www.engneersress.com World of Scences Journal ISSN: 307-307 Year: 03 Volume: Issue: 8 Pages: 48-58 Aahmad Ghanbar,, Arash ahman Deartment of Mechancal Engneerng, Unversty of Tabrz, Tabrz, Iran School of Engneerng Emergng Technologes, Mechatroncs Laboratory, ABSTACT Neural Network Solutons for Forward Knematcs Problem of ybrd Seral-Parallel Manulator Unversty of Tabrz, Tabrz, Iran Ths aer resents forward and nverse Knematcs analyss of a secfc class of seres arallel manulators, known as (6-UPS) manulators. As Forward knematcs roblem of s knd of manulators s a very dffcult roblem to solve because of er hghly nonlnear relatons between ont varables and oston and orentaton of e end effectors, Numercal meods are e most common aroaches to solve. Nevereless, e ossble lack of convergence of ese meods s e man drawback. Therefore, artfcal neural networks (ANN) w er nherent learnng ablty as a strong meod, was used to aroxmate e forward knematcs functon wout any knowledge of manulator structure. In s aer, two tyes of ANN models were used. MLP (mult-layer ercetron network) and BF (radal bass functon network) have been used to solve e forward knematcs roblem of s hybrd manulator and results are obtaned. Smulaton results show e advantages of emloyng neural networks. Also, accordng to average ercentage error, as e erformance ndex, t was found at BF gves better result as comared to MLP. KEYWODS: Artfcal neural networks, knematc Analyss, Mult-layer ercetron, adal bass functon. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 49. INTODUCTION A hybrd manulaton system s a sequence of arallel mechansms whch can overcome e lmted worksace of arallel mechansm and can rovde feature of bo seral and arallel mechansm. They are able to acheve hgh stffness and hgh force-to-weght rato. Many dfferent tyes of hybrd robots have been nvestgated n [-4]. Tano [5] resented a hybrd (arallel seral) manulator consstng of two serally connected arallel mechansm and gave t s closed-form soluton for forward and nverse oston roblems. omdhane s [6] hybrd manulator s also made of a base and two latforms n seres and e moton of e md latform s restrcted only to ree translatons and e second latform rotates shercally w resect to e md latform usng ont connected e md latform and to latform. uang et al. [7] studed e characterstcs of 6 DOF arallel seral hybrd manulators whch features a 3 DOF n seres actuated module mounted on e movng late of anoer 3 DOF n arallel actuated manulator w rsmatc actuators. Yang et al. [8] dscussed e knematcs of hybrd tye manulaton system w 6 DOF, whch consst of a 3-DOF lanar arallel latform and a 3-DOF seral robot arm. uang et al. [9] studed a concetual desgn and dmensonal syness of a 3-DOF arallel mechansm module whch forms e man body of a newly nvented 5-DOF reconfgurable hybrd robot. LangZh et al. [0] studed a hybrd 5DOF manulator based on e novel 3-PS nactuated arallel manulator. In er desgn a DOF seral workng table s laced over e moble latform. Camos et al. [] resented a new meodology to synesze hybrd robots as a whole structure. Ther meod s based on Assure grous as e smlest basc blocks to buld knematc chans. Gallardo-Alvarado et al. [] studed Knematcs and dynamcs of (3-PS) manulators by means of screw eory and e rncle of vrtual work. Mohammadanah et al. [3] Desgn and Analyss of a Novel 8-DOF ybrd Manulator at s arorate for e stated alcatons by connectng two seral and arallel mechansms. Bng L et al. [4] used a hybrd manulator as a mult-dmensonal vbraton solator based on e arallel mechansm. The scheme desgn, nverse knematcs, worksace and dexterty are carred out n er aer. In s aer a novel hybrd robot (6-UPS) s ntroduced at comosed a sequence of two same stewart mechansm modules. The seral form of ese hybrd manulators overcomes e lmted worksace of arallel manulators and mroves overall stffness and resonse characterstcs. Then nverse and forward knematcs soluton s resented for t. Because of er hghly nonlnear relatons between ont varables and oston and orentaton of e end effectors, two tyes of ANN models - MLP (mult-layer ercetron network) and BF (radal bass functon network) - have been used to solve e forward knematcs. Ths aer s organzed as follows: In Secton, descrton of e robot s dscussed. In Secton 3, e nverse and forward knematc solutons are resented n e closed form. MLP network and BF meod to solve forward knematc (FK) are dscussed n secton 4. In secton 5 e results of solvng FK for (6-UPS) manulator robot by ese networks are resented. Comarson of ese networks and concluson are dscussed n secton 6.. DESCIPTION OF TE YBID OBOT The mechansm under nvestgaton n s aer conssts of two same modules at each module s Stewart Platform mechansm w 6 DOFs. In s hybrd mechansm, we have ree latforms and twelve ods. Base latform s statonary and connected to mddle latform va 6 extensble ods. Also, mddle latform s connected to uer latform (as an end effecter) va 6 extensble ods. Each od connects to e latform at ts connecton ont rough a shercal ont, and to e base at ts connecton ont rough unversal ont. Each od conssts of two arts: e uer art and e lower art, whch connect to each oer rough rsmatc ont. Therefore, t s referred to as e (6-UPS) mechansm. Ths manulator s actuated by motors located on e rsmatc onts. Fgure shows e desgn of e mentoned hybrd robot. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 50 Fg.. CAD Fle for (6-UPS) hybrd robot 3. FOWAD AND INVESE KINEMATICS SOLUTION Mechansm knematcs deals w e study of e mechansm moton as constraned by e geometry of e lnks. Tycally, e study of mechansm knematcs s dvded nto two arts: nverse knematcs and drect knematcs. About mentoned hybrd robot, e nverse knematcs roblem nvolves mang a known ose (oston and orentaton) of e movng latforms of e mechansm to a leng of each module s ods. The drect knematcs roblem nvolves e mang from a known leng of each module s ods to a ose of e movng latforms. In s secton e nverse and forward knematcs roblems of roosed mechansm are descrbed n closed form. Fgure shows e vectoral reresentaton of e od at each module. Accordng to fg., e mddle and uer movng latforms frame are shown by {C} and {C} resectvely and base frame w {O}. Also, X ( x, y, z, α, β, γ) and X ( x, y, z, α, β, γ ) resent e locaton and orentaton of e mddle and uer movng latform resectvely. Now, e nverse knematcs of each module s obtaned at frst, and en forward knematcs s consdered. Inverse Knematc roblem of e latforms nvolves determnaton of e lnear oston, of sx Pods for each module rough consderng a secfed oston, of e mddle and uer movng latforms centre. A. Mddle Movng Platform The leng vector of e od n e uer module can be obtaned as: L o b for,,...,6 () o D + (. c) () Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 5 {C} c Uer Module r D r {C} c Base Module D r L r z x {O} y b r B Fg.. vectoral reresentaton of e od at each module Where s e rotaton 3 3 matrx, reresentng e rotaton of frame {C} related to frame {O} and t s defned f X ( x, y, z, α, β, γ) s obvous: 3 Also, 3 3 3 33 (3) Dx bx r r r D D Dy, b by, D z bz c x y z (4) Usng equaton () to (4), e leng of e od, L, for base module can be exressed as: L L L L x y z Dx+ Dy+ Dz + + + + Corresondng auor: Ahmad Ghanbar Emal: A-ghanbar@tabrzu.ac.r 3 x x x 3 y y y + + + 3 3 33 z z z b x b b y z (5) World of Scences Journal
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 5 L L + L + L x y z (6) B. Uer Movng Platform The leng vector of e od n e uer module can be obtaned as: o o for,,...,6 (7) Where: o D + D. +. c for,,...,6 (8) By substtutng equatons () and (8) nto Equaton (7) and consderng: 3 3 3 3 33 (9) And: D x r r D D Dy, D z x c y z (0) can be exressed as: Dx+ Dy+ 3Dz+ x+ y+ 3z x y 3z x + + + + Dx Dy 3Dz x y+ y L 3z x y 3z L z D + + + + + 3 x 3Dy 33Dz 3x 3y 33 z 3x 3y 33z () Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 53 L + + x y z After Inverse Knematcs analyss of (6-UPS), we want to calculate e locaton and orentaton of e mddle and uer movng latform by knowng e leng of ods at each module (Forward Knematcs). As, t s clear, we can rewrte equatons () and (7) as below: f( X) D + c. b L 0 r g( X ) D. +. c c. 0 The soluton of above equatons s e forward knematcs roblem of e mechansms. But, because of hghly nonlnear characterstc of ese equatons, t s so dffcult to solve em. Therefore, ANN s aled to solve forward knematcs of s mechansm. () (3) (4) 4. ATIFICIAL NEUAL NETWOKS Artfcal neural network (ANN) s a arallel-dstrbuted nformaton rocessng system. Ths system s comosed of oerators nterconnected va one-way sgnal flow channels. ANN stores e samles w a dstrbuted codng, us formng a tranable nonlnear system. It ncludes hdden layer(s) between e nuts and oututs. The man dea of e ANN aroach resembles e human bran functonng. Therefore, ANN has a qucker resonse and hgher erformance an a sequental dgtal comuter. Gven e nuts and desred oututs, t s also self-adatve to e envronment so as to resond dfferent nuts ratonally. Varous neural networks have been used to solve e knematcs roblem [5]. They nclude e mult layer ercetron network (MLPN), cerebellar model artculaton controller (CMAC) and radal bass functon network (BFN). ere we use MLPN and BFN for soluton forward knematcs of (6-UPS) and resented a comarson between MLPN & BFN. A. MLPN for Knematc Soluton The multlayer ercetron network conssts of an nut layer, an outut layer and usually one or more hdden layers. Fg.3 shows e archtecture of MLP network emloyed for redcton e forward knematc soluton for uer module. It has an nut layer of 6 neurons, for each module, one hdden layer of 5 neurons w hyerbolc (tanh) actvaton functon, f (n), defned by functon gven n bellow as: f ( n) K tanh( K n ) (5) n N W x (6) K,K are constant coeffcent. For e outut layer, a lnear actvaton functon mlementaton. g ( n) n s used n e The aroxmatng functon Y reresentng Knematc soluton s defned by Eqs. (7) and (8). Ths entals fndng an aroxmatng functon Y reresentng forward knematc soluton s an N-dmensonal vector. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 54 M Y( g A( W ( ) + b N A( f ws( k) W( k) + b( ) (7) (8) W( k) s e weght between k nut and hdden layer neuron and outut summaton node; ( ) b( ) s e term aled to e outut layer neuron; ws (k) s e summaton of hdden layer node; ( ) b s e bas term aled to e W s e weght between hdden layer neuron; k element of nut vector ws; Ms number of hdden layer neurons. The learnng rocess of MLP network nvolves e use of e nut outut data to determne e weghts and bases. The tranng functon udates e weghts and bas values accordng to Back Proagaton algorm: w ( k+ ) w ( k) + w ( k) w ( k) α E( k) + η w ( k ) (9) (0) Where w (k) and w ( k+) are network weghts for each layer n kand k+ ste durng tranng rocess, αlearnng rate, η momentum coeffcent and Es error functon whch we want to otmze t. ere α 0. 55, η 0.75 and error at e end of learnng s 0.00 for tranng set. b f ( n) g x b () b f ( n) g y b () b 3 f ( n 3 ) g 3 z b (3) b 4 5 g 4 b g 5 α β b 6 (n) f M g 6 γ b ( M) Fg.3. Archtecture of MLP network for forward knematcs Soluton for uer module Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 55 B. BF for Knematc Soluton The BF archtecture used for e roosed e rocess s shown n Fg. 4. adal bass functon networks tycally have ree layers: an nut layer, a hdden layer w a nonlnear BF actvaton functon and a lnear outut layer. The actvaton functon n e hdden layer of e BF network s Gaussan, whch s characterzed by ts resonse at decreases monotoncally w dstance from a central ont. ws comonents at were used for MLP network are chosen. In e BF network, Y ( can be exressed as a lnear combnaton of multvarate Gaussan bass functon as follows: Y( M Wh( + b N h( ex ( ws c σ 0 k) () () Where h ( s e outut of node n hdden layer, c s e center of k ws, k σ s e wd of Gaussan functon, W s e weght between BF node for e k nut varable BF unt and outut layer neuron, b s e 0 bas term and Ms e number of hdden layer neurons. The learnng rocess of a radal bass functon network nvolve e determnaton of structural confguratons n terms of e number and oston of bass functon centers s mortant because t drectly affects e qualty of e functonal aroxmaton acheved by an BFN. ere, e values of center are selected randomly from e tranng data set. The wd of Gaussan radal bass functon s exressed n terms of e maxmum dstance between e chosen center d and e number of centers M as: d σ M (3) Thus, after all tranng atterns are resented to e BFN a M P nterolaton matrx Φ s obtaned. Every row corresonds to e resonses of all hdden unts for each attern and every column to each hdden unt rough all atterns. In order to determne an otmal weght vector at mnmzes e cost functon n e least squares manner, e nterolaton meod s used. It can drectly calculate e lnear weghts from target oututs and seudo-nverson of e nterolaton matrx from e followng equaton: W (ΦΦ) T T Φ T (4) Ths soluton roduces an otmal weght vector at mnmzes e cost functon derved from e sum-squared error as descrbed. Based on e gven sread, BF functon teratvely adds one neuron at a tme to e network untl e sum squared error falls below a secfed error goal or a maxmum number of neurons s reached. After several trals, a ree layer network archtecture havng 6 neurons n nut layer neurons for each module, 55 neurons n hdden layer and 6 neurons n outut layer neurons, for each module, s dentfed as e most sutable archtecture. It s bult w e sread constant of e Gaussan set to 0.5 and tranng goal of sum squared errors set to 0.00. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 56 L h ( n) x L h ( n) y L 3 h 3 ( n) z α L 4 L 5 β L 6 h M (n) γ Fg.4. Archtecture of BF network for forward knematcs Soluton for base module 5. ESULTS The data needed for tranng of neural network s obtaned from e nverse knematcs relatons of e hybrd mechansm. The ont sace of e robot can be consdered as an nverse mage of e Cartesan sace and vce versa. Based on s, t s decded to emloy nverse knematcs relatons for determnng e ose of each ods. The ose L and can be used as an nut and e corresondng locaton and orentaton of mddle and uer movng latforms ( X and X ) as e outut for e neural network tranng data. In oer words, L X and X relatonsh s used whle generatng e data whereas X Land X mang s done whle tranng e neural network. ANN traned w such data set s found to redct forward knematc solutons more accurately due to nsgnfcant mang errors between nut and outut data. The ose L and forms e nut and corresondng ( X and X ) forms e outut of e neural network. In order to mantan unformty n tranng of ANN, a smulaton data set us obtaned for tranng and testng of s network also conssts of 04 atterns. In s case also, 80% of total data set was used for tranng and remanng 0% of data was used for testng of e network. Fgs 5, 6 show e result of ANN analyss by consderng at e mddle and uer latforms are arallel n e worksace and er orentaton do not dffer relatve togeer. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 57 Fg.5. ANN esults for Poston of Mddle Platform Fg.6. ANN esults for Poston of Uer Platform 6. CONCLUSION Ths chater has resented a novel hybrd robot w base of stewart mechansm and studed nverse and forward knematcs of t and e general background of artfcal neural networks and alcatons of neural networks to e forward knematcs roblem. The two most oular network tyes (MLPN and BFN), used wdely n knematcs aroxmaton, have been descrbed. The MLPN s a unversal and owerful soluton for almost all functonal aroxmaton alcatons due to ts nherent nonlnear mang caabltes whch can deal w a wde range of rocess features. owever, e tranng rocess s comlex because t nvolves a nonlnear otmzaton algorm whch requres an teratve rocedure and ere s no reasonable mechansm to select a sutable network confguraton. In contrast, e BFN tranng rocess s smle and straghtforward by usng a lnear otmzaton algorm based on e least squares technque. There are several solutons to choose e otmal selecton of e network confguraton from nformaton about e robotc system (tranng data) so at e generalzaton can be mroved. owever, t seems at e erformance of exstng aroaches (bo MLPNs Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r
Neural Network Solutons for Forward Knematcs by Ahmad Ghanbar, et.al. 58 and BFNs) descrbed earler s stll nsuffcently accurate and neffcent for ractcal alcatons. For ese reasons, a novel aroach usng an BFN w regularly saced oston centers has been roosed to solve e knematcs roblem. Ths soluton roduces an BFN w a suffcently small number of centers whlst achevng a satsfactory accuracy for e nverse knematcs aroxmaton. In addton, n order to enhance e generalzaton of BFNs, e concet at e constraned tranng data should be collected closely to e oston of centers has been suggested. EFEENCES [] G. Carbone, M. Ceccarell, A seral arallel robotc archtecture for surgcal tasks, obotca, Vol. 3,. 345 354, 005. [] G. Carbone, M. Ceccarell, A stffness analyss for a hybrd arallel seral manulator, obotca, Vol.,. 567 576, 005. [3] Bng Zhou-Yan Xu, obust control of a 3-DOF hybrd robot manulator, Int J Adv Manuf Technol, Vol. 33,.604 63, 007. [4] J.M. co, J. Gallardo, J. Duffy, Screw eory and hgher order knematc analyss of oen seral and closed chans, Mechansm and Machne Theory, vol. 34,. 559 586, 999. [5] Tano K. Tanev, Knematcs of a hybrd (arallel-seral) robot manulator, Mechansm and Machne Theory, 35,. 83-96,000. [6] L. omdhane, Desgn and analyss of a hybrd seral arallel manulator, Mechansm and Machne Theory, 34, PP. 037-055,999. [7] Mng Z. uang, Shou-ung Lng, Yang Sheng, A study of velocty knematcs for hybrd manulators w arallel seres confguratons, obotcs and Automaton, IEEE Int. Conf. 993. [8] G. Yang, Weha Chen Edwn, u L.eong o, Desgn and Knematc Analyss of a Modular ybrd Parallel- Seral Manulator, Seven Internatonal Conference on Control, Automaton, obotcs And Vson (ICACV'OZ), Sngaore, Dec 00. [9] T. uang, M. L, X.M. Zhao, J.P. Me, D.G. Chetwynd, S.J. u, Concetual desgn and dmensonal syness for a 3-DOF module of e TrVarant a novel 5-DOF reconfgurable hybrd robot', IEEE Trans. ob., vol.,. 449 456, 005. [0] F. LangZh, A.Y. Elatta, L. XaoPng, Knematc calbraton for a hybrd 5DOF manulator based on 3-PS n-actuated arallel manulator, Int J Adv Manuf Technol, vol. 5,. 730 734, 005. [] A. Camos, Ch. Budde, J. esselbach, A tye syness meod for hybrd robot structures, Mechansm and Machne Theory, vol. 43,. 984 995, 008. [] Jame Gallardo-Alvarado, Carlos. Agular-Na era,lus Casque osas,jose M. co-martı nez, Md. Nazrul Islam, Knematcs and dynamcs of (3-PS) manulators by means of screw eory and e rncle of vrtual work, Mechansm and Machne Theory, vol. 43,. 8 94, 008. [3]. Mohammadanah,. Zohoor, Desgn and Analyss of a Novel 8-DOF ybrd Manulator, World Academy of Scence, Engneerng and Technology, vol.34,.37-43, 009. [4] Bng L, We Zhaoa, Zongquan Denga, Modelng and Analyss of Mult-Dmensonal Vbraton Isolator Based on e Parallel Mechsnsm, Journal of Manufacturng System, Vol. 3,.50-58, 0. [5] M. Dehghan, M. Eghtesad, A. A. Safav, A. Khayatan, and M. Ahmad, Neural Network Solutons for Forward Knematcs Problem of EXA Parallel obot, I-Tech Educaton and Publshng,. 498, 008. Corresondng auor: Ahmad Ghanbar World of Scences Journal Emal: A-ghanbar@tabrzu.ac.r