic frulatin ethds fr the serial industrial rbt anipulatrs based n screw thery. The first ne was based n quaternin algebra, the secnd ne was based n dual quaternins, and the last ne was the expnential apping ethd. They fund that the ethd which is based n dual quaternin gives the st cpact and cputatinally efficient slutin. In bth f the previus publicatins the cputatinal cst f the frward kineatic equatins was cnsidered but nt the ne fr the Jacbian atrix calculatin. The unit dual quaternins are used in the design f cntrllers fr rigid bdies and anipulatrs. Erl Özgüra, Yucef Mezuarb [1] used unit dual quaternins t del the kineatics and t design a cntrller fr the pse f a rbt ar. They presented the cputatinal cst fr the Jacbian calculatin based n the expnentials f the unit dual quaternins. They fund that this deling is cpact and cputatinally efficient and then suitable fr real tie cntrl applicatins. Pha and his grup [11] presented an unified psitin and rientatin cntrl fr rbt anipulatr by describing the end-effectr tin as a dual quaternin invlving the translatin and rtatin. Their siulatin and experiental results highlighted the efficiency and perfrance f this cntrller. In a recent paper [1], a rbust cntrller was presented fr rbt anipulatrs using unit dual quaternin that allws an advanced representatin f the end-effectr transfratin withut decupling the rtatinal and translatinal dynaics. S the investigatin f the cputatinal efficiency f the rbt kineatics and dynaics very iprtant fr cntrllers design and ther real-tie cnsideratins. In this paper, the frward kineatics using tw algriths; prduct f expnentials (PE) frula and unit dual quaternin are presented. Matheatical analysis f the ain steps f these algriths perfred in rder t facilitate their cparisn in qualitative and quantitative ters. Cpactness, easiness f understanding and representatin and transfratins, cputatinal csts and ery requireents are cnsidered. In additin, the Jacbian atrix is deterined using bth appraches and fr the first tie a cparisn is ade between these algriths Abstract In this paper, a cparative study f tw ethds are presented t deterine the frward kineatics and Jacbian atrix fr any serial anipulatr, based n prduct f expnentials frula and unit dual quaternin algebra. This cparisn is qualitative and particularly quantitative taken int accunt cputatinal csts and ery requireents that are very critical in the anipulatr cntrl and real-tie algriths. Fr the first tie the cputatinal cst in deterining the Jacbian atrix by using these tw algriths is cpared. The tw algriths are applied t the 7-DOF Kuka Lightweight Rbt using MATLAB and cpared. Keywrds-frward kineatics; jacbian atrix; prduct f expnentials; dual quaternin; cparisn; kuka rbt I. INTRODUCTION The st ppular ethd used in rbt kineatics based n the Denavit-Hartenberg (D-H) ntatin fr definitin f spatial echaniss and n the hgeneus transfratin f pints [1]. Many researchers sught t find algriths based n apprpriate theries fr representing the rigid bdy tinn space and t ake the well understandable and iprve their efficiency. Alternative appraches based n screw thery were prpsed in rbt kineatics and the st iprtant nes are expressed via unit dual quaternins [] r expnentials f twists [3]. In screw thery, every transfratin f a rigid bdy r a crdinate syste with respect t a reference crdinate syste can be expressed by a screw displaceent, which is a translatin alng an axis with a rtatin abut the sae axis [4]. As an efficient tl t describe rigid transfratin, the unit dual quaternin has been applied in varius fields, such as rbtics, echanical design, cputer graphics and visin as well as navigatin [5, 6, 7]. The dual quaternin deals with line representatin and transfratin instead f pint representatin and transfratin under expnentials f twists. Aspragaths and Diitrs [8] cpared unit dual quaternin ethd based n the transfratin and representatin f jint axes with the hgenus transfratin ethd and Lie algebra representatin. They fund that the unit dual quaternin and Lie algebra based ethds ffer a re cpact and cnsistent way fr the definitin f the end-effectr than the hgeneus /17/$ IEEE 179

2 cnsidering the cputatinal tie fr the Jacbian calculatin. The tw ethds are applied t the 7-DOF Kuka LWR Rbt using MATLAB t deterine the frward kineatics and Jacbian atrix and are cpared. II. PRODUCT OF EXPONENTIALS (POE) FORMULA The prduct f expnential frula presents se significant advantages cpared t the D-H frulatin since it uses nly the base and tl fraes and the deterinatin f the twists quite easy and cpact representatin f the jint axes. The well-knwn equatins are presented fr the facilitatin f the cputatinal burden calculatin and the reader can find detailn the presented references. Using the Prduct f Expnentials (PE) frula, the frward kineatics equatins fr a serial anipulatr with θ the vectr f the jint are the fllwing [3]; where g st (θ) = e ξ 1θ 1 e ξ iθ i... e ξ θ g st () e ξ iθ i = [ eω iθ i (I e ω iθ i )(ω i v i ) ] 1 ω x v θ + c θ e ω iθ i = [ ω x ω y v θ + ω z s θ ω x ω z v θ ω y s θ ω x ω y v θ ω z s θ ω y v θ + c θ ω y ω z v θ + ω 1 s θ ω x ω z v θ + ω y s θ ω y ω z v θ ω x s θ ω z v θ + c θ ] g st () is the transfratin between tl and base fraes at the reference cnfiguratin (θ = ), ξ i is the twist f the i th jint axis shwn in Fig. 1 with ω i, q i εr 3, ω i is the unit vectr defining the twist, q i is a pint n the jint axis, and c θ = cs θ i, s θ = sinθ i, v θ = 1 csθ i, is the nuber f degrees f freed, the sybl ~ refers t the twist and skew-syetric atrix and the sybl ^ refers t the unit dual quaternin. Figure 1. The definitin f the twists and unit dual quaternins and vectrn successive jint axes. Accrding t the abve presented frward kineatics the spatial Jacbian is deterined as [3]; J st (θ) = [ξ 1 ξ ξ ] where ξ i = Ad (e ξ 1θ1.e ξ i 1 θ i 1) ξ i Each clun f the Jacbian is frulated by the transfratin f the reference jint twists t the current cnfiguratin f the anipulatr, ts calculatin depends heavily n the calculatin f the adjint f the relative transfratin atrix. The bdy Jacbian can be derived by using an adjin transfratin between the bdy and spatial fraes. III. UNIT DUAL QUATERNION APPROACH The advantages f the spatial representatin and transfratin f the rigid bdies are well justified in the relative literature [13]. Ang the are the fllwing: the singularity-free and un-abiguus representatin, the st efficient and cpact fr fr representing rigid transfrs, and the unified representatin f translatin and rtatin. In additin, it can be integrated with little cding effrt; unit dual quaternins can be used fr sth translatinal and rtatinal interplatin and prper frulatin f the cbined errr in cntrl systes. In [8], a recursive algrith is presented t slve the frward kineatic prble fr any serial rbt based n unit dual quaternins, where the representatin and transfratin f the jint axes are cnsidered rather than pint based appraches. The dual angle α i,i+1 between the unit line vectrs s i and s i+1 is defined as α i,i+1 = α i + εl i and the dual angle θ i between the unit line vectrs n i 1,i and n i,i+1 is defined as θ i = θ i + εd i. Here we present the recursive frulan an analytic way in rder t facilitate the calculatin f the required peratins. The unit dual quaternin fr the transfratin between the unit line vectrs n i 1,i and n i,i+1 is given by Q i = cs θ i + s i sin θ i = (cs θ i + ε( d i sin θ i )) + s sin θ i + ε (d i s cs θ i + s sin θ i ) [ s 3 sin θ i + ε (d i s 3 cs θ i + s 3 sin θ i )] = q + q (3) s 4 sin θ i + ε (d i s 4 cs θ i + s 4 sin θ i ) where q is the scalar part and q is the vectr part f Q i. Since s i is perpendicular t n i 1,i as shwn in Fig.1 then the unit dual line vectr n i,i+1 is deterined by the transfratin f the n i 1,i [14]: N + εn n i,i+1 = Q i n i 1,i = q n + q n = [ N 3 + εn 3 ] (4) N 4 + εn 4 where n is the vectr part f n i 1,i. The unit dual quaternin fr the transfratin between the unit line vectrs s i and s i+1 is given by Q i,i+1 = cs α i,i+1 + n i,i+1 sin α i,i+1 = (csα i + N 1 sin α i + ε(l i N 1 cs α i + N 1 sin α i L i sin α i )) N sin α i + ε (L i N cs α i + N sin α i ) + [ N 3 sin α i + ε (L i N 3 cs α i + N 3 sin α i ) ] (5) N 4 sin α i + ε (L i N 4 cs α i + N 4 sin α i ) 18

3 Then s i+1 = Q i,i+1 s i (6) Equatin (6) is nt presented in the extended fr since the cputatinal cst is equal t the ne f (4). Accrding t [8] fr the cputatin f the psitin vectr f the end effectr nly the priary parts f the dual unit vectrs are required as p = (d i + L i a i,i+1 ) (7) The transfratin atrix T that presents the end effectr frae relative t the base frae is calculated by using nly the priary parts f the dual unit vectrs: T = [ n,+1 s +1 n,+1 s +1 p 1 ] (8) In st f anipulatr cntrl systes the Jacbian is used, ts fast deterinatin is quite critical. The instantaneus first rder kineatics can be slved assuing that the frward kineatics were deterined. T derive the Jacbian atrix using unit dual quaternins, the fllwing equatinllustrate the Velcity relatins Jacbians []. θ i s i = V S = (ω + ε v)(s + ε s ) (9) where θ i = θ i + ε d i and s i = + ε, s (9) can be written as the fllwing equatin: θ i s i = [θ i + ε (θ i + d i )] = ω s + ε (s ω + s v) (1) Fr rtatinal jints, d i =., then ω s = θ i, s ω + s v = θ i (11) s v = s ω + θ i = (p i p +1 ) θ i = (p +1 p i )θ i (1) where s = p +1 s, = p i. Fr (1), the Jacbian Matrix is derived as the fllwing; J i = [ (p +1 p i ) ] = [ p +1 p i s ] i = [ s i p +1 ] (13) where p +1 = n +1 n +1 + ((s +1 s +1 ) T n +1 ) n +1 Jacbian atrix can be deterined als based n the s equatin V st = J s st (θ)θ and ξ s i = s i where V st represents the instantaneus spatial velcity f the end-effectr as the fllwing equatin J = [s 1 s s i s ] (14) where s i = [ ] and this equatin gives the sae results using the prduct f expnential frula based Jacbian atrix given by (). IV. COMPUTATIONAL TIME FOR THE TWO ALGORITHMS Using the equatins presented in the tw previus sectins the cputatinal burden fr each ethd is deterined. Fr the frward kineatics, the entire su f the required ultiplicatins and additins f siple real nubers are calculated and analytically presented fr tw ethdn the Table Ι alng with the respective equatins. TABLE I. NUMBER OF OPERATIONS FOR THE FORWARD KINEMATICS Prduct f Expnential Frula Ter Additins Multiplicatins e ω iθ i (eq.1) (I e ω iθ i ) 3 (ω i v i ) 3 6 (I e ω iθ i )(ω i v i ) 6 9 Overall e ξ iθ i (eq.1) 3 48 Multiplicatin f tw 4 X 4 transfratin atrices Ttal peratins fr Link Rbt = Unit Dual Quaternin Methd Dual Part included = 96 Q i(eq.3) 3 13 n i,i+1 (eq.4) 1 7 Q i,i+1 (eq.5) 3 13 s i+1(eq.6) 1 7 p (eq.7) 6 6 s +1 n,+1 (eq.8) 3 6 Ttal peratins fr Link Rbt Dual Part ignred Q i(eq.3) 3 n i,i+1 (eq.4) 6 9 Q i,i+1 (eq.5) 3 s i+1(eq.6) 6 9 p (eq.7) 6 6 s +1 n,+1 (eq.8) 3 6 Ttal peratins fr Link Rbt

4 Fr the deterinatin f the nuber f peratins required fr the calculatin f the Jacbian atrix it is suppsed that the frward kineatics were dne s the required peratins fr this are nt recalculated. In the prduct f expnential the calculatin f e ξ iθ i and fr the ultiplicatins f the 4 X 4 transfratin atrices are ready fr frward kineatics. Fr the deterinatin f the Jacbian based n unit dual quaternins, and are ready fr the frward kineatics. Fr the Jacbian atrix calculatin, the nuber f peratins fr the tw ethds shwn analytically in the Table ΙΙ. TABLE II. NUMBER OF OPERATIONS FOR THE JACOBIAN MATRIX CALCULATIONS. Prduct f Expnential Frula Ter Additins Multiplicatins Adjint Matrix = Ad g 9 18 Multiplicatin Ad g by ξ i (eq.) Overall Operatins fr Link Rbt ( -1) = 3-3 Unit Dual Quaternin Methd 45 ( -1) = p p +1 ( eq.13) Overall Operatins fr Link Rbt A cparisn f the nuber f atheatical peratins required fr the cputatin f the end-effectr psitin and rientatins and the Jacbian atrix versus the nuber f jints using the tw ethds presented in Fig.. the frward kineatics and Jacbian atrix. Fr the frward kineatics, unit dual quaternin requires and additins using dual part and ignring it respectively cpared with prduct f expnentials that requires 66. Fr ultiplicatins, unit dual quaternin requires and using dual part and ignring it respectively and the prduct f expnential requires 96. Fr the Jacbian atrix, unit dual quaternin requires additins cpared with prduct f expnentials that requires 3-3. Fr ultiplicatins, unit dual quaternin requires and the prduct f expnential requires In case f the Jacbian given by (14) which identical t the ne given by (), the cputatinal cst is zer since the unit dual vectr is calculated in the frward kineatics by (6). Therefre fr real tie applicatins and kineatic cntrl f serial anipulatrs the unit dual quaternin representatin is preferable. It is clear that the nuber f calculatins lwer by using the unit dual quaternin ethd and it is knwn that unit dual quaternin requires 8 ery lcatins fr the representatin f psitin and 3 ery lcatins fr the rientatin despite f the prduct f expnential that is based n the 4 X 4 transfratin atrices that require 16 ery lcatins fr bth the psitin and rientatin s the strage in a ery needs less space. Therefre the cputatinal csts and ery requireents are lwer in unit dual quaternin. The explanatin f the physical eaning f the paraeters and peratinn dual quaternin and PE shws that the intuitive understanding f the cbined rientatin and translatin it is easier in unit dual quaternin based algrith. V. APPLICATIONS OF THE ALGORITHMS IN KUKA LWR ROBOT Figure. Matheatical peratins required vs. the nuber f DOF f a anipulatr. The cparative study illustrates that the unit dual quaternins require cnsiderably lwer nuber f arithetic peratins (additins and ultiplicatins) cpared with the prduct f expnential frula as the nuber f DOF regardless taking int accunt the dual part r nt fr bth Figure 3. Denavit-Hartenberg and the twist f the 7-DOF Kuka Rbt. In this sectin, the tw algriths are applied t 7-DOF Kuka LWR Rbt presented in Fig. 3 and prgraed in Matlab t deterine its frward kineatics and Jacbian atrix fr the verificatin and denstratin f the 18

5 cpared algriths. T ur knwledge is the first tie that the POE and unit dual quaternin are used fr the frward kineatics and the Jacbian deterinatin f the 7-DOF Kuka LWR and in additin that are prgraed and the required peratins are cpared. In KUKA Cntrl Tlbx [15] the frward kineatics are based n hgeneus transfratins and the Jacbian is nt deterined. After the calculatin f frward kineatics, it is fund that the tw algriths give the sae results fr the frward kineatics but with different calculatin tie. There is a difference in the results f the Jacbian atrix given by () and the ne given by (13); since in the case f the prduct f expnential frula, the angular cpnent is the instantaneus angular velcity f the bdy as viewed in the spatial frae, hwever the linear cpnent is nt the velcity f the rigin f the end-effectr frae, rather it is the velcity f a pssibly iaginary pint n the rigid bdy which is traveling thrugh the rigin f the spatial frae at tie t, at is stressed in [3]. If the Jacbian given by (13) is used then the calculated translatinal velcity is the ne f the rigin f the end-effectr frae. The cputatinal tie required by the crrespnding Matlab prgras nt cpared because this depends n the way f prgraing. Fr the general cparisn f the tw algrithllustrated in sectin 4, fr the 7-DOF LWR Kuka rbt (=7), the frward kineatics requires 67 ultiplicatins and 46 additins using the Prduct f Expnential frula and 68 ultiplicatins and 381 additins using the unit Dual Quaternin taking int cnsideratin the dual part while in the case f ignring the dual part, 16 ultiplicatins and 19 additins are required. The Jacbian atrix requires 7 ultiplicatins and 18 additins based n Prduct f Expnential frula and 1 ultiplicatins and 63 additins using the unit Dual Quaternin algrith. Fr the analytic presentatin f the algriths and its prgraing in Matlab, it is clear that unit dual quaternins present higher cpactness and facilitates the understanding f the geetrical eaning f the jint axes. Therefre, the wider use f these ethdnt the rbtics cunity has t be cnsidered particularly the dual quaternin apprach in the anipulatr cntrl f siultaneus rtatin and translatin f the end-effectr. VI. CONCLUSION In this paper, tw ethds are presented in an algrithic way fr the frulatin f the kineatic equatins and Jacbian atrix f serial anipulatrs. The cparative study including the calculatin f the nuber f arithetic peratins (additins and ultiplicatins) as an index f the cputatinal csts fr each algrith are deterined and cpared. The unit Dual Quaternin Methd is prved re effective than the Prduct f Expnential frula as the nuber f the rbt degrees f freed increases. Unit Dual Quaternin ffers a re cpact and cnsistent way than the prduct f expnential frula fr deterining the Jacbian Matrix and needs less cputatinal tie and ery requireents. Calculatins f the frward kineatics and Jacbian atrix using these tw algriths are applied and prgraed in Matlab n the 7-DOF LWR Kuka Rbt and are cpared. Accrding t the quantitative cparisn and the presented qualitative advantages f the unit dual quaternin apprach it is clear that fr real-tie applicatins and anipulatr cntrl the dual quaternins uch preferable than PE. ACKNOWLEDGMENT Abdel-Nasser Sharkawy is funded by the Egyptian Cultural Affairs & Missins Sectr and Hellenic Ministry f Freign Affairs Schlarship fr Ph.D. study in Greece. REFERENCES [1] Dnald L. Pieper, The kineatics f anipulatrs under cputer cntrl. PhD, [] J. Ki and V. R. Kuar, Kineatics f Rbt Manipulatrs via Line Transfratins, J. Rbt. Syst., vl. 7, n. 4, pp , 199. [3] R. M. Murray, Z. Li, and S. S. 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