Fequency Sepaation Actuation Reonance Cancellation fo Vibation Suppeion Contol of Two-Inetia Sytem Uing Double Moto enji Inukai, and Hiohi Fujimoto The Univeity of Tokyo ahiwa, Chiba, Japan Email: inukai@hflab.k.u-tokyo.ac.jp, fujimoto@k.u-tokyo.ac.jp Abtact Uually, a humanoid obot uing moto ha low eonance due to the timing belt at each joint, and the low eonance may deteioate the epone pefomance and poitioning peciion. In thi pape, a joint of humanoid obot which link the moto ide to the load ide via timing belt i tudied. Regading thi ytem a a two-inetia ytem, thi pape popoe a novel contol method which ue multiple moto in both the moto ide and the load ide fo multi-input-ingle-output MISO) ytem. The popoed method achieve vibation uppeion and phae tabilization imultaneouly. The implementation i elatively imple ince thee ae only two paamete equied in the deign. Simulation and expeimental eult alo demontate the effectivene of the popoed contol ytem. I. INTRODUCTION Humanoid obot have been the focu of attention becaue of the low bithate and longevity. Recently, impovement of kinematical pefomance i emakable though the development of technology. Howeve, bandwidth of feedback contol can not be impoved becaue low igidity of the eduction gea of humanoid obot geneate eonance. Theefoe, it i neceay to model a plant by a two-inetia ytem, and intoduce a eonance uppeion contol. In the motion contol field fo two-inetia ytem, many eonance uppeion contol method which aume that poition eno i intalled only on moto ide have been popoed [] [6]. Howeve, thee method cannot uppe load ide poition eo due to emi-cloed ytem. Recently, high peciion poitioning device uch a machine tool which can meaue not only the poition of the moto ide but alo the poition of the load ide i inceaing. In full-cloed loop contol ytem, the cacade contol i often ued. Self Reonance Cancellation Contol SRC) i popoed a a contol method uing multiple eno [7]. Thi method can uppe the eonance peak and widen the bandwidth. Howeve, SRC i not obut againt modeling eo. Theefoe, it cannot apply to humanoid obot becaue moment of inetia, centifugal foce, coioli foce, gavity, and fictional foce depend on potue. The autho eeach goup popoed Self Reonance Ditubance Obeve SRCDOB) which egad all modeling eo a ditubance, nominalize a plant, and impove obutne againt modeling eo. The obut eonance uppeion contol fo two-inetia ytem wa achieved by combining SRC and SRCDOB [8]. Futhemoe, Fequency Sepaation Tao Takahahi Advanced Technology Engineeing Depatment, Patne Robot Diviion, Toyota Moto Copoation Tokyo, Japan Email: tao takahahi@mail.toyota.co.jp Self Reonance Cancellation FS-SRC) wa popoed fo XYganty-tage [9]. It achieve Vibation uppeion and phae tabilization. Conventional method, uch a SRCDOB and FS-SRC ae obut eonance uppeion contol method in ingleinput-multi-output SIMO) ytem uing an actuato and multiple eno. A well, a obut eonance uppeion contol would be achieved in a multi-input ytem. When the obut eonance uppeion in multi-input ytem i applied in a humanoid obot, miniatuization of joint by ditibuted placement of moto i poible. In thi pape, in ode to achieve a obut eonance uppeion contol fo humanoid obot, a novel feedback ytem i popoed in a multi-input-ingle-output MISO) ytem uing double moto and ingle angle eno. The popoed feedback ytem ha two featue. Fit one i that the effect of the vibation uppeion. The othe i that the phae tabilization of the eonance mode can be tuned by only two paamete. Thi popoed method i a baic tudy of multi-input ytem. The compoition of thi pape i hown below. Fitly, the compoition of the 3-joint leg obot ceated fo baic examination of contol deign in a obot leg, and model of twoinetia ytem ae explained. Secondly, the theoy of popoed ytem fo two-inetia ytem uing double moto and ingle angle eno i tated. Finally, imulation and expeiment ae pefomed to how the advantage of the popoed ytem. II. 3-JOINEG ROBOT AND EXPERIMENT EQUIPMENT TO SIMULATE A JOINT A. Baic Compoition of Joint of Humanoid Robot The autho fabicate 3-joint leg obot hown in Fig. in ode to do ome baic examination of contol deign in a obot leg. The 3-joint leg obot ha hip, knee, and ankle joint. The joint of load ide i connected to a hamonic gea though a timing belt fom a moto hown in Fig. 2. Modeling in twoinetia ytem fo each joint i neceay unde the influence of the elaticity of each timing belt. Validity of modeling the joint in a two-inetia ytem ha been hown in peviou eeach [8].
θ L load ide dive moto : Fig.. 3-joint Leg Robot Fig. 2. Compoition of Tanmiion of Joint Fig. 4. T M θ M Model of two-inetia ytem gea atio B L θ L T M B M θ M Fig. 3. Expeiment equipment to imulate joint of humanoid obot Fig. 5. Block diagam of Two-inetia ytem B. Expeiment equipment to imulate a joint of humanoid obot The expeiment equipment to imulate a humanoid obot joint i hown in Fig. 3. In thi equipment, the moto ide and the load ide ae connected though a timing belt, and the dive moto of load ide i mounted to apply the popoed contol ytem, and the load moto i mounted to imulate the potue change of the humanoid obot. The dive moto of load ide and the load moto i connected by igid coupling. Thi expeiment equipment i modeled a a two-inetia ytem hown in Fig. 4. The moto moment of inetia, the moto damping coefficient, the load moment of inetia, the load damping coefficient, and the ping contant ae denoted a, B M,, B L, and epectively. The gea atio i denoted a. The moto ide angle i denoted a θ M, the load ide angle i denoted a θ L, the moto toque i denoted a T M, and the load ide toque i denoted a. The block diagam of two-inetia ytem i hown in Fig. 5. Fo a two-inetia ytem, tanfe function fom moto toque T M to moto angle θ M and load angle θ L, and fom load ide toque to moto angle θ M and load angle θ L epeent by P MM ) θ M 2 B L, ) T M D P LM ) θ L T M D, 2) P ML ) θ M P LL ) θ L D, 3) 2 B M, 4) D whee D 4 B L B M ) 3 B M B L ) 2 B M B L ). 5) Fom the denominato polynomial of )-4) and the numeato polynomial of ), the eonant angula fequency and the antieonant angula fequency given by ω p ω z ), 6). 7) The bode diagam of the fequency epone chaacteitic fom moto toque efeence T M to moto angula velocity θ M and two-inetia ytem fitting model ae hown in Fig. 6. The eonance at 55 Hz come fom the timing belt. Nominal value ae decided by thi fequency epone chaacteitic ae hown in Table. I. Thi model i not exact in the low fequency egion due to nonlinea fiction. In thi pape, thi expeiment equipment i examined. III. FREQUENCY SEPARATION ACTUATION RESONANCE CANCELLATION CONTROL A. Actuation Reonance Cancellation Contol Thi ection decibe about the popoed method called Actuation Reonance Cancellation Contol ARC). Conven-
Magnitude [db] Phae [deg] Fig. 6. 6 4 2 2 2 3 8 9 9 Nominal Model Meauement 8 2 3 Fequency [Hz] Fequency chaacteitic of Expeiment equipment TABLE I. PARAMETER OF EXPERIMENT EQUIPMENT kg m 2 8. 6 B M N m/ad/) 5. 4 kg m 2 2.2 5 B L N m/ad/).8 3 N m/ad).3. tional eonance cancellation contol uch a SRC cancel the eonance by uing encode attached at both the moto ide and the load ide and uing ingle moto. The popoed ytem cancel the eonance of the plant by uing moto attached at both the moto ide and the load ide and uing ingle encode attached at load ide. The block diagam of ARC i hown in Fig. 7. In thi ytem, tanfe function fom input toque to load angle θ L, fom input toque to moto angle θ M ae defined a P L ) θ L P M ) θ M α β ) α β P LM ) ) P MM ) γ δ ) P LL ), 8) γ δ ) P ML ).9) The contolle gain α, β, γ, and δ ae detemined to cancel the eonance of the plant. Although thee ae an infinite numbe of olution, thee paamete ae detemined a follow: α ARC β ARC γ ARC δ ARC J, ) L B M J, ) L B L ), 2) ). 3) The paamete α ARC mean inetia atio. Then, in the cae of the above paamete, the tanfe function P M ) and P L ) Fig. 7. θ L Fig. 8. γ δ α β T M Block diagam of the ARC C) T in F) F) B L B M γ δ α β Block diagam of the FS-ARC ytem ae epeented by P L,ARC ) θ L P M,ARC ) θ M T M P LL P LM θ L θ M θ L, ) 2 4) ). 5) A hown in 4) and 5), the eonance of the plant i canceled. Thi popoed method i the ame a diect diving the load ide. Thi popoed ytem i applicable in the cae of the dive moto of load ide which i enough to diect dive. Thi contol ytem i obut againt ping contant becaue contolle gain do not include. B. Fequency Sepaation Actuation Reonance Cancellation Contol In thi ection, the Fequency Sepaation Actuation Reonance Cancellation Contol FS-ARC) i popoed. Thi popoed method i applicable in the cae of dive moto of load ide which i not enough to diect dive. The block diagam of FS-ARC i hown in Fig. 8. The tanfe function fom T in to the load angle θ L i defined a P FSARC ) θ L F)P T LM ){F)}P L ), 6) in whee F) i the fit ode low pa filte LPF) a follow: F) 2π f LPF 2π f LPF. 7) ARC method cancel the eonance by making the unobevable of the eonant mode fom efeence angle to load
Magnitude [db] P LM ) P L,ARC ) P FSARC ) 2 3 9 Phae [deg] 8 27 36 2 3 Fequency [Hz] Fig. 9. Fequency chaacteitic of Fequency Sepaation Actuation Reonance Cancellation Contol Fig.. Nyquit diagam of the popoed ytem when α i vaiable in the ange fom to and f LPF Hz angle. Theefoe, unexpected vibation occu by ditubance. On the othe hand, FS-ARC doe not cancel the eonance mode completely. Howeve, FS-ARC method can uppe vibation due to ditubance. FS-ARC method can adjut the uppeion pefomance of efeence and ditubance. In FS- SRC method, α of P L ) i vaiable in the ange fom to. β, γ and δ i expeed uing α a follow: β B M α, 8) γ α), 9) δ B L α). 2) The FS-ARC method ue dive moto toque to contol low fequency egion in which the eonance doe not affect, and ue dive moto and dive moto of load ide in high fequency egion whee the eonance exit. FS-ARC method include ARC method. FS-ARC i equivalent to ARC in the cae of α α ARC and f LPF Hz. The bode diagam of the P LM S), P L,ARC S) and P FSARC S) hown in Fig. 9. The olid line i fequency chaacteitic of P LM S), the boken line i fequency chaacteitic of P L,ARC S), and the dot-dahed line i fequency chaacteitic of P FSARC S) at α α ARC. The cut-off fequency of the LPF of FS-ARC i 5 Hz, becaue the eonant fequency of the plant i 55 Hz. The eonance mode of plant i canceled by applying eonance cancellation. The eonance mode of plant i almot canceled by applying FS-ARC. In addition, the band width of feedback ytem can be impoved becaue the phae delay of P FSARC ) at high fequency egion i -8. Next, the tability of the popoed method i analyzed. The feedback contolle i deigned fo P L,ARC ) which i the igid mode of load ide. In thi pape, the feedback contolle i deigned a a PID contolle. The paamete of the contolle i detemined baed on the pole placement deign to pole place at 2 Hz. The paamete ae detemined imply becaue the feedback contolle of thi ytem deigned fo igid plant. Fig.. Nyquit diagam of the popoed ytem when f LPF i vaiable in the ange fom to infinity and α Fig. how the nyquit diagam of FS-ARC ytem when α vaie fom to and f LPF i Hz. Fig. how the nyquit diagam of FS-ARC ytem when f LPF i changed fom to infinity and α i. The olid line in Fig. and Fig. ae the cae that α i and f LPF i. i.e. The eonance mode of P FSARC ) equal to that of P LL ).) The dot line in Fig. i the cae that α i and f LPF i. i.e. The eonance mode of P FSARC ) equal to that of P LM ).) The dot-dahed line in Fig. i the cae that α i and f LPF i infinity. i.e. The eonance mode of P FSARC ) equal to that of P LM ).) Hee, it i noticed that both the amplitude and the diection of the cicle of the eonance mode can be deigned between the own chaacteitic P LL ) and P LM ) in the nyquit diagam. In othe wod, the vibation uppeion and the phae tabilization of the eonance mode can be tuned imultaneouly only by the two paamete α and f LPF. Becaue of thi featue, FS-ARC i the dual contol ytem of FS-SRC which i applied fo SIMO ytem [9]. The tuning of the two paamete α and f LPF i detemined imply. Fit the vibation uppeion i detemined by the
2 Cae Cae2 Cae3 2 Cae Cae2 Cae3 Imaginay Axi 2 Imaginay Axi 2 3 3 4 3 2 2 3 Real Axi 4 3 2 2 3 Real Axi Fig. 2. Nyquit diagam in the imulation Fig. 3. Nyquit diagam in the expeiment tuning of α. Second the tuning of f LPF make bigget the phae magin of the cloed loop. Fo example, the cicle of the eonance mode i located fa fom the point -, ) in the nyquit diagam a much a poible. IV. SIMULATIONS Fequency epone of ARC and FS-ARC ae evaluated in a imulation. The imulation i pefomed in the thee cae a follow: Cae : ARC α α ARC, f LPF Hz), Pole of P L,ARC )C)) : 3 Hz, Cae 2: α α ARC, f LPF. Hz, Pole of P L,ARC )C)) : 25 Hz, Cae 3: α., f LPF 5. Hz, Pole of P L,ARC )C)) : 32 Hz. The contolle i edeigned to equalize the phae magin of Cae 2 and Cae 3 to that of Cae. Phae magin i 43.5 deg. The nyquit diagam of thee cae i hown in Fig. 2. The eonance mode i almot canceled in Cae and Cae 2. The cicle of the eonance mode i located fa fom the point -, ) in Cae 3. The phae tabilization of the eonance mode i tuned. V. EXPERIMENT Fequency epone and time epone of ARC and FS- ARC ae evaluated in expeiment with expeiment equipment hown in Fig. 3. The expeiment i pefomed in the thee cae a ame a the imulation. A. Feqency Repone The nyquit diagam of Fequency Sepaation Actuation Reonance Cancellation Contol i evaluated in expeiment. The expeiment eult i hown in Fig. 3. The eonance mode i almot canceled in Cae and Cae 2. The diection of the eonance mode could be located fa fom the point -, ) in Cae 3. B. Time Repone Time epone of the ARC and FS-ARC ytem ae evaluated by tep epone and load ide ditubance addition expeiment of expeimental equipment a hown Fig.3 ) Step Repone: Fig.4 how the tep epone of load ide angle. Step efeence input i added at.. In Cae, the vibation of the eonance i uppeed. In Cae 2, the vibation of the eonance i uppeed. Howeve, the ovehoot i inceaed fom Cae. In Cae 3, the ovehoot i deceaed, but the time epone i vibationally becaue contolle paamete α i not α ARC. Fig.5 how the toque of load ide dive moto in Cae and Cae 2. By compaing the Cae and Cae 2, peak toque of Cae 2 i malle than that of Cae. The dive moto of load ide can be malle5 % uing FS-ARC ytem at f LPF Hz. 2) Load Side Ditubance Repone: In thi ection, the ARC and FS-ARC ytem i evaluated in load Side ditubance toque addition expeiment. The tep toque i added to load moto. The load angle efeence i et to, and tep ditubance toque.2 N m i added at time.. The expeimental eult i hown in Fig.6. The time epone of Cae and Cae 2 i vibating at pimay eonance fequency. In Cae 3, the ditubance uppeion efficiency i high and the vibation i uppeed. Cae attache weight to the uppeion pefomance in tep efeence epone. Cae 2 attache weight to the miniatuization of load ide dive moto. Cae 3 attache weight to the uppeion pefomance in load ide ditubance uppeion. VI. CONCLUSION Reonance cancellation in MISO ytem which i attached moto at both the moto ide and the load ide i popoed. ARC and FS-ARC can be applied fo two-inetia ytem uch a a joint of humanoid obot. In imulation, the eonance mode i canceled, and the vibation uppeion and the phae tabilization of the eonance mode can be tuned imultaneouly only by the two paamete α and f LPF. The
..8 Cae Cae2 Cae3 Ref 5 Cae Cae2 Cae3 Angle [ad].6.4 Angle [mad] 5.2 5.5..5.2.25 Time [] 2.5..5.2.25.3.35 Time [] Fig. 4. Expeimental eult of tep epone Fig. 6. Expeimental eult of load ide ditubance epone Toque [Nm] Fig. 5..2.8.6.4.2.2 Cae Cae2.4.5..5.2.25 Time [] Toque of load ide dive moto of tep epone [5] G. Zhang, J. Fuuho, Speed Contol of Two-Inetia Sytem by PI/PID Contol, IEEE Tan. Ind. Electon., vol. 47, no. 3, pp. 63 69, 2. [6] Y. Mauhita, H. Ikeda, and H. Sugie, Vibation Suppeion Contol uing the Load-ide Acceleation Feedback, in Poc. the 33th Annual Confeence of IEEE Indutial Electonic Society, pp. 8 85, 27. [7] Y. Seki, H. Fujimoto, and. Saiki, Optimal Mechanical Paamete Deign Uing Self Reonance Cancellation Contol fo Ganty-Type High Peciion Stage, in Poc. The 2th Intenational Wokhop on Advanced Motion Contol, 22. [8] M. Aoki, H. Fujimoto, Y. Hoi, and T. Takahahi, Robut Reonance Suppeion Contol baed on Self Reonance Cancellation Ditubance Obeve and Application to Humanoid Robot, in Poc. IEEE Intenational Confeence on Mechatonic, pp. 623 628, 23. [9]. Sakata, H. Aaumi,. Hiachi,. Saiki, and H. Fujimoto, Fequency Sepaation Self Reonance Cancellation fo Vibation Suppeion Contol of a Lage-Scale Stage Uing Multiple Poition Seno, in Poc. the 39th Annual Confeence of IEEE Indutial Electonic Society, pp.6478 6483, 23. expeimental eult how that vibation uppeion efficiency of tep efeence epone and ditubance epone ae tuned by α and f LPF. The load ide dive moto fo vibation uppeion can be malle uing FS-ARC in compaion to ARC. Conventional eonance cancellation contol method, uch a SRC and SRCDOB in SIMO ytem contol the vitual angle which i canceled eonance. On the othe hand, the popoed ytem in MISO ytem contol the load ide angle θ L. REFERENCES []. Yuki, T. Muakami, and. Ohnihi, Vibation Contol of a 2 Ma Reonant Sytem by the Reonance Ratio Contol, Tan. Int. Elect. Eng. Jpn. vol. 3-D, no., pp. 62 69, 993. [2] Y. Hoi, H. Sawada, and Y. Chun, Slow eonance atio contol fo vibation uppeion and ditubance ejection in toional ytem, IEEE Tan. Ind. Electon., vol. 46, no., pp. 62 68, 999. [3] Y. Hoi, H. Ieki, and. Sugiua, Baic conideation of vibation uppeion and ditubance ejection contol of multi-inetia ytem uing SFLAC tate feedback and load acceleation contol), IEEE Tan. Ind. Appl., vol. 3, no. 4, pp. 889 896, 994. [4]. Ohihi, N. Shimada, and T. Miyazaki, High-pefomance obot motion contol baed on zeo-phae notch filte fo indutial obot, The th IEEE Intenational Wokhop on Advanced Motion Contol, pp.626 63, 2.