H 975 [8] [9] 4 H [10] H [11] [15] H I H II H Fig. 2 H [16] [18] [19] H [8] Fig. 3 3 b, β, δ d Double U Joint [20] a e A G β 2 z C B

974 Vol. 19 No. 8, pp.974 982, 2001 H 1 1 2 3 H Design of Hybrid Compliance using Upper/Lower Bound in the Frequency Domain Shaping and Control of Dynamic Compliance of Humanoid Shoulder Mechanisms Masafumi Okada 1, Yoshihiko Nakamura 1 2 and Shin-ichiro Hoshino 3 Design and control of mechanical compliance would be one of the most important technical foci in making humanoid robots really interactive with the humans. For task execution and safety insurance the issue must be discussed and offered useful and realistic solutions. In this paper,we propose a theoretical design principle of mechanical compliance. Passive compliance implies mechanically embedded one in drive systems and is reliable but not tunable in nature,while active compliance is a controlled compliance and,therefore,widely tunable,but less reliable specially in high frequency domain. The basic idea of this paper is to use active compliance in the lower frequency domain and to rely on passive compliance in the higher frequency. H control theory based on systems identification allows a systematic method to design the hybrid compliance in frequency domain. The proposed design is applied to the shoulder mechanism of a humanoid robot. Its implementation and experiments are to be shown with successful results. Key Words: H Control,Hybrid Compliance,Humanoid Robot,Frequency Dependent Dynamic Compliance 1. [1] [6] f y G G [7] Fig. 1 2000 6 16 1 2 3 NTT 1 Department of Mechano Informatics, University of Tokyo 2 Japan Science and Technology Corporation 3 NTT DATA Corporation Fig. 1 Shaping dynamic compliance JRSJ Vol. 19 No. 8 54 Nov., 2001

H 975 [8] 3 1 2 3 4 [9] 4 H [10] H [11] [15] H I H II H Fig. 2 H [16] [18] [19] H 2. 2. 1 [8] Fig. 3 3 b, β, δ d Double U Joint [20] a e A G β 2 z C B G E D D G 3 E β δ θ 21 θ 22 θ 41 θ 42 Fig. 4 E ( I R y θ 21 Rθ x 22 L z L 3 Rθ x 41 R y θ 42 Rπ) z R z 2 π(i 1)r = li 3 r := [ b 0 0 1 ] T (i =1, 2, 3) 1 2 R ξ θ ξ θ Lξ l ξ l l i E b B D E 1 l 1 l 2 l 3 Fig. 3 Cybernetic shoulder Fig. 2 Shaping dynamic compliance Fig. 4 Definition of rotations 19 8 55 2001 11

976 Fig. 5 The cybernetic shoulder with the rigid or elastic link θ 41 θ 42 l 1 = Js(θ 21,θ 22) l 2 3 L 3 l 3 E l i θ 41 θ 42 L J s Table 1 Spring constant and coefficient of viscosity Type Spring constant Coefficient of [N/m] viscosity [kg/s] 1 1.609 10 3 0.625 2 5.963 10 2 0.45 3 5.963 10 2 1.05 E k 1 k 2 k 3 θ 41 θ 42 L 3 C s 1 k 1 0 0 C s = J s 0 k 2 0 Js T 4 0 0 k 3 E θ 41 θ 42 L 3 2. 2 E Fig. 5 1 7 10 [mm 2 ] Fig. 5 2 φ5 [mm] Fig. 5 3 2 + Fig. 5 3 Fig. 5 1 Fig. 5 Table 1 Fig. 6 Humanoid robot 2. 3 Fig. 6 z PD 1 2 3 Fig. 7 1 [kg] [ e ] [ ] x e 0 y e 0 z 0 = 400 300 270 [mm] 5 JRSJ Vol. 19 No. 8 56 Nov., 2001

H 977 Fig. 10 Generalized control system Fig. 7 Experimental setup Fig. 11 Local feedback system [ z y ] [ G11 G 12 = G 21 G 22 ][ f u ] 8 z : y : f : u : Fig. 10 C f z Fig. 8 Step responses of the strain G zf G zf = G 11 G 12(I + CG 21) 1 CG 21 9 H G zf < 1 10 C H G(s) := sup G(jω) 0<ω< 11 Fig. 9 Frequency analysis Fig. 8 9 W f (s) 100 4 W f (s) = 6 (s + 100) 4 3 25 [rad/s] 3. H 3. 1 H [9] [ ] G11 G 12 G = G 21 G 22 7 H 3. 2 3. 2. 1 P P Fig. 11 C l P θ m θ u θ u θ m C l 19 8 57 2001 11

978 Fig. 12 Disturbance input y = G close f 16 15 G close Fig. 13 C v =0 y := y y 0 17 Fig. 13 Vibration control system θ u y 0 y θ r θ m τ K I k f f Fig. 12 J u J m y ẏ = [ ] [ ] θu J u J m 12 θ m P ] [ ] [ θu θ m P m1 τ u = P τ + τ m [ ] Pu1 P u2 P := P m2 13 14 P u1 P u2 P m1 P m2 K K f y y = G open f 15 Fig. 13 C v y 0 y y y 0 1 [ ] f y y = K(θu,θ m)g G = [ f y 0 y0 u 18 G sg m G ti k G tj G u P u2c l G sg m P u2c l G si k P u2c l G sj G s := (I + P m2c l ) 1 G t := I G s G u := P u1j T u + P u2j T m G m := P m1j T u + P m2j T m I : ] 19 20 21 22 23 18 y 0 C l f u C v 18 y = [ ] [ ] f Ḡ f Ḡ u 24 u C v C v C v 1 2 1 1 I 2 C v G close G close 1 II 3. 2. 2 C v Fig. 14 W 1 W 2 W 3 Fig. 14 C v f z 1 H JRSJ Vol. 19 No. 8 58 Nov., 2001

H 979 Fig. 14 Generalized control system for design of C v 1 G close W 1 < 1 25 G close < W 1 1 26 W 1 1 f z 2 ( ) W2 I G close W 1 < 1 27 G W 1 close > W 1 1 1 W 2 W 2 W 1 1 G close 28 29 W 1 W 2 f z 2 z 1 z 2 f z 3 ( ) 1 W 3 I + Cv Ḡ u Cv Ḡ f < 1 30 W 3 W 3 W 1 W 1 W 2 30 4. 4. 1 P Fig. 14 f u y 4. 2 C v Fig. 14 Ḡf Ḡu Ḡf Fig. 8 Ḡu u y Ḡ u M [21] 0.125 [mm] Ḡu OE Output Error [21] Ḡu Ḡm u Ḡ u Ḡf Fig. 8 Ḡf Ḡm f ARX [21] Ḡ m u = 1.47 10 5 (s + 6975)(s +34.7) (s +2.76 + 23.8j)(s +2.76 23.8j) 31 Ḡ m f = 5.48 10 2 (s +26.7 + 135j)(s +26.7 135j) (s +2.76 + 23.8j)(s +2.76 23.8j) ω =23.8 [rad/s] 4. 3 32 Case 1 31 32 Ḡm u Ḡm f Fig. 14 C v W 1 W 2 W 3 W 1 = 7(s + 15) 3 (s +5) 2 (s + 1000) 33 W 2 =0 34 290(s + 10)3 W 3 = 35 (s + 1000) 3 W 1 W 2 =0 W 3 Fig. 15 C v G open W 1 G close C v C v W 3 19 8 59 2001 11

980 Fig. 15 Gain plots of closed loop systems (Case 1) Fig. 17 Gain plots of closed loop systems (Case 2) Fig. 16 Step responses of the controlled system (Case 1) W 3 G open G close C v H 3. 2. 1 C v C v Fig. 7 Fig. 16 Fig. 8 G open C v 2 C v C v 3. 2. 1 Fig. 16 Fig. 15 Case 2 C v G open < G close 36 Fig. 18 Step responses of the controlled systems (Case 2) 12(s + 20) W 1 = 37 s + 1000 65 W 2 = 38 (s +5) 2 214(s + 10)3 W 3 = 39 (s + 1000) 3 Fig. 17 28 Case 1 W 1 1 G close G open Fig. 16 Fig. 18 Fig. 18 1.5 4. 4 2 1 1.5 JRSJ Vol. 19 No. 8 60 Nov., 2001

H 981 W 1 G close 0 C v C v 1 E Fig. 4 θ 41 θ 42 L 3 H 5. 4 1 2 H 3 4 JSPS RFTF96P00801 [ 1 ] R.P.C. Paul and B. Shimano: Compliance and Control, Proc. of the 1976 Joint Automatic Control Conference, pp.694 699, 1976. [ 2 ] H. Hanafusa and H. Asada: Stable Prehension by a Robot Hand with Elastic Fingers, Proc. of the 7th International Symposium on Industrial Robots, pp.361 368, 1977. [ 3 ] N. Hogan: Mechanical Impedance Control in Assistive Devicesand Manipulators, Proc. of the 1980 Joint Automatic Control Conference, TA10 B, 1980. [ 4 ] J.K. Salisbury: Active Stiffness Control of a Manipulator in Cartesian Coordinates, Proc. of the IEEE Conference on Decision and Control, 1980. [ 5 ] N. Hogan: Impedance Control: An Approach to Manipulation: Part 1 3, ASME Journal of Dynamic Systems, Measurement and Control, vol.107, pp.1 24, 1985. [ 6 ] K.F. L-Kovitz, J.E. Colgate and S.D.R. Carnes: Design of Components for Programmable Passive Imprdance, Proc. of IEEE International Conference on Roboticsand Automation, pp.1476 1481, 1991. [7] 1976. [8] 98 ROBOMEC 98 1BI2 3, 1998. [9] 99 ROBOMEC 99 2P1 78 107, 1999. [10] K. Glover and J.C. Doyle: A State Space Approach to H Optimal Control, in Three Decadesof Mathmatical System Theory (H. Nijmeijer, J.M. Schumacher eds.). pp.179 218, Springer Verlag, 1994. [11] 2 D.D. vol.29, no.12, pp.1421 1426, 1993. [12] µ 11 pp.29 32, 1993. [13] 12 pp.51 52, 1994. [14] 1 vol.32, no.7, pp.1011 1019, 1996. [15] 2 2 vol.32, no.7, pp.1020 1026, 1996. [16] H 20 pp.101 104, 1991. [17] 2 H 21 pp.121 126, 1992. [18] 5 pp.s 176 S 179, 1993. [19] H 13 pp.27 28, 1995. [20] M.E. Rosheim: Robot Evolution, The Development of Anthrobotics. JOHN & SONS, INC., 1994. [21] L. Ljüng: System Identification Theory for the User. 19 8 61 2001 11

982 Prentice Hall, 1987. Masafumi Okada 1969 3 21 1994 3 1996 9 1996 10 PD 1997 2 2000 4 2001 4 IEEE Yoshihiko Nakamura 1954 9 22 82 87 87 91 3 4 CG IEEE ASME Shin ichiro Hoshino 1976 5 25 1999 3 2001 3 NTT JRSJ Vol. 19 No. 8 62 Nov., 2001