Identification of Robot Arm s Joints Time-Varying Stiffness Under Loads

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1 TELKOMNIKA, Vol.10, No.8, December 2012, pp. 2081~2087 e-issn: X ccredted by DGHE (DIKTI), Decree No: 51/Dkt/Kep/ Identfcton of Robot Arm s Jonts Tme-Vryng Stffness Under Lods Ru Xu 1, Qng Chen* 2, Guol Yng 1 1 School of Mechncl Engneerng, Nnjng Unversty of Scence nd Technology 2 School of Mngement nd Engneerng, Nnjng Unversty No. 22, Hnkou Rod, Nnjng, Chn, , +86(25) *correspondng uthor, e-ml: chenq0772@vp.sn.com Abstrct Ths pper s concerned wth the dentfcton problem of two degree of freedom robot rm s jonts tme-vryng stffness under tme-vryng torque lods. The ndustrl robot rms cn be smplfed nto seres hnged lnkge mechnsm. The unrestrnt dynmc equton of two degrees of freedom robot rm cn be obtned by usng nlytcl mechncs method. The model whch ws estblshed n Pro/E hd been mported nto ADAMS for smulton. And the smulton conssted of free vbrton nd forced vbrton. Then by choosng lmted memory lest squre method nd usng the post-processng mesured vrbles, tme-vryng stffness could hve been dentfed. Fnlly, the clcultve stffness ws compred to the rel stffness whch ws smulted n ADAMS. The whole process shows tht the robot rm s dynmc model nd the method of dentfcton re both effectve. Keywords: tme-vryng stffness, robot rm, dentfcton, lmted memory lest squre method Copyrght 2012 Unversts Ahmd Dhln. All rghts reserved. 1. Introducton As becomng new knd of producton pttern, robot rm hs nfluenced the ndustrl producton lot. Most of the robot rms consst of two prts: motor nd reducer. For ts complex structure, the robot rm s jonts stffness s tme-vryng. Therefore, t s dffcult but necessry to dentfy some prmeters of the robot rm s dynmc model whch s the key problem to estblsh ths model. Becuse of lest squre method s smple prncple nd fst convergence, t s wdely used n dentfyng system prmeters. As one of lest squre method s mproved lgorthms, lmted memory lest squre method cn effectvely overcome dt sturton phenomenon when t s ppled to tme-vryng dentfcton. Xnzh C nd Chong C proved ts effect n ther pper [1]. Xopng L, Dh J nd We Z provde mny dentfyng methods of tme-vryng prmeters n ther ppers [2-4]. But, ther robot rm s dynmc equtons lck physcl sgnfcnce nd t s hrd to exmne the result of dentfcton. In ths pper, tkng two degree of freedom robot rm s the reserch object, we cn dentfy the tme-vryng stffness effectvely when combnng robot rm dynmc model wth lmted memory lest squre method. Becuse the robot rm s stffness hs physcl sgnfcnce, the result of dentfcton cn be compred to the sttc mesurement n order to exmne the method s effect on tme-vryng prmeters. 2. Problem Formulton 2.1. Dynmc equtons Anlytcl mechncs use nlytcl mth method to solve mechncl problems. By nlyzng from energy nd work, the reltonshp between moton nd ntercton cn be found. After mkng use of d lembert s prncple nd vrtul dsplcement prncple, the generl dynmcs equtons re estblshed, nd we substtute Lgrnge multpler n order to get unrestrnt Lgrnge equtons [5-6]. Unrestrnt Lgrnge equtons re shown n Eq. 1: d T T V dt q q q ( 1,2,, ) Q N + = = & (1) Receved September 29, 2012; Revsed November 5, 2012; Accepted November 13, 2012

2 2082 e-issn: X Where T s the knetc energy of system, V s the potentl energy of system nd conssts of grvttonl potentl energy nd elstc potentl energy, Q s the work cused by resultnt of non-conservtve force cused by electrc motors, q s generlzed dsplcement nd N s system s degree of freedom. The structure of robot rm s jonts s smplfed nto hnges wth nonlner torson sprngs. So tht, the potentl energy of system whch s cused by these torson sprngs nd the power of grvty cn be expressed n Eq. 2: ( 1, 2) V = K + P = (2) Where K s the elstc potentl energy of the system, P s the grvttonl potentl energy of system. Then we substtute Eq. 2 nto Eq. 1, so robot rm s dynmc equton cn be obtned n Eq. 3: ( K P) d T T + dt q q q ( 1,2 ) + = Q = & The structure of two degree of freedom robot rm s llustrted n Fgure 1: Where φ 1 s the bg rm s generlzed dsplcement, φ 2 s the smll rm s generlzed dsplcement, l 1 s the bg rm s length, l 2 s the smll rm s length, l 1C s the dstnce between the center of mss nd the bg rm s jont, l 2C s the dstnce between the center of mss nd the smll rm s jont, J 1C s bg rm s moment of nert n center of mss, J 2C s the smll rm s moment of nert n center of mss, m 1 s the bg rm s mss, m 2 s the smll rm s mss. Set the ground s the zero potentl energy locton. After beng substtuted ll these prmeters nto Eq. 3, two degree of freedom robot rm s dynmc equtons cn be expressed n Eq. 4 nd Eq. 5: (3) ( J + m l + m l )&& ϕ + m l l && ϕ cos( ϕ ϕ ) m l l & ϕ sn( ϕ ϕ ) C 1 1C C C k ϕ k ϕ ϕ + m gl cosϕ + m gl cosϕ = Q C ( J + m l )&& ϕ + m l l && ϕ cos( ϕ ϕ ) + m l l & ϕ sn( ϕ ϕ ) 2 2 2C 2 2C C C k ϕ ϕ + m gl cosϕ = Q C 2 2 Where k 1 s the bg rm s jont s stffness, k 2 s the smll rm s jont s stffness. (4) (5) Fgure 1. Structure of two degree of freedom robot rm Fgure 2. Mult-rgd-body system s dynmc model 2.2. Dynmc smulton The robot rms cn be smplfed nto seres hnged lnkge mechnsm. Then the 3D model cn be estblshed n Creo Prmetrc. By usng the functon of mss nlyss n Creo Prmetrc, both the mss, moment of nert nd center of mss cn be obtned n Eq. 6: TELKOMNIKA Vol. 10, No. 8, December 2012 :

3 TELKOMNIKA e-issn: X m1 = kg, J1 C = kg/m, l1 C = m 9 2 m2 = kg, J 2C = kg/m, l2c = 0.15 m (6) Then we mport ths model nto ADAMS for smulton. The mult-rgd-body system s dynmc model s bult nd shown n Fgure 2: In Fgure 2, two degree of freedom robot rm conssts of foundton support, bg rm, bg xs, smll rm nd smll xs. All these subssembles re supposed to be mde from 6061 lumnum. The foundton support s fxed on the ground. The bg rm s hnged on the foundton support by the bg xs, nd the bg torson sprng s loded. The smll rm s hnged on the bg rm by the smll xs, nd the smll torson sprng s loded. The bg motor s fxed on bg rm, nd the smll motor s fxed on smll rm. These motors mss wll be equvlent to the robot rms. In order to dentfy the tme-vryng stffness of torson sprngs, the prmeters of torque sprngs re set up s splnes tht descrbe the reltonshp between torson nd deflecton, nd the prmeters of electrc motors re set up s functons tht descrbe the reltonshp between torson nd tme. These reltonshps cn be expressed n Eq. 7: T torque T = f ( t ) motor = g defo Where T torque s the torque cused by torson sprngs, defo s the jonts deflectons, T motor s the torque cused by electrc motors, t s the tme. The splnes tht descrbe the tme-vryng stffness of torson sprngs re llustrted n Fgure 3 nd Fgure 4. The functons splnes tht descrbe the tme-vryng torque cused by electrc motors re llustrted n Fgure 5 nd Fgure 6. (7) Fgure 3. Splne descrbng the bg torson sprng Fgure 4. Splne descrbng the smll torson sprng Fgure 5. Splne descrbng the bg torque cused by motor Fgure 6. Splne descrbng the smll torque cused by motor Identfcton of Robot Arm s Jonts Tme-Vryng Stffness Under Lods (Ru Xu)

4 2084 e-issn: X The dynmc smulton conssts of two dfferent prts: free vbrton nd forced vbrton. Free vbrton: we ssume tht there s no grvty nd no generlzed force. Intl compresson of two torson sprngs s set up s 0.5 degree (vector) from the blnce poston. So tht the robot rm system vbrtes freely. The smulton tme s set s second nd the whole process conssts of 8000 steps. By ADAMS post-processng progrm, the dt of dsplcement, velocty, ccelerton nd vbrton cn be obtned. Then record ll these smulton dt for dentfcton clculton. Forced vbrton: the drecton vector of grvty s set perpendculr to the ground. Intl compresson of two torson sprngs s set up s zero from the blnce poston. The smulton tme s set s second nd the whole process conssts of 8000 steps, too. Smlrly, the dt of dsplcement, velocty, ccelerton, vbrton nd cn be obtned, nd the tme-vryng torque s well. Then record ll these smulton dt for dentfcton clculton Identfcton method When beng dentfed the system prmeters, f the object cn be descrbed s the controlled uto regressve model, the model cn be expressed n Eq. 8: 1 1 ξ A z y k = B z y k d + k (8) Where ξ(k) s whte nose, nd the structure prmeters m, n, d re known. And: A z = 1+ z + z + + z B z = b + z + z + + z m 1 2 m n n The trget of dentfcton s to estblsh m+n+1prmeters ccordng to the mesurble nput nd output. The model cn be descrbed n lest squre method nd s expressed n Eq. 10: = 1 ( 1) n ( ) n b b + ξ T = ϕ ( k ) θ + ξ ( k ) y k y k y k n b u k d b u k d n k Where φ(k) s the dt vector, θ s the prmeter need to be dentfed. And: ( 1 ),,,,, ϕ k = y k y k n u k d u k d nb R T ( n + nb + 1) 1 θ 1,, n, b0,, b = n R b T ( n + n + 1) 1 So the model of two degree of freedom robot rm s jonts cn be expressed n Eq. 12 nd Eq. 13: b (9) (10) (11) k1 2 2 ϕ1 ϕ2 + ϕ1 = ( J1 C + m1l 1C + m2l1 ) ϕ1 m2l1l 2Cϕ2 cos( ϕ2 ϕ1) k && && 2 [ ] t 2 & 2 1 2Cϕ2 sn( ϕ2 ϕ1) 1 1C cosϕ1 2 1 cos ϕ1 ( motor1 ϕ1) ϕ1 0 + m l l m gl m gl + d T d d k1 2 0 ϕ2 ϕ1 = ( J 2C + m2l2c ) ϕ2 m2l1l 2Cϕ1 cos( ϕ2 ϕ1) k && && 2 [ ] t 2 & 2 1 2Cϕ1 sn( ϕ2 ϕ1) 2 2C cos ϕ2 ( motor 2 ϕ2 ) ϕ2 0 m l l m gl + d T d d (12) (13) In ths pper, the lmted memory lest squre method s used to dentfy the stffness. The bsc de of ths method s usng the ltest perod of dt to dentfy the stffness. Once ddng set of dt, nother set should be declned so s to keep the number of dt constnt. Ths method reflects the tme-vryng chrcterstc of prmeters, nd cn solve dt sturton problem. TELKOMNIKA Vol. 10, No. 8, December 2012 :

5 TELKOMNIKA e-issn: X Numercl clculton nd results The source of dt dpted n the dentfcton s from the robot rm s stffness whch s cqured by numercl methods. A reltve percentge error (RPE) s defned to compre the result of the dentfcton whch s expressed n Eq. 14: Σ ktrue kdentfy RPE = 100% Σ k true (14) 3.1. Free vbrton The results of dentfcton bout rms stffness wthout lods re llustrted n Fgure 7 nd Fgure 8. The results of dentfcton bout reltve error of rms stffness wthout lods re llustrted n Fgure 9 nd Fgure 10. Fgure 7. Identfcton of bg rm s stffness wthout lods Fgure 8. Identfcton of smll rm s stffness wthout lods Fgure 9. Reltve error of bg rm s stffness wthout lods Fgure 10. Reltve error of smll rm s stffness wthout lods So, the RPE of bg rm wthout lods s bout 1.093%, whle the RPE of smll rm wthout lods s bout 1.114% Forced vbrton The results of dentfcton bout rms stffness under lods re llustrted n Fgure 11 nd Fgure 12. The results of dentfcton bout reltve error of rms stffness under lods re llustrted n Fgure 13 nd Fgure 14. Identfcton of Robot Arm s Jonts Tme-Vryng Stffness Under Lods (Ru Xu)

6 2086 e-issn: X Fgure 11. Identfcton of bg rm s stffness under lods Fgure 12. Identfcton of smll rm s stffness under lods Fgure 13. Reltve error of bg rm s stffness under lods Fgure 14. Reltve error of smll rm s stffness under lods So, the RPE of bg rm under lods s bout 1.021%, whle the RPE of smll rm under lods s bout 0.995%. 4. Conclusons In the pper, the unrestrnt Lgrnge equtons nd lmted memory lest squre method s ppled to dentfy the two degree of freedom robot rm s jonts stffness. The robot rm s stffness hs physcl sgnfcnce so tht the result of dentfcton cn be compred to the sttc mesurement whch s obtned by ADAMS. Lmted memory lest squre method cn overcome dt sturton phenomenon effectvely. So t cn be ppled to tme-vryng dentfcton n ths stuton. The result reflects tht lmted memory lest squre method nd the process of estblshng robot rm s dynmc model re effectve n both free vbrton nd forced vbrton. Future work of ths study would be dentfyng rel prmeters nsted of smulton nd exmnng ths reserch method s performnce n other models. Acknowledgements Ths work ws supported n prt by the Mjor Stte Bsc Reserch Development Progrm of Republc of Chn (No ) nd the Fundmentl Reserch Funds for the Centrl Unverstes. TELKOMNIKA Vol. 10, No. 8, December 2012 :

7 TELKOMNIKA e-issn: X 2087 References [1] Xnzh C, Chong C. Prmeter dentfcton of tme-vryng systems bsed on the lest-squres modfed lgorthm. Journl of Fuzhou Unversty (Nturl Scence). 2005; 33(2): [2] Dh J, Xopng L. On-lne Fuzzy Identfyng Physcl Prmeters of Jont Movng Stte Robots. Journl of Bejng Unversty of Posts nd Telecommunctons. 2008; 31(1): [3] Dh J, Xopng L. On-lne Identfcton of Tme-vryng Physcl Prmeters of Robot Jont Bsed on Hrmonc Propgton. Journl of Mechncl Engneerng. 2009; 45(3): [4] We Z, Xopng L. Jont prmeters dentfcton of robot bsed on trvelng wve pproch nd neurl network. Journl of Vbrton Engneerng. 2010; 23(2): [5] Theoretcl Mechncs Techng nd Reserch Secton of Hrbn Insttute of Technology. Theoretcl Mechncs. Sxth Edton. Bejng: Hgher Educton Press [6] Ymng Z. Mechncl Vbrton. Frst Edton. Bejng: Tsnghu Unversty Press Identfcton of Robot Arm s Jonts Tme-Vryng Stffness Under Lods (Ru Xu)