ERROR SPACE MOTION CONTROL METHODOLOGY FOR COMPLEX CONTOURS. Robert G. Landers
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1 ERROR SPACE MOION CONROL MEHODOLOGY FOR COMPLEX CONOURS Rb G. L Ai J f C. 7, N., pp. -8, Mh 5
2 Ai J f C,. 7, N., pp. -8, Mh 5 ERROR SPACE MOION CONROL MEHODOLOGY FOR COMPLEX CONOURS Rb G. L ABSRAC Mi i ii p f gi.g., f, bi. M h ipi h f i i ; hf, p f p i i f i g i. Hw, h iiii wh h i g ji gh, i p h i bi i wh ii ipi h p. A i h kw h p i hg i pp i hi pp hi i h b iiz f p. h p i hg i ppi w-i i ii i f i, i, ipi, i. h h k bii f h pp p i hg i ii f p. KW: Mi, p, p. I. INRODUCION Mi i f b f hi wh i i i qi p h i. Mi ii p f gi ppii.g., f, bi. A p, pii f h i ip bw h p i hi pi i qi hi hh ii. If h i i i j, h wih i wi z -. hi h gi i h f Ppi p Ig p Dii PID i i. Hw, PID ih g z - f i p, whih b p b ii. Mip i Api 6, 4; p J 6, 4. h h i wih Dp f Mhi Ap Egi, Uii f Mii-R, 87 Mi Ci, Mhi Egi Bi R, Mii , U.S.A. I ii ii PID, hiq kw Z Ph k E C ZPEC h b p [,]. I h ZPEC hiq, fbk ffw iiz hi g k z ph bw h f h p. hi hiq h b i-, i b iiz pi hiq [3]. A iw f hi i i pi i Ei Lz [4]. I h b hw, hw, h h i f i i f i-i i i h i.., ii i bw h pii h. hi h p f hg kw p [5-8]. I hi hg, ii gih i h hi h, b h, ff f h hi. C hiq qi ii, hb i h pi. h p i hg i p i h i. h hg i i i ppii f w-i i ; hw, i b i iiz f -
3 Rb G. L: E Sp Mi C Mhg f Cp C i wih h w. A ipi h h pi f i i g p i gi bq. Sii i h f i, i, ipi, i i h k bii f h pp i hg. h h h i hg p i hi pp i pb f ifi h wih h f p gih h h hg b i ppi p wh h g i hg. F C = A + B = wh p + B p A ξ 4 5 = [ ] p p = +ω ξ ẋ 6 II. ERROR SPACE MOION CONROL MEHODOLOGY I p p p F = ω I p p p 7 h p pi f h hi i i = + A B = C wh i -ii f hi, i -ii f hi ip, i -ii f hi, A i i, B i ip i, C i p i. A h i, h i ppi i i h p i hg. h i i f, bi wih h i i i g h, i h i g i. F i ji, h i f i ifii, h, h g i i z. F i ji, h i f, h, h g i, i ip i p. Hw, f p, h i i f hg g h. H, h g i hg ii i i g h. If h i i ppi b i, h f i pii if +ω = 3 U h p pph i Fki. [9] i MIMO, h i i p C = C = +ω 8 9 p i h b f, i = i i, i i h pii f h ih i, i -ii f, i p-ii f f, ξ i -ii f p, ij i i f z wih i w j, Ij i ii i wih j w j. h bji i i h f z, hi wi i h iii i z. h, h k pb h b i g pb. A hi g f h, i f i g hiq.g., p p, Li Qi Rg b f h f = G h phi hi f b +ω = G h p p hiq i iiz i hi pp. h gi i i fi f h hi p, i -p p i, h i g i. hf, h gi i wi b i-, i.., G = G f p i h g i i g h.
4 Ai J f C,. 7, N., Mh 5 III. INERPOLAION OF COMPLEX CONOURS Ip g i f pii h p pi. Ipi h f i i i i. p, h i f ppi b i f i g i. Hw, hi i iiii wh h i g ji bi i h i p h if ii i pfi iiz. Fh, i i g h, whih i ip f ppii, wi b ii. hi i i i i ipi f i i h i ip ipi h f p wh h i i i ppi g h. hi h i bq ppi ip i. h f pii f i f w-i i gi b = + θ = + θ 3 i hi ip wi hi i i f g h i. h f pii f i f w-i i gi b X R [ ] = + 4 = + 5 i[ ] Y R If h p g j i h h h b i f p. Ni h f =, h bii i, w b ppib f h -i qi ip b i. hf, h i ppi h i i h ihip = + [ ] [ ] 8 If h f pii gi i p i, h p g i gi b = + wh i gi b Eq. 8. hi ip hi i f g h. hi ipi h i w ppi ipi. h -i f pii, pi, X [ ] = + 9 = + Y b i[ ] h p g i gi b = + i b + N h Eq. Eq. 6 i h f i wh = b = R. h ipi h i w ppi i. h -i f pii, pi, X b = + + [ ] [ ] R = + 6 h i f g h i wi b hi. F p, h p g b i b ib ii i i g h. h ihip bw h i i p g ip i 7 = = [ ] + [ ] = Y b [ ] i [ ] h p g i gi b = + wh ] + [ ] [ = 4i[ ][ ] 4 bi[ ] 5 wh = =. h i Eq. 7 = [ ] i [ ] b[ ] 6
5 Rb G. L: E Sp Mi C Mhg f Cp C 3 I. WO-AXIS ERROR SPACE MOION CONROLLER h p hg i b i w i i ppii w-i i i f w i hg,. h -i qi f w-i hi, h i i p i h f h h hi i p, [] τ + = K 7 τ + = K 8 h p pi f h w-i hi i gi b Eq. wh = 9 [ ] = +ω +ω ξ [ +ω +ω ] 37 I F = ω I C 4 = C = +ω 4 = 4 = 4 = 3 [ ] = 3 [ ] h phi f b +ω = 43 G A = τ τ K τ B = K τ C = h i SIMULAION CASE SUDIES Sii i w f i f i g, i, ip, i. h hi p f b-g hi []: τ =.55, τ =.56, K = 3.68, K = A Rg-K fh i i wih i i f. i iiz h hi i qi. h i -p p i, b i, f h ii, 5,, 83.33, ± j, 5 ± j, h f i i i = 5. F i i i, h i pf i, wi b z h bw. h i h ii i f h pii h. 5. Di = + 35 wh F C 4 44 A ξ B = 36 h p i i w ppi f i g h f i. I hi, h g i i z i h i f i ifii h gi i i. h f -i pii gi, pi, b Eq. 3 wh θ = 45. h hw i F..
6 4 Ai J f C,. 7, N., Mh 5 i f i p - - pii. i i i -. i i i i - 5 i 5 µ i f i p - - pii -.5 i -. i i -.5 i i i i i 5 5 µ F.. Di ii. F. 3. Ci ii µ µ i i F.. Di -. F. 4. Ci -. Si h iffi qi h fi h f, Eq. 3, i f i, h - i z. h i, hw, z h i i iiii h pi. 5. Ci h p i i w ppi i. Si h i f i f i, h g i, whih, i hi, i qi h p g i, i. hf, h gi i i. h f -i pii gi, pi, b Eq. 4 5 wh X =, Y =, R = 5, h p g i gi b Eq. 6. h hw i F h i - i 48, whih i bw h i f i. Ahgh h iffi qi h fi h f, Eq. 3, i f i, h - i w -z i i ppii. 5.3 Eipi h p i i w ppi ipi. h f -i pii gi, pi, b Eq. 9 wh X =, Y =, =, b =. h p g i gi b Eq.. h ipi i p i h h i f hg g h. h, h g i wi b ω = ρ wh h i f i +.5 i [ ] b [ ] ρ = b 44 Si h g i hg, h gi i wi wih i. h ii hw i F. 5 ~ 7. h i i.7µ. I ii, h -
7 Rb G. L: E Sp Mi C Mhg f Cp C 5 i i f p - - pii -.5 i -. i i -.5 i -.5 µ i i i i F. 5. Eip ii wih ib i F. 6. Eip - wih ib. i 3 ρ f i p i F. 7. Eip ipi wih ib µ i g p pi k f h f i i g h wih i - f 4.6µ. hf, h p i hg i b pi k f p wh h g i hg whi ii i i g h. A pi, ii i wh p g i f.97 i i h ipi ip. hi p g i w h h h h i p h wi b h f bh ii. Si h p g i i, h gi i wih i. h ii h p g i ip hw i F. 8 ~. h i - i 43, whih i h f h h iffi qi h fi h f i, Eq. 3, i wh p g i i i h ipi ip. Agi, h - i -z i i. h i i, hw, i i f p - - pii -.5 i i i -. i i i i i F. 8. Eip ii wih = µ i µ F. 9. Eip - wih =.97.
8 6 Ai J f C,. 7, N., Mh 5 5 i 5 f 3 ρ i p i i g i i f p pii -.5 i -. i i -.5 i i i i i µ F.. Eip ipi wih =.97. F.. Li ii wih ib. i ifi, whih wi p pb f ppii wh i i g h i qi. 5.4 Li h p i i w ppi ii. Sii ip, i i p i h h i f hg g h. h, h g i wi gi ii i i g h. I ii, h i h w pi f h wh h g i g z i h i f g ifii. h -i f pii gi b Eq. 3, pi, wh = 5 b = 5, h p g i gi b Eq. 4. h g i wi b ω = ρ wh h i f i ρ = +.5 [ ] [ ] 45 gi b Eq. 5 6, pi, = = 4 i [ ] [ ] b[ ] = = 8i[ ][ ] bi[ ] Si h g i hg, h gi i wi wih i. h ii hw i F. ~ 3. h w pi f h i b h w g f h i f i F. 3. h i - i 5.6µ. I ii, h µ i F.. Li - wih ib. ρ i f i p i F. 3. Li ipi wih ib. i g -p pi k f h f i i g h wih i - f 4.8µ. Agi, h p i hg i b pi
9 Rb G. L: E Sp Mi C Mhg f Cp C 7 i k f p wh h g i hg pi f h, whi ii i i g h. A pi, ii i wh p g i f.897 i i h i ip. hi p g i w h h h h i p h w b h f bh ii. Si h p g i i, h gi i wih i. h ii h p g i ip hw i F h i - i 4.87µ. A, h i i i ifi, whih i wbk f ppii wh i i g h i qi. i f i p pii -. i i -. i i i i i i F. 4. Li ii wih = µ -6 µ ρ 3 i f i p i i g F. 6. Li ipi wih =.897. I. SUMMARY AND CONCLUSIONS h p i hg w p i hi pp. h hg w ppi w i i ii i f i, i, ipi, i w. h ii h k pbiii, i f i i, f h p i, f p wh h g i hg g h. A, h ii h h pi f p w qi bi, f p. I h b h h hi pii i h ii i w hh p h hi ii pbiii, h, b w h b iz wih ii-g i. Whi h hi iiz i hi pp pi f i ppii, hh hi i b iiz i h p i hg. h pizi iiz i hi pp w i ; hw, h pizi b iiz wih ii hg. A, whi h p i hi pp w f w-i i, h p i hg p i hi pp i g gh b ppi i wih b f i F. 5. Li - wih =.897. NOMENCLAURE, b p [] -i -i pii [],
10 8 Ai J f C,. 7, N., Mh 5 ė, G K, K R,, X, Y ė -i -i i [] gi i -i -i i gi [] i i [] -i -i f pii [] giz i pii [] i [] -i -i g [] f i i [] -i -i pii [] giz i f pii [], -i -i pii [] ẋ, ẋ -i -i ii [] ẋ, ẋ -i -i i [ ] 8. K, Y. C-C. L, A C f F Di, A CIRP,. 4, N., pp Fki, G.F., J.D. Pw, A. Ei-Nii, 3 E., Fbk C f Di S, Ai W, R, Mh, U.S.A Sii, K. -C., Mhi F Di hi C A S f h S f h A, ASME J. Mf. Si. Eg.,. 9, N. 4B, pp L, R.G., Spi Mhi C: A D Apph P F C Ch Ai Cp, Ph.D. Dii, Dp f Mhi Egi Appi Mhi, Uii f Mih, A Ab, Mih, U.S.A ẋ, ρ τ, τ θ ω ẋ -i -i jk [ 3 ] p pi [] i f [] -i -i i [] g f pii -i i [] p g [] g i [] REFERENCES. izk, M., Z Ph E k Agih f Di C, ASME J. D. S. M. C.,. 9, N., pp izk, M., M.S. Ch, S. R, -C., Pii f Ni C wih Lh, ASME J. D. S. M. C.,. 9, N., pp , -C. M. izk, Api Z Ph E k Agih f Di C, ASME J. D. S. M. C.,. 9, N. 4, pp Ei, G. R.D. Lz, Cpi f Mi C Lp f Ii Appii, P. IEEE I. App. Cf.,. 4, pp K, Y., C-Cp Bii Cp C f Mf S, ASME J. D. S. M. C.,., N. 4, pp Sii, K. P.K. Kki, C-Cp C f Bii F Di Shi, ASME J. D. S. M. C.,., N., pp K, Y. C-C. L, ib Gi C Cp C f C, A CIRP,. 4, pp Rb G. L Ai Pf f Mhi Egi h Uii f Mii R. Pi, h w i h ii i h Egi Rh C f Mhi S h Uii f Mih. D. L i hi B.S. g f h Uii f Okh i 99, M.E. g f Cgi M Uii i 99, Ph.D. g f h Uii f Mih i 997, i Mhi Egi. Hi h h i i h f, i, i, f f p.g.,, fii i, pii, pi fz pp, f ibgi,, i ppii. D. L h f hi pbii h i f f h Ni Si Fi, US Dp f Ei, Si f Mf Egi, Mii Rh B, i pi. H i h Si f Mf Egi M. Eg Mh O Yg Mf Egi Aw i 4 i b f ASEE, ASME, IEEE, SME.
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