DIFFERENTIAL KINEMATICS OF CONTEMPORARY INDUSTRIAL ROBOTS

Transcription

1 Int. J. of Appled Mechancs and Engneerng 0 vol.9 No. pp.-9 DOI: 0.78/jame-0-00 DIFFERENTIAL KINEMATICS OF CONTEMPORARY INDUSTRIAL ROBOTS T. SZKODNY Insttute of Automatc Control Slesan Unversty of Technology ul. Akademcka -00 Glwce POLAND E-mal: tadeusz.szkodny@polsl.pl The paper presents a smple method of avodng sngular confguratons of contemporary ndustral robot manpulators of such renowned companes as ABB Fanuc Mtsubsh Adept Kawasak COMAU and KUKA. To determne the sngular confguratons of these manpulators a global form of descrpton of the end-effector knematcs was prepared relatve to the other lnks. On the bass of ths descrpton the formula for the Jacoban was defned n the end-effector coordnates. Next a closed form of the determnant of the Jacoban was derved. From the formula sngular confguratons where the determnant s value equals zero were determned. Addtonally geometrc nterpretatons of these confguratons were gven and they were llustrated. For the exemplary manpulator small correctons of jont varables preventng the reducton of the Jacoban order were suggested. An analyss of postonal errors caused by these correctons was presented. Key words: knematcs manpulators mechancal system robot knematcs.. Introducton The controllers of such renowned companes as ABB Fanuc Mtsubsh Adept Kawasak COMAU and KUKA make possble movement programmng n jont space or the Cartesan space. The followng commands PTP LIN and CIRC can be appled to programmng n the Cartesan space. The mentoned commands requre a start pont and end pont. For programmng n the Cartesan space these ponts must be descrbed n Cartesan coordnates relatve to the base frame connected to the base of a manpulator. These coordnates can be obtaned by a vson system. Durng the realzaton of such programmed movement the robot happens to stop before reachng the border area and before reachng the start or the end pont. The entrapment takes place when the manpulator reaches the sngular confguratons. It s wthout doubt the major problem of modern ndustral robots whch makes t mpossble for the robots wth a vson system to cooperate wth the cameras properly. The lnear system of ordnary dfferental equatons that descrbes the knematcs can be appled to programmng the robots n the Cartesan space. In ths system a manpulator Jacoban s present. For the succesve va ponts nterpolatng trajectory programmed n the Cartesan space jont varables of these ponts can be computed. To these calculatons the standard algorthm for solvng the system of lnear equatons whch s one of the elements of a computer software lbrary can be appled. The lack of protecton n software aganst the reducton of the Jacoban rank n ths algorthm can cause the nterrupton of the calculatons and the robot can stop performng ts operatons. Ths rank decreases n sngular confguraton. The problem of the nverse knematcs solutons n a dfferental form for the sngular confguraton s presented n Chaccho (99) Kozłowsk (00) Nakamura (009) Sclano (00) Spong (997) Tchoń (997). In Chaccho (99) Kozłowsk (00) Nakamura (009) Sclano (00) for a sngular confguraton t s proposed to use: sngular value decomposton technques SVD of the Jacoban the damped least-squares nverse of the Jacoban DLS or sngularty robust nverse of the Jacoban. In the work

2 T.Szkodny (Tchoń 997) a method of avodng sngulartes by usng of dynamcal systems of the own moton s presented. The same work presents a soluton of the nverse knematcs n sngular confguratons obtaned usng: the method a normal form Jacoban attached the zero and the Jacoban space. A common feature of knematc sngularty avodance methods presented n Chaccho (99) Kozłowsk (00) Nakamura (009) Sclano (00) Tchoń (000) s ther hgh computatonal complexty. A very smple approach to the problem of determnng the knematc sngulartes based on the dfferental descrpton s presented n Spong (997) on the examples of manpulators wth three degrees of freedom. Smple because based on closed forms of the determnants of manpulators Jacoban allowng a very smple determnaton of jont varables descrbng sngular confguratons. Smple rules of lnear algebra are recommended n Spong (997) for solvng the nverse knematcs n the feld of speed. Only the methods whch do not requre a large number of calculatons can be of practcal use. Ths paper presents a very smple method of correcton allowng to avodance of the sngular confguraton of modern ndustral robots. In ths method the values of thrd and ffth jont varables are corrected. These values were determned from the closed form of the Jacoban determnant of modern ndustral robots (Spong 997). The closed form of formulas descrbng the thrd and ffth jont varables for the sngular confguratons of manpulators was determned. On the bass of these formulas one s able to determne whether n the next step of the numercal calculatons the freezng of the program caused by a reducton of the Jacoban rank wll take place. These correctons prevent the reducton of the Jacoban rank and allow the use of any method of solvng the nverse knematcs ncludng the methods descrbed n the aforementoned works (Chaccho 99; Nakamura 009; Sclano 00; Tchoń 000). In the second chapter the knematc structure and a descrpton of the end-effector knematcs n relaton to the other lnks ncludng the base of the manpulator are presented. The formulas consttutng the dfferental descrpton of the knematcs are presented n the thrd chapter. A general descrpton of sngular confguratons ncludng ther llustraton and the examples of the calculatons of jont varables correctons preventng the reducton of the Jacoban rank are presented n the fourth chapter. The ffth chapter summarzes the paper.. The global descrpton of the knematcs Fgure llustrates the robot manpulators of the majorty of modern robots wth numbered lnks. The lnk 0 s attached to the ground other lnks - are movable. The lnk 0 wll be called a base lnk the last lnk wth number wll be called an end-effector. The grpper or another tool s attached to ths lnk. Neghborng lnks are connected by revolute jonts. Fgure llustrates the manpulator knematcs schema wth the co-ordnate systems (frames) assocated wth lnks accordng to a Denavt-Hartenberg notaton (Jezersk 00; Kozłowsk 00; Szkodny 0a; 0b). The x 7 y 7 z 7 frame s assocated wth the grpper. The poston and orentaton of the lnks and tool are descrbed by homogenous transform matrces. Matrx A descrbes the poston and orentaton of the -th lnk frame n relaton to --st. T s a matrx that descrbes the poston and the orentaton of the end-effector frame n relaton to the base lnk. Matrx E descrbes the grpper frame n relaton to the end-effector frame. Matrx X descrbes the poston and the orentaton of the grpper frame n relaton to the base lnk. Parameters l are used n the Denavt-Hartenberg notaton. For further descrpton of the knematcs jont varables wll be used. Varables for = and 90. To smplfy the followng denotatons wll be used: S sn C cos j S sn C cos j j j j j. Matrces A are descrbed by Eq.(.a). C 0 S lc S 0 C ls A S C 0 ls C S 0 lc A

3 Dfferental knematcs of contemporary ndustral robots C 0 S lc C 0 S 0 S 0 C ls S 0 C 0 A A lnk lnk lnk lnk lnk lnk 0 lnk Fg.. The manpulator. O x x λ x x x lnk lnk y y z y z x A l O=O A z O A λ x y lnk z λ7 E x7 z7 O7 y7 l A y lnk lnk O y z A y O lnk x x0 T X grpper λ y0 z0 O0 l x0 z A lnk 0 Fg.. Knematc scheme of the manpulator Denavt-Hartenberg parameters jont varables and jont frames.

4 T.Szkodny C 0 S 0 S 0 C 0 A C S 0 0 S C 0 0 A. (.a) Matrx E s descrbed by Eq.(.b) E. (.b) Matrx - T descrbes the end-effector frame knematcs n relaton to --st frame. Equaton (.a) (Szkodny 0a) descrbes these matrces. T A. (.a) j From Eqs (.a) and (.a) one obtans j C S 0 0 S C 0 0 T CC CS S S SC SS C C T S C CCC SS CCS SC CS CS SCC CS SCS CC SS SS T SC SS C C C C C C S S S S C C C C S S C S S S S C C C S S C S C S C C S S C C S S T SCC CS SCS CC 0 0 CCS SC lc CS CCS SCS CC ls CC SCS SS SS 0

5 Dfferental knematcs of contemporary ndustral robots 7 S C C C S S C S C S C C S S C C S S C C C C S S S S C C C C S S C S S S T SCC CS SCS CC 0 0 SCS CC ls ls C C SCS CCS SC lc lc CS CCS SS SS 0 C C C C S S S S C C S CCC SS CSC S SCC CS S S CCC SS CSC C SCC CS T 0 CSCCS SC CSS SSCS CC SSCCS SC CSS CSCS CC C CCS SC SSS 0 CSCS CCSSS SSCS CCCSS CCS SC 0 lc SS S ls ls CC SCSC ls CSS ls ls CC SCS S. (.b) lc CS lc CS These matrces are necessary to the dfferental descrpton of the end-effector knematcs n relaton to the base frame. The descrpton s presented n the next chapter.. The dfferental descrpton of the knematcs To make the descrpton of the manpulator knematcs ndependent of the shape of the grpper the requred poston and orentaton of the grpper frame x 7 y 7 z 7 (descrbed by the matrx X ) are converted to the end-effector frame x y z. Correlaton Treq XreqE makes the converson possble. Therefore n further consderatons we wll focus on the end-effector knematcs. The movement of the x 7 y 7 z 7 frame n relaton to the x 0 y 0 z 0 frame wll be descrbed by usng dsplacement dfferentals 7 dx 7 7 dy 7 7 dz 7 of orgn O 7 and x-y-z current angle dfferentals 7 d 7x 7 d 7y 7 7z d (Crag 99; Szkodny 0). These dfferentals are descrbed n the x 7 y 7 z 7 frame. req

6 8 T.Szkodny Smlarly the movement of the x y z frame n relaton to the x 0 y 0 z 0 frame wll be descrbed by usng the dsplacement dfferentals dx dy dz of the orgn O and x-y-z current angle dfferentals d x d y d z. These dfferentals are descrbed n the x y z frame. Further we wll focus on the movement descrpton of the end-effector and therefore one t s necessary to convert the grpper 7 dfferentals to end-effector dfferentals. The dfferental equaton T E X whch connects 7 end-effector and grpper dfferentals results from the work (Szkodny 0a). In ths equaton and 7 7 are dfferental transformaton matrces respectvely of the end-effector and grpper frame. These matrces have the followng forms 0 dz dy dx dz 0 dx dy dy dx 0 dz d7z d7y dx d7z 0 d7x dy d7y d7x 0 dz7 From the above Eq.(.) results 7 7 T X 7E E 7E. (.) From Eqs (.) and (.b) results Eq.(.) whch makes t possble to compute the end-effector dfferentals from grpper dfferentals. 7 d x d 7x 7 7 dx dx7 7 d 7y 7 d y d 7y 7 7 dy dy7 7 d 7x 7 d z d 7 dz dx7. 7z (.) To descrbe the end-effector knematcs one wll apply the Cartesan dfferental matrx D and the jont dfferental matrx dx dy dz dx dy dz T T dq d d d d d d (Szkodny 0). Equaton (.) connectng these matrces s a dfferental descrpton of the end-effector knematcs. d D J q. (.) In ths equaton an end-effector Jacoban J s present descrbed n the x y z frame whch has the form of Eq.(.).

7 Dfferental knematcs of contemporary ndustral robots 9 x x x x x x ' ' ' ' ' ' y y y y y y ' ' ' ' ' ' z z z z z z ' ' ' ' ' ' J x x x x x x ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' y y y y y y z z z z z z. (.) Fgure llustrates the dfferental dsplacement and the rotaton of the end-effector caused by the dfferental dsplacement and the rotaton of the x y z frame. The vectors dr and d respectvely descrbe the dsplacement and rotaton of the x y z frame n relaton to the x - y - z - frame caused by a dfferental ncrease of the jont varable d. The x y z frame s connected wth the -th lnk and concdes wth x - y - z - frame for d 0. The dsplacement of dr of x y z frame s caused by the dsplacement of d r and rotaton of vector d by an angle d. Therefore T z- z d dr z d z d y y x- d x dr x y y- dr x d d Fg.. Dfferental dsplacements and rotatons of -th and -th frames. d d d r r d. (.) Ths movement wll be recalculated n relaton to the x y z frame usng Eq.(.). dx d a r dy b dr dz c dr. (.) The vectors a b c are the versors of the x y z frame descrbed n the x - y - z - frame. The rotaton of the x y z frame s the same as the rotaton of x y z frame n relaton to the x - y - z - frame

8 0 T.Szkodny whch s a vector d. Ths vector can be recalculated n relaton to the x y z frame by the followng Eqs (.7) x d d a dy b d dz c d. (.7) In Fg. one can see that all lnks are connected by revolute jonts. Therefore the angles d are vectors drected along the z - axs havng a length equal to the dfferental d. Therefore d kd. (.8) k s the versor of the z - axs descrbed n the x - y - z - frame. For such jonts no dsplacement of the coordnate system x y z n relaton to the x - y - z - frame takes place therefore dr 0. Thus from Eqs (.) and (.8) one obtans the followng dr d d dy j dx d. (.9) The vectors and j are versors of the x - and y - axes respectvely descrbed n x - y - z - frame. dx and dy are the x and y - coordnates of the vector d n the x - y - z - frame. After takng nto account Eqs (.9) and (.) one obtans the followng dx ax dy ay dx d dy bx dy by dx d (.0) dz cx dy cy dx d. a x ay are the x and y - coordnates of the versor a n x - y - z - frame. The coordnates of the versors b and c were marked n a smlar way. From Eqs (.7) and (.8) one obtans the followng correlatons d x az d d y bz d d z cz d. (.) The dfferental of each coordnate n the Cartesan matrx D s the sum of the correspondng coordnate dfferentals caused by dfferental changes n jont varables d. For example dx dx. In the Jacoban matrx J the dervatves of Cartesan coordnates of the x y z endeffector frame wth respect to the jont varables are present. The dervatve x x only because the

9 Dfferental knematcs of contemporary ndustral robots dfferental dx depends on the value of d. The stuaton s smlar wth other dervatves n ths matrx. Therefore the elements of matrx J can be presented n the form of Eqs (.) and (.) takng nto account the relatons (.0) and (.). x x ax dy ay dx y y z z b d b d x y y x x y y x c d c d (.) x x y y a z b z z z c z. (.) Matrces - T descrbed by Eq.(.b) can be presented n the form Eq.(.). T. (.) ax bx cx dx ay by cy dy az bz cz dz Quanttes appearng n formulas (.) and (.) can be replaced by the correspondng elements of the matrx - T resultng from the form Eq.(.b) and correspondng to the form Eq.(.). It s easy to notce that the ndex n Eqs (.)-(.) s the number of column n the matrx J. After usng Eqs (.b) (.)-(.) and smplfcatons one obtans the followng elements of the end-effector Jacoban J x l ( S C C C S )+l S (S C C +C S )+l S (S C C +C S )+ -λ C (S C C +C S )+λ ( C C S C S C C +S S S ) y l (S C S C C ) l S (S C S C C ) l S (S C S C C )+ +λ C (S C S C C )+λ (C S S +S C S C C C C )

10 T.Szkodny z l S S +l S S S +l S S S λ S S C x C S S S C S +C C C C y C S C C C S C +S S S z C C S +S C ; x l ( C C C C +C S S +C S S )S +(S C C C S S S +C C S )C )+ +l C S +λ (C C C S S ) λ (S S C C C ) y l (C C C S +C S C S S S )S (S C C S +S S C +S C S )C )+ -l S S +λ ( C C S S C ) λ (S C C +C S ) z l ((S C S C C )C (C C S +S C )S ) l C +λ C S x S C C +C S y S S C +C C z S S ; x l C S +λ (C C C S S )+λ ( S S C +C C ) y l S S +λ ( C C S S C )+λ ( C S S C C ) z l C +λ C S x S C C +C S y S S C +C C z S S ; x λ S S y λ S C z 0 x S C y S S z C ;

11 Dfferental knematcs of contemporary ndustral robots x λ C y λ S z 0 x S y C z 0; x 0 y 0 z 0 x 0 y 0 z. (.) Equaton (.) wth dependences (.) s a dfferental descrpton of the end-effector knematcs.. The sngular confguratons For the sngular confguratons of manpulators the J determnant equals zero. Thus to determne such confguratons one needs to calculate the determnant. The closed form of the determnant was obtaned by Symbolc Math Toolbox lbrary of Matlab. It s descrbed by Eq.(.). det J = l S (λ C l S )( l S l S +λ C +l ). (.) O O O Fg.. Confguraton at whch 0. The z and z axes are collnear. O * O O O O =l O O =λ O Fg.. Confguraton at whch C ls 0. Ponts O O and O are stuated on a straght lne.

12 T.Szkodny The sngular confguratons appear n zero values of the factors on the rght sde of Eq.(.). The jont varables at whch t wll take place are descrbed by Eq.(.). * * C l S 0 arc tg l S 0 0 ls ls C l 0 ** l l ls l () arctg ls l (.) O O ** () z 0 O l O O O =l O O =l O O =λ ** ** ( ) ( ) ** ** ( ) ( ) O z 0 ** ( ) O O O Fg.. Confguraton at whch ls ls C l 0. Pont O s placed on the lne passng through the z 0 axs. Fgures and llustrate these confguratons. For example let us analyze knematc sngulartes of the manpulator IRB-00. The manpulator has the followng parameters (Szkodny 009) l 0mm l 00mm l 0 mm 7 mm 70 mm 8 mm. The ranges of angle changes and are the followng: 70. The calculatons below were mode n Matlab on a PC from an Intel Pentum processor wth a frequency of GHz. Let us assume that for the confguraton from Fg.. For ths confguraton * 0 0. In ths case the det J =0 and the rank J =. The rank of the Jacoban J s less than and therefore a standard numercal soluton algorthm of Eq.(.) wll nterrupt the calculatons. One can avod zerong the determnant by ncreasng or decreasng the angle by a mnmum ncrement mn resultng from the resoluton of the encoder and the gear rato. Let us assume that a typcal resoluton of

13 Dfferental knematcs of contemporary ndustral robots encoders s the followng / rad. and the value of the gear rato equals 00. For such data mn. 0 0 rad. After the correctons mn one obtans det J. 70 and rank J =. Let us estmate the grpper poston error caused by ths correcton. The knematc scheme n Fg. results n the error not greater than. Let us assume that a typcal value of the mn 7 parameter grpper s the followng 7 0 mm. For such a grpper mn causes a change of the grpper poston not greater than. 0 mm. * For the confguraton from Fg. the jont varable The manpulator IRB-00 cannot reach ths varable because t s outsde the range of ts changes. Let us assume that for a confguraton from Fg.. ** For these jont varables one obtans (). 7 ** ( ) **. The jont varable ( ) ** s outsde the range of ts changes. For the jont varable () det J and rank J =. The rank of the Jacoban J s less than and therefore a standard numercal soluton algorthm of Eq.(.) wll nterrupt the calculatons. In order to prevent a decrease n the rank of the Jacoban J one wll ncrease and reduce the jont varable ** () by a mnmum ncrement mn equal to mn. After correctons ** mn of the jont varable () one obtans det J. 8 0 and rank J =. The change n mn causes an error of the grpper poston whch equals max mm.. Summary A dfferental descrpton of the manpulator knematcs presented n ths paper s the bass for desgn of contemporary software drvers of ndustral robots ndependent of the software manufacturers robots. Ths descrpton can be appled n IRB manpulators seres ; n Fanuc manpulators M M M70 M0 M900; n a KUKA manpulators KR KR KR KR; n a Mtsubsh manpulators RV-A RV-A RV-S RV-S RVS; n Adept manpulators s00 s0 s80 s700; n Kawasak manpulators seres M FS00N ZHE00U KF. For KUKA KRC Adept s00 and Mtsubsh RV-AJ manpulators the orgn of second and thrd lnk frame s at the same pont O =O (see Fg.). Therefore for these manpulators one has to assume that l 0. In modern ndustral robot controllers actuator varables calculaton s realzed n two stages n master-level of control and n a sngle stage n the slave-level control. In the frst step n the master-level the trajectory generator calculates the Cartesan coordnates of the trajectory va-ponts. The trajectory s specfed by a sutable Cartesan programmng command (for example LIN). The calculated va-ponts n the Cartesan space are pre-assgned tmes. The tmes resultng from the speed n the Cartesan space declared n a sutable command parameter are assgned to the ponts. In the second stage jont varables are determned (Crag 99; Szkodny 0) for the earler computed va-ponts n the Cartesan space. To determne these varables the teratve calculaton algorthm of solutons Eq.(.) s used. Next actuator varables are calculated for jont varables of these va-ponts. Such calculated actuator varables are sent from the master level nto the slave level of control wth constant frequency. The actuator varables of the va-ponts are calculated at the tme of the robot moton to the prevously calculated va-ponts. If a va pont determned by the trajectory generator n the Cartesan space s sngular calculatng jont varables usng the teratve algorthm wthout a protecton aganst the Jacoban rank reducton results n an undesrable

14 T.Szkodny nterrupton of the calculatons and stoppng the robot. The robot stops before the a sngular va-pont when approachng the prevously calculated nonsngular va-ponts. It may happen that the va-pont obtaned n the frst stage of the calculaton s near a sngular vapont. In ths case the teratve calculaton algorthm of jont varables may be convergent but the resultng actuator varables wll be very dfferent from the correspondng varables of the prevous va-pont. For large ncrements of jont varables and already pre-declared tmes of the prevous and current va-ponts t may happen that the average rate of change of actuator varables s much greater than the maxmum actuator speed. Furthermore n ths case the actuator varables acceleratons may occur above the maxmum acceleraton that can be reached by the actuators. The problem of these exceedances s solved by the trajectory generator n the slave-layer of control. Command moton parameters (e.g. LIN) are sent from the master level nto the slave level control. These are among others the declared translatonal movement speed and acceleraton. In general the acceleraton s declared n percent of the maxmum permssble values. These parameters are the bass for the generaton of actuator varables n the slave level of control. Waveforms of actuator varables are generated at a constant speed sectons connected wth fxed segments acceleratons. If the movement command e.g. LIN declares translatonal movement speed and acceleraton at the level of 0% of ther maxmum values the result s that the n slave level of control the waveforms wth speed and acceleraton rates of not more than 0% of the maxmum actuator speed and acceleraton wll be generated. In the controllers of contemporary ndustral robots one can wrte hs/her own applcatons. These applcatons can be wrtten usng Eq.(.) derved from ths work. In ths equaton dfferentals d should be replaced by the ncrements and dfferentals dx dy dz d x d y and d z by correspondng ncrements x y z x y and z. Such dscretzed Eq.(.) s the bass for the second stage of the calculaton at master level of control. Durng ths stage the jont varables of the va-pnts are teratvely calculated from the dscretzed Eq.(.). Recall that at ths level of control n the frst stage the Cartesan coordnates of the va-ponts are calculated. Assume that the jont varables of the prevous va-pont of trajectory were calculated teratvely and ( ) denote them by. For the calculaton of the jont varables of the current va-pont we wll also apply the teratve calculaton algorthm. In the next for example the k-th teraton of calculatons the ntal values of the lnk varables ( ) ( ) are equal to the end values ( ) ( ) from the prevous teraton. We start k ( ) start ( k ) calculate the Jacoban J and ncreases end k ( ) k ( ) end k from the dscretzed Eq.(.). The end values of start k the jont varables n the k-th teraton ( ) ( ) are the sum ( ) ( ) +. Subsequent teratve ( ) calculatons cause that the end values end( k) of teraton approach values representng the soluton of the nverse knematcs of the current va-pont. If the current va-pont s sngular the approachng t gves rse to the rsk of a reducton of the Jacoban ( ) start ( k ) ( ) k ( ) J rank. These protecton measures are presented n the fourth chapter by usng the example of the IRB-00 manpulator. They consst n the correcton of the angles and by mnmum ncrements mn and mn seen by the servos. ( ) ( ) * If durng the teratve calculatons satsfes the nequalty we assume: ( ) end ( k ) = ** () ( ) end ( k ) = * sgn ( ) k start( k) sgn ( ) k ( ) mn. If ( ) mn ( ) ** start( k) ( ) start k start( k) mn mn we substtute. Also for the angle ( ) ( ) satsfyng the nequalty

15 Dfferental knematcs of contemporary ndustral robots 7 ( ) * ( ) start( k) start( k) mn we assume: ( ) end = sgn ( ) ( ) k ( ) ( k ) mn. For the other varables end ( k ) = ( ) start( k). Usng such protecton measures one should check the range of the changes n the grpper poston caused by above angles correctons. For the sample IRB-00 manpulator suggested correctons can at most result n changes of the poston whch s the sum of errors.e.. 0 mm mm mm. The typcal postonng accuracy of ndustral robots are of the order 0 mm. Thus the errors caused by such correctons are acceptable. If the grpper poston errors caused by such correctons were equal to or greater than the requred postonng accuracy nverse knematcs equatons n a global form presented n Szkodny (00) would need to be used. For these equatons we can apply smple crtera for selectng solutons at the jont varables. Such crtera are much smpler and faster to mplement than the crtera proposed n the Jacoban methods (Chaccho 99; Nakamura 009; Sclano 00; Tchoń 000). The dfferental descrpton of the knematcs ncludng correctons to prevent the loss of the Jacoban rank presented n ths paper consttutes the bass for the creaton of our own software free from the basc defect namely undesred stoppng n sngular postons. Nomenclature A homogeneous matrx descrbng the -th lnk frame n relaton to the --st lnk a versor of x axs descrbed n --st frame ax ay az coordnates of versor a n --st frame b versor of y axs descrbed n --st frame bx by bz coordnates of versor b n --st frame c versor of z axs descrbed n --st frame c c c coordnates of versor c n --st frame x y z D Cartesan dfferental matrx d vector descrbes the poston of the -th lnk frame orgn n relaton to --st frame descrbed n --st frame dx dy dz the coordnates of the vectors d n --st frame d d d r dfferental dsplacement of the x y z frame orgn n relaton to the x - y - z - frame caused by a dfferental ncrease of the jont varable r dfferental dsplacement of the -th lnk frame orgn n relaton to the x - y - z - frame caused by a dfferental ncrease of the jont varable d d dfferental rotaton of the x y z frame n relaton to x - y - z - frame caused by a dfferental ncrease of the jont varable d d dfferental rotaton of the -th lnk frame n relaton to x - y - z - frame caused by a dfferental ncrease of the jont varable d dq jont dfferental matrx E homogeneous matrx descrbng the grpper frame n relaton to the end-effector frame

16 8 T.Szkodny versor of x - axs descrbed n --st frame J end-effector Jacoban J descrbed n the -th lnk frame j versor of y - axs descrbed n --st frame k versor of z - axs descrbed n --st frame l dsplacement along x axs T homogeneous matrx descrbng the end-effector frame n relaton to base frame T req requred matrx T T homogeneous matrx descrbng the end-effector frame n relaton to --st frame X homogeneous matrx descrbng the grpper frame n relaton to base frame X req requred matrx X x yz base frame x yz -th lnk frame x' yz ' ' frame connected wth -th lnk concdng wth --st frame for d 0 rotaton angle of the -th lnk frame n relaton to --st frame about the x axs Δ dfferental transformaton matrces of the end-effector descrbed n the -th frame 7 Δ 7 dfferental transformaton matrces of the grpper descrbed n the7-th frame rotaton angle of the -th lnk frame n relaton to the --st frame about z axs the -th lnk varable dsplacement along z axs References Chaccho P. Chavern S. and Sclano B. (99): Drect and nverse knematcs for coordnated moton tasks of two manpulator system. Journ. of Dynamcs Systems Measurement and Control. vol.8 No. pp Crag J.J. (989): Introducton to Robotcs. New York: Addson-Weseley Publ. Comp. charter. Jezersk E. (00): Dynamcs and Control of Robots. Warsaw: WNT 00 ch. (n Polsh). Kozłowsk K. Dutkewcz P. and Wróblewsk W. (00): Modelng and Control of Robots. Warsaw: PWN ch (n Polsh). Nakamura Y. and Hanafusa H.(009): Inverse knematc solutons wth sngularty robustness for manpulator control. Journ. of Dynamcs Systems Measurement and Control. vol.08 No. pp.-7. Sclano B. Scavcco L. Vllan L. and Orolo G. (00): Robotcs Modellng Plannng and Control. Sprnger Verlag Berln 00 chapter.. Spong M.W. and Vdyasagar M. (997): Robot Dynamcs and Control. Warsaw: WNT chapter..-. (n Polsh). Szkodny T. (009): Basc component of computatonal ntellgence for IRB-00 robots. Man-Machne Interactons. Berln Hedelberg: Sprnger-Verlag part.xi pp.7-. Szkodny T. (00): Inverse Knematcs Problem of IRB Fanuc Mtsubsh Adept and Kuka Seres Manpulators. J. of Appled Mechancs and Engneerng. Zelona Góra. Unversty Press vol. No. pp.87-8.

17 Dfferental knematcs of contemporary ndustral robots 9 Szkodny T. (0): Foundaton of Robotcs. Glwce. Slesan Unversty of Technology Publ. Company ch.. (n Polsh). Szkodny T. (0a): Knematcs of Industral Robots. Glwce. Slesan Unversty of Technology Publ. Company ch.8. (n Polsh). Szkodny T. (0b): Foundaton of Robotcs Problems Set. Glwce. Slesan Unversty of Technology Publ. Company ch. (n Polsh). Tchoń K. Mazur A. Dulęba I. Hossa R. and Muszyńsk R. (000): Moble Manpulators and Robots. Academc Publ. Company PLJ 000 chapter.. (n Polsh). Receved: May 0 Revsed: June 8 0