Visual Servoing Based on Multirate Sampling Control
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1 (), () Visual ervoing Based on Multirate ampling Control tability Improvement by Intersample Observer and Periodi Disturbane Rejetion Hiroshi Fujimoto (Nagaoka University of Tehnology), Yoihi Hori (The University of Tokyo) Abstrat- In this paper, novel multirate sampling ontrollers are proposed for digital ontrol systems, in whih the sampling period is restrited to be longer than the ontrol period The proposed feedbak ontroller assures perfet disturbane rejetion (PDR) at M intersample points in the steady state Moreover, the intersample observer is developed in order to redue the phase delay aused by zero-order hold and to inrease the stability margin by estimation and ompensation of the intersample response Net, the novel sheme of repetitive ontrol is proposed based on the open-loop estimation and swithing funtion, whih enables the rejetion of periodial disturbane without any sarifie of the losed-loop harateristis Finally, the proposed ontrollers are applied to visual servo system by introduing the workspae ontroller and nonlinear perspetive transformation The advantages of these approahes are demonstrated by simulations and eperiments using a robot manipulator Key words: multirate sampling ontrol, disturbane rejetion, repetitive ontrol, visual servoing 1 Fig1 r(t) y(t) u(t) H T r,t y,t u T u D/A CPU T y A/D T u T y T y T u [1, 2 T y (T u <T y) D/A A/D T u <T y [3 [6 [7 [10 [11 [9 1 N M [12 [13 r(t) (T r ) r[i C[z y[i H (T u ) (T y ) u(t) d(t) Fig 1: Digital ontrol system P (s) 2 T u <T y M T y >T u T f = T y T f 1 M Fig2 T y N M = N/n N,M n Fig2 µ j(j =1,,N),ν k(k =1,,M) (1),(2) T f µ j = j/n,ν k = k/m y(t) 0=µ 0 <µ 1 <µ 2 <<µ N = 1 (1) 0 <ν 1 <ν 2 <<ν M = 1 (2) P (s) [14 [1 n(t)
2 [i u 1 [i (i ν 1 )T f (i ν M 1 )T f 1 [i M 1 [i M [i u 2 [i u N 1 [i u N [i H M d(t) u (t) - y (t) y[i P (s) [  ˆb Ĵ Ĉ ˆd O (i µ 1 )T f (i µ N 1 )T f it f (i µ 2 )T f (i 1)T f Fig 2: Multirate ampling ontrol 21 n ẋ(t) =A (t)b u(t), y(t) = (t) (3) Fig2 T y(= T f ) N [i =(it f ) [i 1 =A[i B, y[i =C[i (4) [ [ A B e = AT f b 1 b N (5) C O 0 0 b j = (1 µ(j 1) )T f (1 µ j )T f e Aτ b dτ, u =[u 1,,u N T (6) t =(i ν k)t f [i =Ã[i B (7) à 1 b11 b1n [ = à B (8) à M bm1 bmn à k = e A νk T f, =[ T 1,, T M T (9) k[i =[i ν k=((i ν k)t f ) (10) (νk µ µ j <ν k : (j 1) )T f e Aτ b (ν k µ j )T f dτ b kj = (νk µ µ (j 1) <ν k µ j : (j 1) )T f e Aτ b 0 dτ ν k µ (j 1) : 0 22 ẋ p(t) =A p p(t)b p(u(t) d(t)) (11) y(t) = p p(t) (12) d(t) ẋ d(t) =A d d(t), d(t) = d d(t) (13) A d =0, d =1 ω d [ 0 1 A d = ωd 2, 0 d =[1, 0 (14) F Fig 3: Multirate ontrol with disturbane observer (11), (13) ẋ(t) = A (t)b u(t) (15) y(t) = (t) (16) [ [ [ Ap b p d bp A =, b =, p = O A d 0 d =[p, 0 (15) t =(i ν k)t f [i ν k (7) k [i ν k = Ãk[i B k (17) [ [ Ãpk à pdk à k =, O à B Bpk k = dk O (15) (4) ˆv[i 1 = ˆv[iˆby[iĴ (18) ˆ[i = Ĉ ˆv[iˆdy[i (19) Fig3 =u p[iu d[i =F p ˆ p[if d ˆ d[i =F ˆ[i (20) F =[F p, F d e v e v = ˆv v ˆ[i =[iĉev[i (17) (20) [ p[i ν k d[i ν k = (21) e v[i 1 à pk B pkf p à pdk B pkf d BpkF Ĉ [ p[i O à dk O d[i O O  e v[i B pk [14, 15 k = 1,,M (1,2) à pdk B pkf d = O (22) F d [1, 9 k =1,,M à pd BpF d = O (23) [ = Ãpd Bp à pd1 à pdm ˆ Bp1 BpM (24)
3 F d 1 F d = 1 B p à pd (25) t =(i ν k)t f d[0 d[i p[i ν k p[i, e v[i ÃpM B pm F p  p[i ν k=0 (18) (20) [ ˆv[i 1 [  ĴFĈ = ˆb ĴFˆd F Ĉ F ˆd [ ˆv[i y[i (26) M = N/n d p 1 M [16 M N/n p 23 F p (4) (MIO) F p ÃpM B pm F p u p[i u p[i = u p1[i u pn [i = f y f y ˆ p[i (27) f y T y T u T y F p (19) 2 ˆ p[i =à p ˆ p[i Bp (28) à p, Bp (7) (18) (20) u p[i u p[i = u p1[i u pn [i = f u O O f u ˆ p[i ˆ p[i µ N 1 f u T u u p[i =F u ˆ p[i (28) ˆ p[i (29) 1 B p [14, 15 2 (28) ˆ ˆ d [i 1 = A dd ˆ d [i ˆd [i with F d ˆ d [i C 2[z H M Observer P (s) Fig 4: Feedforwad repetitive ontrol u p[i =(I F u Bp) 1 F u à p ˆ p[i (30) F p F p =(I F u Bp) 1 F u à p (31) y[i 32 [11 [7, 8 (31) F p [1, T 0( =2π/ω 0) d(t) =a 0 a k os kω 0t b k sin kω 0t (32) k=1 (13) [13 Fig4 d[i t = t 0 ON ˆ d[t 0 OFF ˆ d[t 0
4 Y o Y y amera q 2 l 1 l 2 q 1 objet Camera Coordinate ystem Camera Image plane ξ ξ y f z y Objet O X X o Fig 7: Perspetive model Fig 5: Two-link DD robot with amera ref ẍ ref K p J 1 ao(q) q ref Robot with DOB ξ ref Inner-loop H M ref ι 1 (θ) C[z P (s) (T u ) o e sk d J ao (q) q (T y ) e std I 2 ξ Camera ι(θ) Fig 6: Workspae Controller (Inner-loop) ˆ d[i 1=A dd ˆ d[i (33) A dd = e A dt f (25) F d 1 M C 2[z 3 Fig5 33 [ms 2 2 z PC(CPU: Celeron 400MHz) RT-Linu l 1 =359[m, l 2 =240[m [piel 31 Fig6 [18 J ao ẍ ref (= [X,Y T ) 2 ref u ( Fig 8: Visual servo system 1[ms ) (s) =P (s)u(s), P (s) = K p s 2 K ds K p I 2 (34) Fig5 K p, K d K p = 100,K d =20 Fig6 (, y) o Fig7 (, y) ο ο = f z [ y = f [ os θ sin θ z sin θ os θ [ Xo X Y o Y (35) f z Z θ = q 1 q 2 ο = ι(θ)( o )=ι(θ) e Fig8 ο ref o e (11) (35) ι 1 (θ) Fig8 y T y 100 [ms ref T u 25[ms N 4 T d 1 (T d = T y) Fig4 1 [17
5 Table 1: stability margin (f nyq =50[Hz) Closed-loop poles f l [Hz Input multipliity N Gain margin [db deg ross-over [Hz Phase margin [deg db ross-over [Hz (a) Feedbak (b) Feedforward Fig 10: Frequeny responses ([z and T [z) ξ (a) f l = 10 [Hz ξ (b) f l = 20 [Hz Fig 9: Multirate vs ingle-rate (eperiment) (13) 1 s 1 Table ep( 2πf lt y) Table1 f l 1[Hz (N =1) (N =4) f nyq = 5[Hz f l 2[Hz Fig9 f l 1[Hz f l 2[Hz T 0 =05[s (32) k =1, 3, 5 Fig3 (FB) Fig4 (FF) Fig10 (a) FB 2 FF C 2[z (b) Fig11 05[s (a) Feedforward vs Feedbak ξ (b) Multirate vs ingle-rate Fig 11: Position Error X o X (simulation) Fig 12: Feedforward vs Feedbak (eperiment) X o X (a) FB FF t 0 = 10[s (b) M(= 2) Fig12, 13 3 Fig12 FB Fig11(a) t = 8[s FF t 0 = 10[s Fig13 (a) (b) 3 Fig13 T y = 400[ms 100[ms T y = 100[ms
6 ξ (a) Error in ξ (b) Control input X ref () Camera position X Fig 13: Multirate vs ingle-rate (eperiment, T y = 400 [ms, N = 4) Fig 14: Error ratio E R(k) () Fig14 () E R(k) X o(t) E 2 R(k) = tskt 0 t s (X o(t) X (t)) 2 dt tskt 0 t s X 2 o (t)dt (36) t s Fig14 (k =10 ) 4 T u <T y [1 H Fujimoto: General Framework of Multirate ampling Control and Appliations to Motion Control ystems, PhD thesis, The University of Tokyo (2000) [2,,, 1, pp (2001) [3 K Hashimoto and H Kimura: Visual servoing with nonlinear observer, IEEE Int Conf Robotis and Automation, pp (1995) [4 M Nemani, T C Tsao and Huthinson: Multi-rate analysis and design of visual feedbak digital servo-ontrol system, AME, J Dynam yst, Measur, and Contr, 116, pp (1994) [5 J T Feeddma and O R Mithell: Vision guided servoing with feature-based trajetory generation, IEEE Trans Robotis and Automation, 5, 5, pp (1989) [6 H Fujimoto and Y Hori: Visual servoing based on multirate sampling ontrol, IEEE Int Conf Robotis and Automation, pp (2001) [7 W-W Chiang: Multirate state-spae digital ontroller for setor servo systems, Conf Deision Contr, pp (1990) [8 T Hara and M Tomizuka: Performane enhanement of multi-rate ontroller for hard disk drives, IEEE Trans Magnetis, 35, 2, pp (1999) [9 H Fujimoto, Y Hori, T Yamaguhi and Nakagawa: Proposal of perfet traking and perfet disturbane rejetion ontrol by multirate sampling and appliations to hard disk drive ontrol, Conf Deision Contr, pp (1999) [10,,,, D, 120, 10, pp (2000) [11 Y Hori, T Umeno, T Uhida and Y Konno: An instantaneous speed observer for high performane ontrol of d servomotor using DP and low preision shaft enoder, 4th European Conf on Power Eletronis, Vol 3, pp (1991) [12 Hara, Y Yamamoto, T Omata and M Nakano: Repetitive ontrol system a new-type servo system, IEEE Trans Automat Contr, 33, pp (1988) [13 C Kempf, W Messner, M Tomizuka and R Horowitz: Comparison of four disrete-time repetitive algorithms, IEEE Contr yst Mag, 13, 5, pp (1993) [14, N-delay, D, 117, 5, pp (1997) [15 M Araki and T Hagiwara: Pole assignment by multiratedata output feedbak, Int J Control, 44, 6, pp (1986) [16 H Fujimoto and Y Hori: Vibration suppression and optimal repetitive disturbane rejetion ontrol in semi-nyquist frequeny region using multirate sampling ontrol, Conf Deision Contr, pp (2000) [17 H Fujimoto and Y Hori: Visual servoing based on intersample disturbane rejetion by multirate sampling ontrol time delay ompensation and eperimental verifiation, Conf Deision Contr, pp (2001) [18,, D, 113-D, 5, pp (1993)
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