Synthesizing and Modeling Human Locomotion Using System Identification

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1 yz d Md H U y Id W UIMAN, Adé MONIN d J-P AUMOND AA - CNR 7 v d C R, 3077 T Cdx 4, F {,, Ab yz d b -d d. Hwv, yzd, w d. U y, w w d d d dy d. T d jy v d d w bdy. T dy y, w dd bk-bx, w w dd d d. W w dd d w v d wy. I. INTRODUCTION dd b. Hwv, w d v, y d b, 2, b 3 d 4. M yz d. T bjv vd d w w dd. yz v. Wy 5 db yz w by d q. P d d 6 q d v d, d v ( ). A d w d by 7. T d -, w b. Kw d 8 v d d bd d bjv yz -. Uk v, w w d v b yz d d. T d v d y d dy y. T d d d d 9. Gy k, d w d d kw y bj d -. I, d d v q. T d d. O bjv b - d. T jy v R 3 d d w bdy. A d b vy b y d -d. T b w b dd by vb y k d bdy k d x- z. T dv d y d xy vy, b d v. Mv, w d q y x, wk qd xd -x. A. Bk-bx d P v J II. DFINING MOD TRUCTUR C 4 C 5 C 2 C 3 C w k j F.. A d v k 3D d W d 3D d d F.. I 23 j, j d d (DOF). Ty d by. T dd v k. dd by, w w b dd y. F. 2 w by, y, y d w d by b. T y d x vb. I b bd b d. Hwv, w d v y. T w d y d d. d b bd by bk-bx

2 T j y B C P y d B k b x v D O F F. 2. A d d by d b 0,. T d kw. I k w x + = (x, ) y = (x, ) + v () w d kw, x R y, R, y R d v. T v dd b dd. I, dy d q (), d b y. I, k d d vb b d k b d. T v dy, w x- z 2. T j- q by vb d 3. T d b x = Ax + B y = Cx + D + v (2) w A, B, C, D y. T y d k b vd by y dd d b 9, b d (.. N4ID MATAB TM y d bx). Uy d y d (, y d jy). A w d d y d, w w w w d PO-MOP d 4 ( (III-A)) d w d. B. x- z T x- v R 3 db x d d DOF d. A v x 5, w v d 2. 3 b - q d O(3) b w d + ( ). I, w R 3 3, dd q v O(3). W x- R 3 3 w 0,0,0T = 0, 0, 0, T d v 0 v = ( 2 θ)ˆv, ( 2 θ) T = q v, q w T (3) w q v v q, θ = v d ˆv = v v. T v q θ b ˆv. T v x-, 3 R 3 v w v = ( v ) = 2 (q w ) q v (4) q v A d d d by, v d y. T v b bd x 6. C. I/ I w kw v d d, d b y xd. T d v d dy d d. Hwv, by yz, w v bvd v d b d dd,2 d 3, d w k b d dd 4 d 5 (F. ). T bd k d qy. T d b d F. 3, w P d y by. T by w v, y d x- y by, 2 d 3. T 4 d 5 w k x-. T b bd dd d 3. T vd bv, w d b b dy d, w x by () y. III. IDNTIFYING MOD A. vvw PO-MOP d T PO-MOP d b d d 4. Gv d x = Ax + B y = Cx + D + v (5) w x,, y d v dd (II-A). T d qd A, B, C, D. F, d kd Hk 2 N α+ 2 3 N α+2 U,α,N α α+ N

3 D O F x M C v P P v x 2 D O F 2 x M 3 D O F 3 x M w k x 4 D O F 4 x M 5 D O F 5 x M F. 3. A vvw d w b dx d, α d b w x, d N dx d x. T b w d b b y d 7. T Hk x Y,α,N d Hk x V,α,N dd y. T w - q y dvd y d (5): Y,α,N = Γ α X,N α+ + Φ α U,α,N + V,α,N (6) W Γ α xdd bvby x y, Φ α bk- x C CA Γ α =. CA (α ) D CB D 0 0 Φ α = CAB CB D CA (α 2) B CB D d X,N α+ = x x 2 x N α+. PO-MOP d d d vb v. Cd w QR z: U +α,α,n U,α,N α Y,α,N α Y +α,α,n R = R 2 R R 3 R 32 R 33 0 R 4 R 42 R 43 R 44 QT (7) w R w- d y x Q (Q T Q = I) d d d w- R Q = Q Q 2 Q 3 Q 4 I w d qy R42 R 43 = Y+α,α,N Q2 Q 3 d N = (Γ α X α,n α+ + V +α,α,n ) Q 2 Q 3 N V +α,α N Q2 Q 3 = 0 w.. w w.. d w bby. Cqy, w v, Γ α xd x R 42 R 43 v d (VD) w R42 R 43 = U V T + U 2V w x v ( d), d x yd ( d x ). T x Γ α b d w T (8) Γ α = U /2 (9) d w C A dy Γ α C =Γ α ( :, :) Γ α ( : (α ), :)A =Γ α ( + : α, :) (0) I d d B d D, w d -q vdd y q (U ) T R3 R 32 R 4 = (U ) T Φα R2 R 22 R () w vd Φ α, w B d D y d. B. D w M D I d (6) d by -z d. F, -z d v qd A, B, C, D. A, d w

4 d d d dd b. A w v d, dy d vv d d v jy. T w d PO-MOP d w d w. Cd w d {, y} N d =, 2,, K (2) = w d d jy (x). F / d, w b w d q Y,α,N = Γ α X,N α+ + Φ αu,α,n + V,α,N (3) T d q y b bd w Y,α,N Y,α,NK = Γ α X,N α+ XK,N K α+ + Φ α U,α,N U,α,N K K + V,α,N V,α,N K K (4) T q d q (6), w bdy PO-MOP by QR z w x U+α,α,N U+α,α,N 2 2 U K +α,α,n K U,α,N α U2,α,N 2 α UK,α,N K α Y,α,N Y 2 α,α,n Y K (5) 2 α,α,n K α Y+α,α,N Y+α,α,N 2 2 Y+α,α,N K K d q (7). C. d T d v (x 0 ). x 0 y v v. Hwv, d b d d ( jy). Cd d {, y }, w x 0 w q y y 2. = Γ 2 αx 0 + Φ α. y α α (6) T q vd x 0 dv x. D. D w b d O v d, w y b d b d. T v b w d b d vy: ρ(a) <, w ρ(a) d d A. y 8 v d d z d y by dd d x b d d. T d ρ(a) < d by qv yv q ρ(a) < P δi : P APA T δi w δ > 0. O w b xdd bvby x Γ α, z b z bj J Γ (A) (Γ,α Γ 0,α A)R Γ 2 2 (7) P δi X T 0 X P 2 (8) w Γ,j R (j +) v by C(A) C(A) + Γ,j.. C(A) j d X AP. W Â ˆX d ˆP Â = ˆX ˆP. I w R Γ P b vd z b vv z v y ( 8 d). IV. IDNTIFICATION AGORITHM T dy d, w d dy d by P d ( : =,, 5). T d d F. 3. N w d d d d k, w x PO-MOP d d w d. W z d w ) Idy d P: T d v X = x, y, z T d Pv Θ = θ x, θ y, θ z T. Θ b d dy jy v (. θ x jy {y,z}). 2) dy d : T v b, w PO-MOP d w d d F. 3. F x, dy by. F, w d d -. By v dd d P, w v, d x- y. T x- d (F. ). d, w y PO-MOP d y (A, B, C, D). N by bvy b, w dd d b b by y d xd (III-D). T bv xd d vd d, w k jy ( jy bd ) v d v w bdy. A d w b d jy X = x, ȳ, z T (.. Béz v),

5 jy dd. I, jy b dd w w X = x y x = ȳ x + ỹ = X + X (9) z z z w X jy v d X d z-z v jy 9. F, jy b d by dy X. x X j b d by x, y d z w, w b X. A X d z-z v jy, X d. b dd by b w q. T b b w z = Az X = Cz + v (20) w z. T A d C, w PO-MOP d. A w kw d b, w d b b. V. XPRIMNTA RUT I, w bd by d. W v d db (0 d j), w d dvd. F. 4 d dd d. W d d d, P d ( : =, 2,, 5) by 0, 0, 5, 5, 0, 0, 0 vy, d d d yz w 250 PC w.5 GHz d 52 MBy RAM. W v vdd bd d w x ) F x: I x, jy v b d y ( d ). F. 5 bd. 2) d x: I d vy y dd d, w d jy d. T jy b zd v b dd. N, x by d. F. 6 bd. T bvd vb k k d by v dd dd d. Hwv, b d by x d d 3, 20. Idd, k y v -, w v. Vd d wk vb :// VI. CONCUION I, w v d w d d. T b d d d 3D bdy v k, dy vb d by d. T bd, b b d wk. Fy, w bv b q d d v q d b. VII. ACKNOWDGMNT T d d w bd...d. T db w d w d NF IA RFRNC T. A. MM, M, I J Rb R, N. B, T d d v. Oxd: P Oxd: P P, R. Bk d K. C, A v k by v k by y, J B, v., , F. M,. F, M.-P. C, d G. Db, C wk: vy, J Vz d C A, D. Wy d J. H, I y d, Pd I V Ry A I y (VRAI 97), J. P d J. d, A bd - wk q, T J C A d V Wd. 7 H. C. d D. N. Mx, A, ACM IGGRAPH, ( A, UA), AUGUT T. Kw d. Y., M d - y, /ACM IGGRAPH y A, j, y Id: Ty U. P H I d y, 2d d., J J. JOBRG, N bk-bx d y d: M d, A, v. 3,. 2, , 995. J. JOBRG d, N bk-bx d y d: d vvw, A, v. 3,. 2, , F.. G, P z x, J G T, v. 33, H, K. P, d J. Pv, y, ACM T G, M. V, Id d d v v - d, A, v. 30,.,. 6 74, M. Rd,. Zx, d. k, A M Id Rb M. CRC P, K. k, A w q v, ACM IGGRAPH, v. 9, , Jy M. Vb, b-bd d d -v y, A, v. 3,. 2, , y d D.. B, b d w d by d z, I T, v. 48, , JUY M. G, M d, ACM y Iv 3D G, Kv d M. G, Fk d, ACM IGGRAPH y C A, , 2002.

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