Inverse Kinematic Solution of Robot Manipulator Using Hybrid Neural Network

Transcription

1 Inverse Knematc Soluton of Robot Manpulator Usng Hybrd Neural Network Panchanand Jha Natonal Insttute of Technology, Department of Industral Desgn, Rourkela, Inda Emal: Bbhut B. Bswal and Om Prakash Sahu Natonal Insttute of Technology, Department of Industral Desgn, Rourkela, Inda Emal: {bbbswal, Abstract Inverse knematcs of robot manpulator s to determne the jont varables for a gven Cartesan poston and orentaton of an end effector. There s no unque soluton for the nverse knematcs thus necesstatng applcaton of approprate predctve models from the soft computng doman. Although artfcal neural network (ANN) can be ganfully used to yeld the desred results but the gradent descent learnng algorthm does not have ablty to search for global optmum and t gves slow convergence rate. Ths paper proposes structured ANN wth hybrdzaton of Gravtatonal Search Algorthm to solve nverse knematcs of 6R PUMA robot manpulator. The ANN model used s mult-layered perceptron neural network (MLPNN) wth back-propagaton (BP) algorthm whch s compared wth hybrd mult layered perceptron gravtatonal search algorthm (MLPGSA). An attempt has been made to fnd the best ANN confguraton for the problem. It has been observed that MLPGSA gves faster convergence rate and mproves the problem of trappng n local mnma. It s found that MLPGSA gves better result and mnmum error as compared to MLPBP. Index Terms forward Knematcs, Inverse knematcs, mult-layered neural network, D-H parameters, gravtatonal search algorthm, back propagaton algorthm I. INTRODUCTION An ndustral robot conssts of a set of rgd lnks connected together by set of jonts. To control the overall moton of mechansm for each lnks connected by varous jonts lke revolute or prsmatc s performed by motors. Generally tool or end effector performs tasks n the Cartesan coordnate system whch s controlled by jont coordnate system. For better poston and orentaton of robot end effector to perform stated task, t s essental to understand knematcs relatonshp between the jont Manuscrpt receved xxxxxx 014; revsed xxxx 014; accepted xxxx, 014. Copyrght credt, OM00, correspondng author: Panchanand Jha, Bbhut Bhusan Bswal, and Om Prakash Sahu. coordnate system and the Cartesan coordnate system. Generally there are two types of knematc analyss whch s forward knematcs and nverse knematcs. Forward knematcs s converson of jont spaces varables nto end-effector poston and orentaton. Converson of the poston and orentaton of robot manpulator end-effectors from Cartesan space to jont space s called as nverse knematcs problem. Ths s of fundamental mportance n calculatng desred jont angles for robot manpulator desgn and postonng. The correspondng jont values must be computed at hgh speed by the nverse knematcs transformaton [1]. For a manpulator wth n degree of freedom, at any nstant of tme jont varables s denoted by = (t), = 1,,...n and poston varables by x j = x(t), j = 1,,...m. The relatons between the end-effectors poston x(t) and jont angle (t)can be represented by forward knematc equaton x( t) f ( ( t)) (1) where, f s a nonlnear contnuous and dfferentable functon. On the other hand, wth the desred end effectors poston, the problem of fndng the values of the jont varables s nverse knematcs, whch can be solved by, ' ( t) f ( x( t)) () Inverse knematcs soluton s not unque due to nonlnear, uncertan and tme varyng nature of the governng equatons [-]. The dfferent technques used for solvng nverse knematcs can be classfed as algebrac, geometrc and teratve. The algebrac methods do not guarantee closed form solutons. In case of geometrc methods, closed form solutons for the frst three jonts of the manpulator must exst geometrcally. The teratve methods converge to only a sngle soluton dependng on the startng pont and may not work near sngulartes. In case of numercal method the major dffculty of nverse knematcs s that, when the Jacoban

2 matrx s sngular or ll-condtoned, t does not fnd a soluton. In addton, f the ntal approxmaton of the soluton vector (.e. the vector of jont varables) s not suffcently accurate, ths method may become unstable [4-5]. Because of the above mentoned reasons, varous authors adopted ANN.The smulaton and computaton of nverse knematcs usng multlayer feed perceptron network s partcularly useful where less computaton tmes are needed, such as n real-tme adaptve robot control [6-7]. If the number of degrees of freedom ncreases, tradtonal methods wll become more complex and qute dffcult to solve nverse knematcs [8]. Although the use of ANN s not new n the feld of mult-objectve and NP-hard problem to arrve at a very reasonable optmzed soluton, the MLPGSA has not been tred to solve nverse knematcs problem for 6R PUMA robot manpulator. Therefore, the man am of ths work s focused on mnmzng the mean square error of the neural network-based soluton of nverse knematcs problem usng GSA. The tranng data of neural network have been selected very precsely. Especally, unlearned data n each neural network have been chosen, and used to obtan the tranng set of the last neural network. The sectonal organzaton of the paper henceforth s as follows: secton pertans to the mathematcal modellng of 6R PUMA robot manpulator. Introducton to standard MLP and GSA have been brefed usng flowcharts and assocated equatons n secton along wth the method of applyng GSA to ANN as evolutonary tranng algorthms. The expermental results as obtaned from smulatons are dscussed elaborately n Secton 4. II. MATHEMATICAL MODELLING OF 6R PUMA ROBOT MANIPULATOR Denavt-Hartenberg (DH) algorthm s used to calculate the ndvdual homogeneous transformaton matrces whch then use to derve the forward and nverse knematcs of 6R PUMA robot manpulator. DH parameters and assocated values for PUMA manpulator have gven n table 1 and assgned coordnate frames are shown n Fg. 1, TABLE I. D-H PARAMETERS Inverse knematcs of PUMA manpulator s gven below: 1 a tan ( Px Py d, d) a tan( Px, Py) () a tan ( Pz, a tan ( a d 4 Px Py d ) b, b a ) where, p a a d d4 b, p Px Py Pz a can also be expressed n other form: a tan( Frame a tan( (degree) a d Px Py 4 b d (m) d, b, Pz,) a) a (m) (4) (5) a tan ( a d4 b, b ) a tan ( d4, a) (6) We can separate the arm and wrst f the manpulator has sphercal wrst. Therefor rotaton matrx for arm can be gven by: c1c s1 c1s R A s1c c1 s1s (7) s 0 c Poston matrx for arm can be gven by: (degree) 0 θ θ θ d =0.144 a 1= θ 4 d 4=0.418 a = θ Px c1 ( ca sd4 ca ) s1d P A Py s1( ca sd4 ca ) c1d (8) Pz s a cd4 sa Now general equaton for sphercal wrst can be evaluated from mappng of z-y-z Euler angle nto gven rotaton matrx: T Where, G R R Z-Y-Z ( 4, 5, ) =G 6 (9) A Therefore we can evaluate elements of matrx G from equaton (9), Fgure 1. Model and coordnate frames of manpulator g1 ( c1c) r1 ( s1c) r sr (10)

3 g s1r1 c1r (11) g ( c1s) r1 ( s1s) r cr Therefore, (1) a tan ( g, g1), f 0 4 a tan ( g, g1), f 0 Where, g1 g (1) 5 a tan (, g) (14) a tan ( g, g1), f 0 6 a tan ( g, g1), f 0 ( x y z (15) Where p,p,p ) represents the poston and ( nx,ny,nz), (ox,oy,oz), (ax,a y,az) the orentaton of the end-effector. It s obvous from the equatons () through (15) that there exst multple solutons to the nverse knematcs problem. By comparng the errors between these four generated postons and orentatons and the gven poston and orentaton, one set of jont angles, whch produces the mnmum error, s chosen as the correct soluton. III. APPLICATION OF GSA FOR TRAINING MLP Analytcal soluton of nverse knematcs problem s hghly non-lnear and mathematcally complex n nature. An ANN model does not requre hgher mathematcal calculatons and complex computng program. ANN requres ntal selecton of weght whch s vgorous to yeld local optma, convergence speed and tranng tme for the network. Generally weght s randomly selected n the range of 0 to 1, after actvaton functon weght of each neurons adjusted for the next teraton. The heurstc optmzaton algorthm optmzes the weghts of the neural networks. When certan termnaton crtera are met, or a maxmum number of teratons are reached, the teratons cease. From the prevous research hybrd optmzaton algorthm started evolvng wth hgh and remarkable advances n ther performances [9, 10]. These technques produces better outflow from local optmum and testfed to be more operatve than the standard method. In ths paper we have optmzed weght and bas for each neuron usng GSA as shown n Fg.,. For the tranng of network t s mportant to have all connecton weghts and bases n order to mnmze the mean square error. To optmze MLP neural network t s mportant to have ftness functon GSA and then t s requred to defne the ntal weght and bas for the tranng of MLP neural network. The basc steps and flow chart of MLPGSA has gven n Fg.,. (A) ftness functon Fgure. Flow chart for MLPGSA From [5-7,9], n each epoch of learnng, the output of each hdden node s calculated from equaton (16). 1 (16) f ( nk ), j 1,,,..., h n (1 exp( ( w. x b ))) Where n k n 1 1 j j j w. x b,n s the number of the nput nodes, w j s the connecton weght from the th node n the nput layer to the jth node n the hdden layer, b j s the bas (threshold) of the jth hdden node, and x s the th nput. After calculatng outputs of the output nodes from equaton (17). h ok wkj. f ( nk ) bk, k 1,,..., m 1 j (17) where w kj s the connecton weght from the jth hdden node to the kth output node and b k s the bas (threshold) of the kth output node. Fnally, the learnng error E (ftness functon) s calculated from equaton (18-19). E k E I =I +1 m 1 q k 1 k k ( o y ) (18) E k ( ) (19) q Start Generate ntal populaton I=1 Evaluaton of ftness for each agent Update the Gbest and worst of the populaton Calculate M and a for each agent Update velocty and poston d v ( t 1) rand d d v ( t) a ( t) d d d x ( t a) x ( t) v ( t a) No Gen.>Max Gen. k where q s the number of tranng samples, y s the desred output of the th nput unt when the kth tranng Yes Stop

4 k sample s used, and o s the actual output of the th nput unt when the kth tranng sample s used. Ftness functon can be calculated from equaton (0). Where the number of nput nodes s equal to n, the number of hdden nodes s equal to h, and the number of output nodes s m. Therefore, the ftness functon of the th tranng sample can be defned as follows: Ftness X ) E( X ) (0) ( IV. RESLUTS AND DISCUSSION The proposed work s performed on the Matlab R01a. Back-propagaton algorthm was used for tranng the network and for updatng the desred weghts. In ths work the tranng data sets were generated by usng equaton () through (17). A set of 1000 data sets were frst generated as per the formula for the nput parameter px, py and pz coordnates n mm. These data sets were the bass for the tranng, evaluaton and testng the MLP model. The followng parameters were taken: learnng rate 0.45, momentum parameter 0.86, number of epoch 500, number of hdden layer, number of nputs and number of output 6. The MSE for MLPBP algorthm shown n Fg., the used soluton method gves the chance of selectng the output, whch has the least error n the system. So, the soluton can be obtaned wth less error. Table gves the expermental results and comparson between the MLPBP algorthms wth respect to hybrd MLPGSA for two hdden layers. Fg., (a), (b), (c), (d), (e) and (f) shows the selected best mean square curve of MLPBP for all jont varables. Smlarly best chosen mean square curve of MLPGSA from table depcted n Fg. 4, (a), (b), (c), (d), (e) and (f) for all jont varables. TABLE II. MEAN SQUARE ERROR FOR ALL JOINT ANGLES FOR MLPBP AND MLPGSA Sn. Mean square error of MLPBP Mean square of MLPGSA e e e e e e e-15 (a)

5 (b) (c) (d)

6 (e) (f) Fgure. Mean square error for all tranng samples of MLPBP. Fgures (a), (b), (c), (d), (e) and (f) are mean square error curve for angles,,, and respectvely. 1 4, 5 6 (a)

7 (b) (c) (d)

8 (e) (f) Fgure 4. Mean square error for all tranng samples of MLPGSA. Fgure (a), (b), (c), (d), (e) and (f) are mean square error curve for angles,,, and respectvely. 1 4, 5 6 V. CONCLUSION In ths paper, two methods have been consdered whch are MLPBP and MLPGSA to obtan the soluton of nverse knematcs of 6R manpulator. In ths approach forward and nverse knematc model of 6R manpulator s used to generate the data set for tranng the MLP. The dfference n desred and predcted data wth MLPBP, gves poor results as compared to MLPGSA because of the fact that MLPGSA accumulates small number of epoch compared to that done by MLPBP wth hybrd learnng algorthm and hence MLPGSA provdes a better soluton for nverse knematc problems. It has also been observed that t gves faster convergence rate and mproves the problem of trappng n local mnma. Therefore, MLPGSA can be used for accurate and fast soluton of nverse knematcs. Future research wll revse the rules, nputs, number and type of membershp functons, the epoch numbers used, and tranng sample to further refne the MLPGSA model. REFERENCES [1] D. Xu, C. A. A. Calderon, J. Q. Gan amd H. Hu, An Analyss of the Inverse Knematcs for a 5-DOF Manpulator, Internatonal Journal of Automaton and Computng vol., pp , June 005. [] S. S Chddarwar and N. R Babu, Comparson of RBF and MLP Neural Networks to Solve Inverse Knematc Problem for 6R Seral Robot by a Fuson Approach, Engneerng Applcatons of Artfcal Intellgence vol. pp , March 010. [] S. Alavandar and M. J. Ngam, Neuro-Fuzzy based Approach for Inverse Knematcs Soluton of Industral Robot Manpulators, Int. J. of Computers, Communcatons & Control, vol., pp. 4-4, 008.

9 [4] M. L. Husty, M. Pfurner and H. P. Schrocker, A New and Effcent Algorthm for the Inverse Knematcs of a General Seral 6R Manpulator, Mechansm and Machne Theory vol. 4, pp , February 006. [5] A. Olaru and S. Olaru, Optmzaton of the Robots Inverse Knematcs Results by usng the Neural Network and LabVew Smulaton, IPCSIT, 011, pp [6] S. Mrjall, S. Z. M Hashm and H. M. Sardroud, Tranng Feedforward Neural Networks usng Hybrd Partcle Swarm Optmzaton and Gravtatonal Search Algorthm, Appled Mathematcs and Computaton vol. 18, pp , July 01. [7] E. Rashed, H. Nezamabad-pour and S. Saryazd, GSA: A Gravtatonal Search Algorthm, Informaton Scences vol. 179, pp. 48, June 009. [8] J. R. Zhang, J. Zhang, T. M. Lok and M. R. Lyu, A Hybrd Partcle Swarm Optmzaton Back-Propagaton Algorthm for Feed Forward Neural Network Tranng, Appled Mathematcs and Computaton vol. 185, pp , February 007. [9] S. Sarafraz, H. Nezamabad-pour and S. Saryazd, Dsrupton: A New Operator n Gravtatonal Search Algorthm, Scenta Iranca D vol. 18 (), pp , June 011. [10] P.D. Wasserman, Neural Computng, Van Nostrand Renhold, NewYork, 1989, pp Panchanand Jha graduated n Producton Engneerng n the year 007 from ITGGU, Blaspur Inda. He completed Masters n Mechancal Engneerng wth Specalzaton n Producton Engneerng n 009 from Natonal Insttute of Technology, Rourkela, Inda. After a short stnt as a Lecturer n Mechancal Engneerng at RCET, Bhla he joned Department of Industral Desgn, Natonal Insttute of Technology, Rourkela as a Research Fellow. Hs research nterests nclude Robotcs, Manufacturng Processes, soft computng technques and Development of Optmzaton tools. Bbhut B. Bswal graduated n Mechancal Engnnerng from UCE, Burla, Inda n Subsequently he completed hs M.Tech. and Ph.D. from Jadavpur Unversty, Kolkata. He was n faculty of Mechancal Engneerng at UCE Burla from 1986 tll 004 and then joned Natonal Insttute of Technology, Rourkela as Professor and currently he s the Professor and Head of Department of Industral Desgn. He has been actvely nvolved n varous research projects and publshed more than 90 papers at Natonal and Internatonal levels, the areas of research beng robotcs, automaton, mantenance engneerng and ndustral organzaton. He was a vstng Professor at MSTU, Moscow and a vstng scentst at GIST, South Korea. Om Prakash Sahu receved the B.E. degree n Electroncs and Telecommuncaton n 005 and M- Tech. n Instrumentaton and control system n 008 from the Unversty of CSVT, Bhla, Inda and joned as a faculty n the same nsttute n 006. Currently he s pursung PhD at Natonal nsttute of technology Rourkela, Inda. He has been publshed more than 1 techncal research papers at Natonal and Internatonal levels. Hs areas of research nterest nclude ndustral robotcs, control system and Instrumentaton, Computer ntegrated manufacturng and automaton.