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2 Indutril Robot Control Sytem Prmetric Deign on the Be of Method for Uncertin Sytem Robutne All A. Neenchuk nd Victor A. Neenchuk Open Acce Dtbe 1. Introduction Indutril robot often operte in condition of their prmeter ubtntil vrition tht cue vrition of their control ytem chrcteritic eqution coefficient vlue, thu generting the eqution fmilie. Anlyi of the dynmic ytem chrcteritic polynomil fmilie tbility, the tble polynomil nd polynomil fmilie ynthei repreent complicted nd importnt tk (Polyk, 00, ). Within the prmetric pproch to the problem the erie of the effective method for nlyi hve been developed (Bhtthryy et l., 1995; Polyk, 00, ). In thi wy, V. L. Khritonov (Khritonov, 1978) proved tht for the intervl uncertin polynomil fmily ymptotic tbility verifiction it i necery nd enough to verify only four polynomil of the fmily with the definite contnt coefficient. In the work of Y. Z. Typkin nd B. T. Polyk the frequency pproch to the polynomilly decribed ytem robutne w offered (Polyk & Typkin, 1990; Polyk & Scherbkov, 00; Typkin & Polyk, 1990; Typkin, 1995). Thi pproch comprie the robut tbility criteri for liner continuou ytem, the method for clculting the mximl diturbnce wing for the nominl tble ytem on the be of the Typkin Polyk hodogrph. Thee reult were generlized to the liner dicrete ytem (Typkin & Polyk, 1990). The robut tbility criterion for the rely control ytem with the intervl liner prt w obtined (Typkin, 1995). The uper-tble liner ytem were conidered (Polyk & Scherbkov, 00). The problem for clculting the polynomil intbility rdiu on the be of the frequency pproch i invetigted (Krev & Furov, 00). The technique for compoing the tbility domin in the pce of ingle prmeter or two prmeter of the ytem with the D-decompoition pproch ppliction i developed (Gryzin & Polyk. 006). The method for definition of the nominl polynomil coefficient devition limit vlue, enuring the hurwitz tbility, h been offered (Brmih, 198). The tk here i reduced to the ingle-prmeter optimiztion problem. The imilr tk re olved by A. Brtlett (Brtlett et l., 1987) nd C. Soh (Soh et Source: Indutril-Robotic-Theory-Modelling-Control, ISBN , pp. 96, ARS/plV, Germny, December 006, Edited by: Sm Cubero 895

3 896 Indutril Robotic: Theory, Modelling nd Control l., 1985). Condition for the generlized tbility of polynomil with the linerly dependent coefficient (polytope) hve been obtined (Brtlett et l., 1987; Rntzer, 199). One of the mot importnt tge, while clculting dynmic ytem with uncertin prmeter, i enuring robut qulity. The control proce qulittive chrcteritic re defined by the chrcteritic eqution root loction in the complex plne (the plne of ytem fundmentl frequencie). In thi connection, three min group of tk being olved cn be ditinguihed: determining the ured root loction domin (region) for the given ytem, finding condition of whether root get into the given region or not (determintion of the Λ-tbility condition) nd locting root in the given domin (enuring Λ- tbility). The frequency tbility criteri for the liner ytem fmilie nd lo the method for finding the lrget diturbnce rnge of their chrcteritic eqution coefficient, which gurntee the ytem ymptotic tbility, re conidered by B. T. Polyk nd Y. Z. Typkin (Polyk & Typkin, 1990). The ured domin of the intervl polynomil root loction i found in (Soh et l., 1985). The root locu theory i ued in (Givoronky, 006) for thi tk olution. Condition (Vicino, 1989; Shw & Jyuriy, 199) for the intervl polynomil root getting into the given domin of ome convex hpe re defined. The prmetric pproch to robutne, bed on the root locu theory (Rimky, 197; Rimky & Tborovetz, 1978; Neenchuk, 00; Neenchuk, 005), i conidered in thi chpter in ppliction to the indutril nthropomorphou robot control ytem prmetric deign. The developed technique llow to et up the vlue of the prmeter vrition intervl limit for the ce when the tbility verifiction howed, tht the given ytem w untble, nd to enure the ytem robut qulity by locting the chrcteritic eqution fmily root within the given qulity domin.. Indutril robot nd it control ytem decription Mot indutril robot re ued for trnporttion of vriou item (prt), e. g. for intlling prt nd mchine tool in the cutting mchine djutment, for moving prt nd unit, etc. During the robot opertion due to ome internl or externl reon it prmeter vry, cuing vrition of the ytem chrcteritic eqution coefficient. Thi vrition cn be rther ubtntil. In uch condition the ytem i conidered, the uncertin ytem..1 Generl decription of the nthropomorphou indutril robot The indutril robot conidered here i ued for opertion n integrted prt of the flexible indutril module including thoe for tmping, mechnicl -

4 Indutril robot control ytem prmetric deign on the be of method for uncertin 897 embly, welding, mchine cutting, cting production, etc. The indutril robot i hown in fig. 1. It comprie mnipultor 1 of nthropomorphou tructure, control block including periphery equipment nd connecting cble. Mnipultor h ix unit (1 8 in fig. 1) nd correpondingly i of ix degree of freedom (ee fig. 1): column turn, houlder 5 wing, rm 6 wing, hnd 7 wing, turn nd rottion. The rm i connected with the joining element 8. Controlling robot of uch type, belonging to the third genertion, i bed on the hierrchicl principle nd feture the ditributed dt proceing. It i bed on ppliction of pecil control proceor for utonomou control by every degree of freedom (lower executive control level) nd centrl proceor coordinting their opertion (higher tcticl control level).. Indutril robot mnipultor unit control ytem, it tructure nd mthemticl model Executive control of every mnipultor unit i uully executed in coordinte of thi unit (Nof, 1989) nd i of the poitionl type. It i the cloed-loop ervocontrol ytem not depending on the other control level. Although rel unit control i executed by digitl device (microproceor, controller) in dicrete wy, the effect of digitiztion i uully neglected, the digitiztion frequency i high enough to conider the unit nd the controller the nlog (continuou) ytem. A for the tructure, the unit control loop re lmot imilr nd differ only in the prmeter vlue. Therefore, ny unit of the indutril robot cn be conidered for invetigting the dynmic propertie. Figure 1. Anthropomorphou indutril robot

5 898 Indutril Robotic: Theory, Modelling nd Control The tructure of the mnipultor unit ubordinte control i hown in fig.. The implified verion of the tructure i preented in fig. In fig. the plnt i repreented by element 1 ( DC motor); 5 i the enor trnforming the nlog peed ignl into the peed code (photo-pule enor), 6 i the element combining the peed PI regultor, code-pule width trnformer nd cpcity mplifier, 7 i the trnformer of nlog poition ignl into the poition code (photo-pule enor), 8 i the proportionl regultor of the mnipultor houlder poition, 9 i the trnfer mechnim (reducer). In fig. the trnfer function ' Wp ( ) = Wp ( ) where W p () i the plnt trnfer function. Subtitute correponding prmeter nd expre the plnt trnfer function follow: W p where ϕ 1 ( ) = =, (1) U L R g A A ( jm jl ) ( jm jl ) Ce C C М U g i the input voltge, ϕ i the object hft ngle of rottion. М Figure. Control ytem for the indutril robot mnipultor houlder unit On the bi of (1) write the mnipultor unit control ytem chrcteritic eqution

6 Indutril robot control ytem prmetric deign on the be of method for uncertin 899 R L A A CeC j L m М A C j М m K L 1 A K T C M K j m L K A p T K = 0 or 0 1 = 0, () where 0 = 1; R = AL 1 ; A = e M ; ( j CC m j l ) L A CМK1K = ; ( j j ) L T m l A CMK KpK = ; ( j j ) L T m l A - R A i the motor nchor reitnce; - L A i the nchor inductnce; - j l i the lod inerti moment; - j m i the nchor inerti moment; - C e i the electric-mechnicl rtio of the motor; - CM i the contructive contnt of the motor; - T i the time contnt of the PI regultor; - K 1 nd K re photo-electric enor coefficient; - K nd K p re gin of regultor by peed nd poition correpondingly. Suppoe the robot unit h the following nominl prmeter: - R A = 0,6 Ω; - L A = 0,001 henry; 5 - j l =,0 10 kg /m 5 - j m = 0,8 10 kg /m V - C e = 0,16 ; rd C = C ; - M e ; - T= 0, ; - K 1 = 66,7, K = 50; K 0,078, K =,5 - = p.

7 900 Indutril Robotic: Theory, Modelling nd Control Figure. Structure of the poition control ytem loop for the mnipultor houlder unit After ubtitution of the nominl vlue into () rewrite the unit chrcteritic eqution 5 7 0,5 10 0,7 10 0,6 10 0,56 10 = 0 () The coefficient of () re the nominl one nd while robot opertion they often vry within the enough wide intervl. For thi reon when clculting the robot control ytem it i necery to conider the prmeter uncertinty nd enure the control ytem robutne. 8. The technique for robut tbility of ytem with prmetric uncertinty The method i decribed for ynthei of the intervl dynmic ytem (IDS) tble chrcteritic polynomil fmily from the given untble one, bed on the ytem model in the form of the free root locu portrit. Thi method llow to et up the given intervl polynomil for enuring it tbility in ce, when it w found, tht thi polynomil w untble. The ditnce, meured long the root locu portrit trjectorie, i defined the etting up criterion, in prticulr, the new polynomil cn be elected the neret to the given one with conidertion of the ytem qulity requirement. The ynthei i crried on by clculting new boundrie of the polynomil contnt term vrition intervl (tbility intervl), tht llow to enure tbility without the ytem root locu portrit configurtion modifiction.1 The tk decription While invetigting uncertin control ytem for getting more complete repreenttion of the procee, which occur in them, it eem ubtntil to dicover correltion between lgebric, frequency nd root locu method of in-

8 Indutril robot control ytem prmetric deign on the be of method for uncertin 901 vetigtion. Such correltion exit nd cn be pplied for finding dependence between the ytem chrcteritic eqution coefficient vlue (prmeter) nd it dynmic propertie to determine how nd wht coefficient hould be chnged for enuring tbility. One of the wy for etblihing the bove mentioned correltion cn be invetigtion of the ytem root locu portrit nd Khritonov' polynomil root loci (Khritonov, 1978). Conider the IDS, decribed by the fmily of chrcteritic polynomil n P() = j j= 0 n j = 0, () where j [ j, j ], 0 > 0, j = 0,, n; j nd j re correpondingly the lower nd upper boundrie of the cloed intervl of uncertinty, [, ]; = iω. The coefficient of polynomil () re in fct the uncertin prmeter. The tk conit in ynthei of the tble intervl fmily of polynomil () on the bi of the initil (given) untble one, i. e., when the initil ytem tbility verifiction by ppliction of Khritonov' polynomil gve the negtive reult. Clcultion of new prmeter vrition intervl boundrie i mde on the be of the initil boundrie in correpondence with the required dynmic chrcteritic of the ytem. The new boundrie vlue definition criteri cn be different, in prticulr they cn be elected the neret to the given one. In thi ce the ditnce, meured long the ytem root trjectorie, i ccepted to be the criterion of uch proximity. j j. The intervl ytem model in the form of the root locu portrit Introduce the erie of definition. Definition 1. Definition. Definition. Definition. Nme the root locu of the dynmic ytem chrcteritic eqution (polynomil), the dynmic ytem root locu. Nme the fmily (the et) of the intervl dynmic ytem root loci, the root locu portrit of the intervl dynmic ytem. The lgebric eqution coefficient or the prmeter of the dynmic ytem, decribed by thi eqution, being vried in definite wy for generting the root locu, when it i umed, tht ll the ret coefficient (prmeter) re contnt, nme the lgebric eqution root locu free prmeter or imply the root locu prmeter. The root locu, which prmeter i the coefficient k, nme the lgebric eqution root locu reltive to the coefficient k.

9 90 Indutril Robotic: Theory, Modelling nd Control Definition 5. Definition 6. Remrk 1. Remrk. The root locu reltive to the dynmic ytem chrcteritic eqution contnt term nme the free root locu of the dynmic ytem. The point, where the root locu brnche begin nd the root locu prmeter i equl to zero, nme the root locu initil point. One of the free root locu initil point i lwy locted t the origin of the root complex plne. The bove remrk correctne follow from the form of eqution (). The free root locu poitive rel brnch portion, djcent to the initil point, locted t the origin, i directed long the negtive rel hlf-brnch of the complex plne to the left hlf-plne. Remrk i correct due to the root loci propertie (Udermn, 197) nd becue rel root of eqution with poitive coefficient re lwy negtive (ee fig. ). The peculirity of the free root loci, which ditinguihe them from nother type of root loci, conit in the fct, tht ll their brnche trive to infinity, pproching to the correponding ymptote. For crrying on invetigtion pply the Teodorchik Evn free root loci (TEFRL) (Rimky, 197), i. e. the term "root locu" within thi ection will men the TEFRL, which prmeter i the ytem chrcteritic eqution contnt term. To generte the IDS root locu portrit pply the fmily of the mpping function n 1 n 1 n n n 1 = u(,ω) iv(,ω) = n, (5) where u(,ω) nd v(,ω) re hrmonic function of two independent vrible nd ω; n i the root locu prmeter; = iω. Anlyticl nd grphicl root loci re formed uing mpping function (5). The root locu eqution i follow: iv(,ω) = 0 (6) nd the prmeter eqution (Rimky, 197) follow: u(,ω) = n. (7)

10 Indutril robot control ytem prmetric deign on the be of method for uncertin 90 The frgmentry root locu portrit for the IDS of the forth order, which i mde up of four Khritonov' polynomil free root loci, i hown in fig.. The Khritonov' polynomil h 1, h, h nd h in thi figure re repreented by point (root), mrked with circle, tringle, qure nd pinted over qure correpondingly. There re the following deigntion:, i = 1,,,, the cro center of ymptote for the root loci of every polynomil h i, t l, l = 1,,, cro point of the root loci brnche with the ytem ymptotic tbility boundry, xi i. The root loci initil point, which repreent zeroe of mpping function (5), re depicted by X-. Becue in fig. ll root of the Khritonov' polynomil re completely locted in the left hlf-plne, the given intervl ytem i ymptoticlly tble (Khritonov, 1978). hi

11 90 Indutril Robotic: Theory, Modelling nd Control Figure. Root loci of the Khritonov' polynomil for the ytem of cl [;0]

12 Indutril robot control ytem prmetric deign on the be of method for uncertin 905. Invetigtion of the chrcteritic polynomil fmily root loci brnche behvior t the ymptotic tbility boundry of the ytem The brnche of the IDS root locu portrit, when croing the tbility boundry, generte on it the region (et) of cro point. Nme thi region, the cro region nd deignte it R ω. According to the theory of the complex vrible (Lvrentyev & Shbt, 1987) nd due to the complex mpping function (5) continuity property, thi region i the mny-heeted one nd i compoed of the eprte heet with every heet (continuou ubregion), formed by the eprte brnch while it move in the complex plne following the prmeter vrition. The cro region portion, generted by only poitive brnche of the ytem root locu portrit, nme the poitive cro region nd deignte it R ω. R ω R ω. (8) Define lo the ubregion r ω (either continuou or dicrete one) of the cro region R ω (8) generted by the root loci brnche of ny rbitrry ubfmily f of the intervl ytem polynomil fmily (), nd nme it the (poitive) cro ubregion, thu, r ω R ω. (9) Introduce the following et: W r = { ri} A r = { ri} (10) (11) where W r i the et (fmily) of the cro ubregion r ω (9) point coordinte ω ri ; A r i the et (fmily) of vlue ri et W r point. of the root locu prmeter а n t the Define the miniml poitive vlue cro ubregion r ω : r of the root locu prmeter within the min r = inf A r. (1) min

13 906 Indutril Robotic: Theory, Modelling nd Control Peculiritie of the IDS root loci initil point loction mke it poible to drw concluion bout exitence of it chrcteritic eqution coefficient vrition intervl, enuring ymptotic tbility of the given ytem. Sttement. If the initil point of the IDS chrcteritic polynomil rbitrry ubfmily f free root loci, excluding point lwy ituted t the origin, re locted in the left complex hlf-plne, there exit the intervl d of the root loci prmeter n vlue, enuring ymptotic tbility of the ubfmily f. d = (0, r ), (1) min Proof. The ubfmily f free root loci generte t the ytem tbility boundry the cro ubregion r ω (9) of cro point, which i formed by the et (10) of the cro point coordinte nd correponding et (11) of the prmeter vlue. If the initil point re locted, it i defined by the ttement, on every i-th brnch of every polynomil root loci there exit n intervl r i = ( li,0) of root vlue (trting from the brnch initil point with coordinte li until the point, where it croe the tbility boundry, xi iω of the complex plne), which i completely locted in the left hlf-plne. Therefore, there exit lo the pproprite mximum poible common intervl d m (which i common for ll the brnche) of the root loci prmeter n vlue (beginning from zero up to the definite mximum poible vlue n = ), correponding to the vlue of root within ome intervl r k = ( lk,0), which enure the ytem tbility. Nme thi intervl d m the dominting intervl nd define it d m = (0, r m ). Deignte the root i coordinte vlue intervl, locted on every poitive i-th brnch of the fmily nd correponding to the dominting intervl, r d = ( li, ri ). It i evident, tht r m will be mximum poible t the tbility boundry, i. e. t ri = 0. Then, ri [ r m = r min ri 0], i. e. the dominting one i the intervl d m = (0, r min ), which repreent itelf the intervl d (1). Hence, the ttement i correct. r m Definition 7. The intervl of polynomil () root loci prmeter vlue nme the polynomil tbility intervl by thi prmeter or imply the polynomil tbility intervl, if the polynomil ymptotic tbility property hold within thi intervl. In ce, if ome initil point re locted t the tbility boundry (excluding the point, which i lwy locted t the origin), nd on the umption, tht ll the ret point re locted in the left hlf-plne, the dditionl nlyi i required for finding the tbility intervl exitence. For thi purpoe it i necery to define the root loci brnche direction t their outcome from the initil

14 Indutril robot control ytem prmetric deign on the be of method for uncertin 907 point, locted t the tbility boundry, i. e. jut to determine wht hlf-plne they re directed to: left one or right one. Obviouly, uch tbility intervl exit in the following ce: ) ll the root loci brnche with initil point, locted t the tbility boundry, re directed from thee point to the left hlf-plne; b) ll poitive root loci brnche with initil point, locted t the tbility boundry, re directed from thee point to the left hlf-plne. To determine the bove indicted brnche direction t the initil point, it i enough to define the root locu enitivity vector (Neenchuk, 005) direction t them. A reult of the IDS root locu portrit nlyi everl generl regulritie hve been dicovered, being inherent in Khritonov polynomil free root loci: pired convergence of the root loci brnche t the complex plne imginry xi (point t 1, t, t, t in fig. ); pired convergence of the correponding ymptote t the rel xi of the complex plne (point h1, h, h, h in fig. ); the tendency for the ytem robut propertie vrition while vrying it chrcteritic polynomil coefficient vlue. It give the poibility to fix the fct of exitence of the ytem chrcteritic eqution coefficient vrition intervl, enuring it robut tbility nd lo to determine how the coefficient vlue hould be chnged for the ytem dynmic chrcteritic correction, if it i untble. The IDS root locu portrit invetigtion, which h been crried out, confirm tht they cn be uccefully pplied for the in-depth tudying robut propertie of thee ytem.. Prmetric ynthei of tble uncertin ytem The condition for exitence of the polynomil () fmily coefficient tbility intervl were formulted in the previou ection. Here we define wht thee intervl vlue hould be. For thi purpoe conider the polynomil () ubfmily f, coniting of the ytem Khritonov polynomil, nd develop the procedure for ynthei of the tble Khritonov polynomil on the be of the untble one, which depend on the root loci initil point loction in reltion to the ymptotic tbility boundry. For the ynthei procedure development pply the Khritonov polynomil free root loci. Conider the ce, when initil point re locted in the left hlf-plne. In thi ce the lgorithm of ynthei cn be divided into the following tge.

15 908 Indutril Robotic: Theory, Modelling nd Control Stge 1. Obtining the Teodorchik Evn free root loci eqution (6) for ech one of the IDS four Khritonov polynomil. A the Khritonov polynomil repreent the ubfmily of the IDS polynomil fmily, they generte the bove decribed cro ubregion r ω (9) on the tbility boundry, which i formed by the et (10) of the cro point coordinte. Stge. Clculting coordinte ri ω of the et (10) by olution of the TEFRL eqution, obtined in tge 1, reltive to ω in condition, tht = 0. In thi wy the et W r (10) i formed. For every obtined vlue of ri ω from W r the correponding vlue of the vrible coefficient n i clculted by formul (7), thu, forming the et A r (11). Stge. Definition of the tbility intervl by the coefficient n. For thi purpoe, uing (1), define the miniml one, r, of the prmeter vlue t point of the et A r. Thu obtin the intervl d (1) of the prmeter n vrition, which enure tbility of the Khritonov polynomil nd, therefore, the ytem in whole. Before decribing the next tge of ynthei formulte the following theorem. Theorem. For robut tbility of the polynomil fmily () it i necery nd enough to enure the upper limit of the contnt term n vrition intervl to tify the inequlity n < r min, (1) if the fmily i tble t n = 0. Proof. Let the coefficient n to be the polynomil () root locu prmeter. Under the theorem condition fmily of () i tble t n = 0, i.e. the root loci initil point re locted in the left hlf-plne. Therefore, in view of ttement 1 the theorem i vlid. Stge. Compring the obtined tbility intervl (1) with the given intervl n [ n, n ] of the prmeter n vrition in correpondence with inequlity (1). In ce, if condition (1) i not tified, the upper limit n of the prmeter vrition intervl i et up in correpondence with thi inequlity. When the power n of the polynomil i le or equl thn, n, the bove given theorem i pplied without ny condition, i. e. it i not required to t- min

16 Indutril robot control ytem prmetric deign on the be of method for uncertin 909 ify condition of the Khritonov polynomil root rel prt negtivity t n = 0, becue in thi ce the coefficient poitivity lwy gurntee negtivity of the root rel prt. The bove decribed lgorithm llow to crry on the prmetric ynthei of the tble intervl ytem without modifiction of it root locu portrit configurtion, by imple procedure of etting up the chrcteritic polynomil contnt term vrition intervl limit. The numericl exmple, demontrting the reult obtined, i given below Conider the intervl ytem, decribed by the initil chrcteritic polynomil = 0, (15) where the rel coefficient re: а 0 = 1; 8, а 1 11,6; а 8; 6,5 а 8,1; 8,99 а 50,. Let the coefficient to be the root locu prmeter. Then, define the mpping function: ω ω δ ω δ ω ω ω ω ω = i i i i i i Write correpondingly the TEFRL nd the prmeter eqution::. 6 0; ) ( = ω ω ω = ω ω ω Define the Khritonov polynomil for the intervl ytem with the initil chrcteritic polynomil (15): 50,. 6,5 11,6 ) ( 8,99; 8,1 8 8, ) ( 8,99; 6,5 8 11,6 ) ( 50,; 8,1 8, ) ( = = = = h h h h The root loci of thee polynomil re repreented in fig., decribed bove. Number of ymptote n (in fig. they re indicted 1,,, 6 ) i contnt for every one of Khritonov polynomil nd i equl to n = n m = 0 =, where m i the number of pole for function (5).

17 910 Indutril Robotic: Theory, Modelling nd Control The center of ymptote re locted on the xi nd hve coordinte: h1 =,10; h =,90; h =,10; h =,90 (ee fig. ). The ymptote center coordinte coincide in pir: for the pir h ( ) nd h ( ), nd lo for the pir h ( ) nd h ( ). The inclintion ngle of ymptote for the given root loci re correpondingly the following: 1 ϕ = 0 ; 0 ϕ = 5 ; ϕ = 15 ; 0 ϕ = 180. According to fig., every pir of the root loci trive to the me ymptote, i.e. the pir re formed by thoe root loci, which ymptote center coincide, it w indicted bove. For definition of eqution (15) coefficient intervl, enuring the ytem tbility, tbility condition (1) i pplied. Thu, the following vlue A r hve been defined: ri of the et r 1 r r r = 19,67 for the polynomil h 1 ; = 116, for the polynomil h ; = 77,75 for the polynomil h ; = 5,89 for the polynomil h. The miniml vlue i r = r = 5,89. min Becue < 5,89, in correpondence with (1) the given intervl ytem i ymptoticlly tble.. The method for enuring uncertin ytem qulity In thi ection the tk i olved for locting the uncertin ytem root within the trpezoidl domin. The method llow to locte root of the uncertin ytem chrcteritic eqution fmily within the given qulity domin, thu enuring the required ytem qulity (generlized tbility). The tk i olved by incribing the ytem circulr root locu field into the given qulity domin. The trpezoidl domin, bounded by the rbitrry lgebric curve, i conidered. Peculirity of the method conit in the root locu field ppliction.

18 Indutril robot control ytem prmetric deign on the be of method for uncertin 911 The ytem with prmetric uncertinty re conidered, decribed by the fmily of chrcteritic polynomil p() = n 1 n 1 n 1 n (16) where 1,..., n re coefficient, which depend linerly of ome uncertin prmeter k, nd cn be either rel or complex one. For election of the uncertin prmeter k, trnform eqution (16) nd rewrite it in the following form: φ() kψ() = 0 (17) where φ() nd ψ() re ome polynomil of the complex vrible ; k i the ytem uncertin prmeter. Bed on (17), derive the expreion for k in the form φ() k = f() = = u(, ) iv(, ) (18) () where u(,ω), v(,ω) re hrmonic function of two independent rel vrible nd ω. Conider ome proviion bout the root locu field. Definition 8. The root locu field of the control ytem i the field with the complex potentil ϕ( ) = u(, ω) iν(, ω), tht i defined in every point of the extended free prmeter complex plne by etting the root locu imge exitence over the whole plne (Rimky & Tborovetz, 1978). Then, et the root locu imge by the rel function h = h( u, ν, t), where t i the contnt vlue for every imge. Nme t, the imge prmeter. Suppoe the imge i defined over the whole free prmeter plne by etting the correponding boundrie of the prmeter t. Thu, uing mpping function (18), define in the generl form the clr root locu field function f *= f *(,ω) (19)

19 91 Indutril Robotic: Theory, Modelling nd Control nd the root locu field level line eqution f *(,ω) = L, (0) where L = cont = t j, t j i the prmeter of the j-th imge, t j, j = 1,,,.1 The tk formultion Define the qulity domin Q (fig. 5) in the left complex hlf-plne of the ytem fundmentl frequencie (root plne), bounding the eqution (16) root loction by the line L η ' nd L η '' of the equl degree of tbility (tbility mrgin) nd the line L β nd L β of contnt dmping, tht i equivlent to etting permiible limit for the following ytem qulity indictor: degree of the ytem tbility η nd ocilltion β. In fig. 5 the qulity domin Q h the hpe of trpezoid. The tk conit in locting the chrcteritic eqution (16) root within the domin Q, i. e. in determintion of uch domin D of the uncertin prmeter k vlue, which enure loction of thi eqution root (e. g., p 1, p, p, p in fig. 5) within the given domin Q, when the ytem qulittive chrcteritic do not get beyond the preet limit for η nd β, enuring thu the ytem Q- tbility nd fulfillment of the condition. bounded by the line of equl degree of tbility nd contnt dmping k D i Q, (1) where i = 1,,,, n.

20 Indutril robot control ytem prmetric deign on the be of method for uncertin 91 Figure 5. The domin Q of the deired chrcteritic eqution root loction, bounded by the line of equl degree of tbility nd contnt dmping. For olving the tk, pply the root locu field of the circulr imge (circulr root locu field CRLF) (Rimky, 197; Neenchuk, 005). The field function (19) nd the level line eqution (0) for the CRLF in the generl form: f* = f*(, ω,, b) ()

21 91 Indutril Robotic: Theory, Modelling nd Control f*(, ω,, b) = ρ. () where nd b re the imge center coordinte by xe u nd υ correpondingly, = cont nd b = cont; ρ i the circulr imge rdiu. The circulr root locu field for the ytem of cl [;0] re repreented in fig. 6 nd 7. The CRLF loction in the complex plne to the gret extent i defined by the given circulr imge center loction, which i mpped onto the complex plne by the field locliztion center (ee definition. in (Neenchuk, 005)). Locliztion center of the field, decribed by the level line L 1 (L 1 ', L 1 '', L 1 '''), L, L, L, re locted in the point 1,, (fig. 6, b). The level line bound the correponding domin (W 1, W, W, W in fig. 6, b) in the plne. Every uch mny-heeted domin W repreent the mpping of the root locu level line dik-imge of the certin rdiu.. Locting root in the given domin The given tk i olved by incribing the level line of the CRLF, previouly oriented in pecil wy in the complex plne, into the given qulity domin of the ytem. Thi level line imge in the free prmeter plne k, tht repreent ome circle of the rdiu r, will be the boundry of the required domin D (the required dik). Then, in ce, if the circulr imge center i locted in the origin, the following condition hould be tified: k r. The field orienttion For reliztion of the bove indicted tk olution lgorithm, t firt it i necery to et orienttion (loction) of the clr CRLF in reltion to the ytem qulity domin in uch wy to enure the poibility of the field level line incription into thi domin. Aume the circulr imge center i locted on the poitive rel xi u, including the origin. The deired loction of the circulr field i ttined, when ll it locliztion center (i. e. the point, which repreent mpping of the circulr imge center onto the complex plne ) re locted inide the qulity domin. The enough condition for enuring uch orienttion of the field locliztion center i loction of function (18) zeroe within thi domin. A it w initilly umed, tht the circulr imge center w locted on the rel xi, the locliztion center cn be et in two wy: - in zeroe of function (18), i. e. in pole of the open-loop ytem trnfer function; - on the brnche of the invetigted control ytem Teodorchik Evn root locu (TERL).

22 Indutril robot control ytem prmetric deign on the be of method for uncertin 915 (а) b) (b) b) Figure 6. Circulr root locu field when etting the imge center in the origin of the vrible prmeter plne k (а) b) (b) b) Figure 7. Circulr root locu field when hifting the imge center in reltion to the origin of the vrible prmeter plne k

23 916 Indutril Robotic: Theory, Modelling nd Control In the firt ce the circulr imge center will be locted in point C, where k = 0 (fig. 6, а). In the econd ce the field locliztion center hould be locted on the TERL poitive brnche egment being completely locted within the given qulity domin. Coordinte u = nd υ = b (fig. 7, а) of the correponding imge center re determined from formul (18). The level line incription After etting the field locliztion center it i poible to trt incription of it level line into the given qulity domin. The incription procedure conit in finding uch level line, which completely belong to the given qulity domin nd which repreent itelf the mpping of the circulr imge with the mximl poible rdiu, tht evidently will gurntee the required Q-tbility of the fmily (16). Conditionlly divide the tk into two ubtk of the level line incription into the domin, bounded only by: - the verticl line of equl degree of tbility; - the inclined line of contnt dmping. Conider the firt ubtk. For it olution find the extreme point of contct of the CRLF level line nd the line L η ', L η '' of equl degree of tbility (fig. 5). Apply the formul for the grdient of the root locu field: grdf f = f i ω j, where f * (,ω) i the field function; i, j re projection of the identity vector, directed long the norml to the field level line, onto the xe nd ω correpondingly. Becue in the deired point of contct the grdient () projection onto the xi iω re equl to zero, determine thee point coordinte by compoing two ytem of eqution: () f * (, ω) = 0 ω ; = η ' (5) f * (, ω) = 0 ω, = η '' (6)

24 Indutril robot control ytem prmetric deign on the be of method for uncertin 917 where the firt eqution of every ytem repreent projection of the grdient onto the xi ω; η ' nd η '' re coordinte of cro point of the xi nd the line L η ' nd L η ' correpondingly. From the firt ytem of eqution the coordinte ω of the extreme contct point of the line L η ', bounding the qulity domin from the right ide, nd the CRLF level line i determined. The econd ytem llow to determine the coordinte ω of the extreme contct point (e. g., point t in fig. 8) of the line L η '', bounding the domin Q on the left ide, nd the CRLF level line. Turn to the econd ubtk conidertion. For it olution it i necery to find the extreme contct point (point) of the CRLF level line nd the line L β or L β (fig. 5) of contnt dmping. The only one line, L β or L β, i choen becue when the imge center i et on the xi u of the free prmeter plne, the CRLF i ymmetric in reltion to the xi iω. The line L β will be conidered tngent to the CRLF level line. Figure 8. The domin of root loction, incribed into the given qulity domin

25 918 Indutril Robotic: Theory, Modelling nd Control Write the eqution of tngent to the clr CRLF level line ( tngent to the curve) in the generl form: f * (, ω) f ( ) * (, ω) ( Ω ω) = 0, ω (7) where, Ω re current coordinte of point on the tngent;, ω re the point of contct coordinte. A in thi ce the tngent to the level line pe through the origin, et coordinte nd Ω to zero nd rewrite (7) in the following form: f * (, ω) f ( ) * (, ω) ( ω) = 0. (8) ω On the other hnd, the eqution of the level line L β i ω = µ, where µ i the rel contnt, µ = tg β (fig. 5), β i the ngle between the contnt dmping line nd the xi ω. By compoing on the bi of the lt two eqution the ytem f * (, ω) f ( δ) δ * (, ω) ( ω) = 0 ω ω = µ (9) nd olving (9), obtin coordinte nd ω of the deired contct point. It i necery to note, tht when olving both the firt nd the econd ubtk, everl point of contct to every qulity domin boundry cn be found. It i explined by the fct, tht contct point re determined for both globl nd every locl field level line. In thi ce the level line correponding to the circulr imge of the miniml rdiu i lwy choen. Thu, from three point t 1, t nd t (fig. 8), found by the bove decribed method, the point t 1 locted on the level line L 1, correponding to the circulr imge of the miniml rdiu, i choen. Thi line repreent itelf the boundry of the deired domin D of the uncertin prmeter k vlue, enuring the required ytem opertionl qulity indictor. Conider the numericl exmple. The ytem qulity domin Q (ee fig. 5) i bounded by the line of equl degree of tbility, decribed by eqution = 1., =.7,

26 Indutril robot control ytem prmetric deign on the be of method for uncertin 919 nd the line of contnt dmping with eqution = ω, = ω. Set the chrcteritic eqution, decribing the dynmic ytem of cl [;0] nd elect polynomil φ() nd ψ() (ee (17)): φ() = 7,5 17,8 1,1; (0) ψ() = 1. (1) Suppoe, tht the polynomil contnt term n i the uncertin prmeter. It i required to determine the domin of the perturbed coefficient n vlue, belonging to the given qulity domin Q. Evidently, the pole p 1 = 1.5, p =.5 nd p =.5 (in fig. 8 re mrked by X-) of the open loop trnfer function re locted inide the qulity domin Q. Define the circulr root locu field by etting the root locu imge exitence region over the whole plne of the free prmeter n. For thi purpoe et the circulr field loction by defining circulr imge center in the point C with coordinte = 5, b = 0 (fig. 7, а) in the free prmeter plne n. Then, it locliztion center re locted in point C 1, C nd C on the brnche of the ytem Teodorchik Evn root locu, hown in fig. 7, b. Clcultion were crried on with ppliction of the computer progrm for enuring the required qulity of control ytem with prmetric uncertinty, developed for the bove decribed method reliztion. Polynomil (0), (1) nd the domin Q boundrie eqution were entered the input dt. The following reult hve been obtined. The circulr imge root locu eqution for the given ytem: ω 6,ω ω 15ω ,8ω ω 91,8 0 0ω ω 6 0 = 0. The CRLF function, pplied for the ytem invetigtion: f * (, ω) = ω 158ω 6 ω 5,ω 6 15ω 0,7ω 15, 5 91,9 ω 0 0ω ω 11ω 6 8. For determintion of the CRLF level line, incribed into the qulity domin, the following ytem of eqution (5), (6) nd (9) were olved:

27 90 Indutril Robotic: Theory, Modelling nd Control 6ω 6ω 1ω 1ω 60ω 8,8ω 60ω 8,8ω ,6 = 0 ; = 1, ,6 = 0 ; =,68 5 (6ω 15ω 1ω 90ω 6ω 158ω )( ) (6ω 1ω 60ω. 8,8ω ,6)( ω) = 0 ω = The firt eqution of the firt nd the econd ytem repreent the CRLF grdient vlue in the contct point of the field level line nd the line, bounding the qulity domin from the left nd right (the line of equl degree of tbility), the econd eqution repreent the eqution of the line of equl degree of tbility. The firt eqution of the third ytem repreent the eqution of tngent to the CRLF level line, which pe through the origin. A reult of thee eqution three point of contct of the CRLF level line nd the line L η ', L η '' nd L β, bounding the qulity domin, re defined. In fig. 8 thee point re t 1, t for contct of level line L 1 '', L 1 ' correpondingly nd the contnt dmping line L β nd point t for contct of the level line L '' nd the line L η '' of equl degree of tbility. It h been found, tht the point t belong to the level line, incribed into the domin Q, nd the line L ', L '', which correpond to the contct point t, nd the level line L ''' get beyond thi domin (the line L ', L '' nd L ''' repreent mpping of ingle circulr imge). Thu, three imply connected cloed region (in fig. 8 they re cro-htched) re formed, bounded correpondingly by three level line L 1 ', L 1 '' nd L 1 '', repreenting three heet of the three-heeted domin, defined by mpping of the imge dic onto the plne uing three brnche of the three-vlued mpping function. Thi three-heeted domin repreent the domin of the chrcteritic eqution root, tifying the required qulity. The imge of thi domin boundry onto the plne n i the circle of rdiu r =, bounding the deired cloed domin D of the free prmeter n vlue, which comply with the given condition of the ytem tbility. The developed method for prmetric ynthei of the dynmic ytem, meeting the robut qulity requirement, i bed on the circulr root locu field ppliction. It llow to elect ome region of the ytem chrcteritic eqution root loction, belonging to the given qulity domin, which define the required qulity indictor vlue (degree of tbility nd ocilltion), nd lo to define the correponding region of the vrible prmeter vlue, enuring the ttu when the ytem qulity chrcteritic do not get beyond the boundrie et. The min dvntge of the method i, tht it llow to determine the ytem prmeter vlue, which enure the required qulity indic-

28 Indutril robot control ytem prmetric deign on the be of method for uncertin 91 tor for ce when the given ytem doe not comply with the requirement, i.e. to crry on the dynmic ytem prmetric ynthei. The method cn be generlized to the tk of root loction within the domin of other hpe. 5. Prmetric deign of the indutril robot robut control ytemon the be of the method for intervl control ytem ynthei Opertion of the indutril robot (ee ection ) in condition of uncertinty i conidered, when it prmeter re ubject to ubtntil vrition. The bove decribed technique i pplied for olving the tk of the nthropomorphou robot mnipultor unit control ytem prmetric ynthei. It llow to find nlyticlly the mnipultor prmeter vlue vrition rnge, which will enure mintining the ytem tbility property nd the required opertionl qulity within their limit, i. e. to enure the ytem robutne. 5.1 Control ytem model for the ce of opertion in condition of uncertinty Robot lod chnge with vrition of the weight of the item they crry, tht cue vrition of the lod inerti moment j l, which i linerly included into the chrcteritic eqution coefficient (ee (1) nd ()), generting their vrition intervl. Currently during the deign procedure robot prmeter vlue in the ce of ubtntil prmeter vrition re obtined by the technique of tet nd mitke. Conduct prmetric ynthei of the mnipultor houlder control ytem in condition of it prmeter uncertinty uing the nlyticl method decribed in. Let coefficient of the chrcteritic eqution () for the mnipultor houlder unit vry within the following limit: 0 = 1; 0, 10 0, ,5 10 ; 0,7 10 0, ,7 10 ; 0, , ; Suppoe ny of the coefficient, e. g., vrie continuouly long the rel xi in the plne of ytem fundmentl frequencie. Tking into ccount expreion (1), the complex mpping function (5), tht determine root loci of the intervl ytem reltive to, i defined φ( ) f ( ) = = ( 1 ). () ψ( ) The control ytem chrcteritic eqution i

29 9 Indutril Robotic: Theory, Modelling nd Control ( 1 = φ ) ψ( ) = 0. () The limit vlue of eqution () coefficient vrition intervl re entered to the input of the pckge ANALRL for computer-ided invetigtion of control ytem with vrible prmeter. During the pckge functioning the Khritonov' polynomil of the ytem chrcteritic eqution re formed: 1 h ( ) = h ( ) = h ( ) = h ( ) = 0, 10 0,5 10 0, 10 0,5 10 0,7 10 0,7 10 0,7 10 0, , ,5 10 0, ,5 10 0, , , ,56 10 = 0, = 0, = 0, = 0. Thee four eqution form the bi for genertion of the mthemticl model for the robot intervl control ytem in the form of the Khrotiniv polynomil root loci. Conidering preence of the lod inerti moment j l ubtntil vrition, it i required to find the coefficient vrition intervl, enuring tbility of the chrcteritic eqution fmily. 5. Procedure of the control ytem prmetric ynthei For the tk olution pply the method, decribed in ection, which llow to clculte the chrcteritic eqution fmily coefficient intervl, enuring the ytem robut tbility. Firt, zeroe of function () (the pole of the open loop trnfer function) re clculted for the bove Khritonov' polynomil nd, if they re locted in the left-hlf plne of the plne (ee ttement given in ubection.) or on the tbility bound iω, the root loci of Khritonov' polynomil re generted on the bi of thee function. A for our exmple one of zeroe of function () i locted on the tbility bound (in the point = 0), the direction of the correponding root locu i verified. It i tted tht the poitive brnch i directed from thi zero to the left hlf plne tht, ccording to the bove given ttement (ee.), men the exitence of poitive tbility intervl of the ytem invetigted. After contructing the prmeter function ccording to the technique uggeted bove (ee ection.), the vrible coefficient vlue from the et A r (11)) in the cro point of the Khritonov' polynomil root loci brnche with the ytem tbility bound iω re determined. For the given ce the following vlue hve been obtined:

30 Indutril robot control ytem prmetric deign on the be of method for uncertin 9 r 1 r r r 9 =0, 10 (polynomil h 1); 9 =0,6 10 (polynomil h ); 9 =0,1 10 (polynomil h ); 9 = 0,80 10 ( polynomil h ). According to the correponding tk lgorithm nd the obtined vlue, the r miniml poitive vlue min = i determined. The intervl (1) of vlue vrition i clculted tht enure ytem ymptotic tbility: d = (0; )). On the bi of the theorem, formulted in., the following tbility condition of the intervl ytem i formed: 0 < < A the root locu prmeter vrie within the limit = , nd = , the upper one doen t comply with the tbility condition. For enuring tbility of the conidered intervl control ytem the upper limit hould be et to = The limit of the cceptble intervl of the coefficient vrition re the following: = , = 0. From the bove decribed clcultion it i evident tht the developed method cn be pplied not only for invetigting the intervl ytem tbility, but lo for clculting intervl of it vrible prmeter in ce the initil ytem i not tble. It i worth to pinpoint tht the method llow to enure the ytem ymptotic tbility by etting up only one coefficient of it chrcteritic eqution.

31 9 Indutril Robotic: Theory, Modelling nd Control 6. Concluion Indutril robot repreent device, which uully operte in condition of ubtntil uncertinty. Therefore, in thi chpter the problem of uncertin control ytem tbility nd qulity i conidered in ppliction to the indutril robot nlyi nd ynthei tk olution. The tk for ynthei of the intervl control ytem tble polynomil h been olved. For it olution the invetigtion of the ytem root locu portrit behvior t the ymptotic tbility boundry h been crried out. On thi bi the ytem robut tbility condition w formulted. The method h been developed for etting up the intervl polynomil for enuring it tbility in ce, when the tbility verifiction howed, tht the initil polynomil w untble. If the ytem order n >, thi method i pplicble when the Khritonov polynomil free root loci initil point re locted in the left complex hlf-plne, becue in thi ce the root locu portrit configurtion enure exitence of the tbility intervl on every brnch of the root loci. When n, the method i pplied without ny condition (limittion). The lgorithm conidered llow to crry on prmetric ynthei of the tble intervl ytem without it root locu portrit modifiction by etting up the limit vlue of the chrcteritic polynomil coefficient vrition intervl. Thu, the tbility intervl for the initilly untble polynomil i defined. The obtined tble polynomil cn be elected to be the neret to the initil (given) one in the ene of the ditnce, meured long it root trjectorie with conidertion of the pproprite ytem qulity requirement. The root locu method h lo been developed for enuring the required qulity (Q-tbility) of the uncertin control ytem. It i bed on incription of the circulr root locu field level line into the given qulity domin. Currently during the indutril robot deign procedure in the ce of ubtntil prmeter vrition the robot control ytem prmetric ynthei i often conducted by the method of tet nd mitke. The technique, conidered here, llow to crry on the nlyi nd prmetric ynthei of the robot control ytem, operting in condition of prmetric uncertinty, uing nlyticl procedure.

32 Indutril robot control ytem prmetric deign on the be of method for uncertin Reference Brmih, B. (198). Invrince of the trict hurwitz property for polynomil with perturbed coefficient. IEEE Trn. Automt. Control, Vol. 9, Nо. 10, (October 198) 95 96, ISSN Brtlett, A.; Hollot, C. & Lin, H. (1987). Root Loction of n entire polytope of polynomil in uffice to check the edge. Mth. Contr., Vol. 1, Nо. 1, (1987) 61 71, ISSN Bhttchryy, S.; Chpellt, H. & Keel, L. (1995). Robut Control: the Prmetric Approch, Prentice Hll, ISBN X, New York. Givoronky, S. (006). Vertex nlyi for the intervl polynomil root locliztion in the given ector [in Ruin], Thee of the Third Interntionl Conference on Control Problem, p. 95, ISBN , Mocow, June 0, 006, Intitute of Control Science of the Ruin Acdemy of Science, Mocow. Gryzin, E. & Polyk, B. (006). Stbility region in the prmeter pce: D-decompoition reviited. Automtic, Vol., No. 1, (Jnury 006) 1 6, ISSN Hndbook of Indutril Robotic (1989). Edited by Nof, S. John Wiley nd Son, ISBN , New York. Khritonov, V. (1978). About ymptotic tbility of the liner differentil eqution fmily equilibrium [in Ruin]. Differentyl'nye Urvneniy, Vol. XIV, No. 11, (November 1978) , ISSN Krev, A. & Furov, А. (00). Etimting the intbility rdii for polynomil of rbitrry power [in Ruin]. Nonliner Dynmic nd Control. Iue : Collection of Pper / Ed. Emelynov, S. & Korovin, S. FIZMATLIT, ISBN , Mocow, pp Lvrentyev, M. & Shbt, B. (1987). Method of the Complex Vrible Function Theory, Nuk, Mocow. Neenchuk, A. (00). Prmetric ynthei of qulittive robut control ytem uing root locu field. Proceeding of the 15th Triennil World Congre of IFAC, vol. E, pp. 1 5, ISBN X, Brcelon, Spin, July 1 6, 00, Elevier Science, Oxford. Neenchuk, A. (005). Anlyi nd Synthei of Robut Dynmic Sytem on the Bi of the Root Locu Approch [in Ruin], United Intitute of Informtic Problem of the Belruin Ntionl Acdemy of Science, ISBN , Mink. Polyk, B. & Typkin, Y. (1990). Frequency criteri for robut tbility nd periodicity [in Ruin]. Avtomtik i Telemekhnik, No. 9, (September 1990) 5 5, ISSN Polyk, B. & Scherbkov, P. (00),. Robut Stbility nd Control [in Ruin], Nuk, ISBN , Mocow.

33 96 Indutril Robotic: Theory, Modelling nd Control Polyk, B. & Scherbkov, P. (00), b. Supertble liner control ytem I, II ([n Ruin]. Avtomtik i Telemekhnik, No. 8, (00) 7 5, ISSN Rntzer, A. (199). Stbility condition for polytope of polynomil. IEEE Trn. Automt. Control, Vol. 7, Nо. 1, (Jnury 199) 79 89, ISSN Rimky, G. (197). Bic of the Generl Root Locu Theory for Control Sytem [in Ruin], Nuk i Tekhnik, Mink. Rimky, G & Tborovetz, V (1978). Computer-Aided Invetigtion of the Automtic Sytem, Nuk I Tekhnik, Mink. Shw, J. & Jyuriy, S. (199). Robut tbility of n intervl plnt with repect to convex region in the complex plne. IEEE Trn. Automt. Control, Vol. 8, Nо., (Februry 199) 8-87, ISSN Soh, C.; Berger, C. & Dbke, K. (1985). On the tbility propertie of polynomil with perturbed coefficient. IEEE Trn. Automt. Control, Vol. 0, Nо. 10, (October 1985) , ISSN Soh, Y. (1989). Strict hurwitz property of polynomil under coefficient perturbtion. IEEE Trn. Automt. Control, Vol., Nо. 6, (June 1989) 69 6, ISSN Typkin, Y. & Polyk, B. (1990). Frequency criteri of robut modlity for the liner dicrete ytem [in Ruin]. Avtomtik i Teemekhnik, No., (April 1990) 9, ISSN Typkin, Y. (1995). Robut tbility of the rely control ytem [in Ruin]. Dokldy RAN, Vol. 0, No. 6, (Februry 1995) , ISSN Udermn, E. (197). The Root Locu Method in the Control Sytem Theory [in Ruin], Nuk, Mocow. Vicino, A. (1989). Robutne of pole loction in perturbed ytem. Automtic, Vol. 5, No. 1, (Jnury 1989) , ISSN

34 Indutril Robotic: Theory, Modelling nd Control Edited by Sm Cubero ISBN Hrd cover, 96 pge Publiher Pro Litertur Verlg, Germny / ARS, Autri Publihed online 01, December, 006 Publihed in print edition December, 006 Thi book cover wide rnge of topic relting to dvnced indutril robotic, enor nd utomtion technologie. Although being highly technicl nd complex in nture, the pper preented in thi book repreent ome of the ltet cutting edge technologie nd dvncement in indutril robotic technology. Thi book cover topic uch networking, propertie of mnipultor, forwrd nd invere robot rm kinemtic, motion pth-plnning, mchine viion nd mny other prcticl topic too numerou to lit here. The uthor nd editor of thi book wih to inpire people, epecilly young one, to get involved with robotic nd mechtronic engineering technology nd to develop new nd exciting prcticl ppliction, perhp uing the ide nd concept preented herein. How to reference In order to correctly reference thi cholrly work, feel free to copy nd pte the following: All A. Neenchuk nd Victor A. Neenchuk (006). Indutril Robot Control Sytem Prmetric Deign on the Be of Method for Uncertin Sytem Robutne, Indutril Robotic: Theory, Modelling nd Control, Sm Cubero (Ed.), ISBN: , InTech, Avilble from: ytem_prmetric_deign_on_the_be_of_method_for_uncertin_ytem_robu InTech Europe Univerity Cmpu STeP Ri Slvk Krutzek 8/A Rijek, Croti Phone: 85 (51) Fx: 85 (51) InTech Chin Unit 05, Office Block, Hotel Equtoril Shnghi No.65, Yn An Rod (Wet), Shnghi, 0000, Chin Phone: Fx: