5D1 SY4/14/-485 214 SICE 1, 2 Dynamically Consistent Motion Design of Humanoid Robots even at the Limit of Kinematics Kenya TANAKA 1 and Tomomichi SUGIHARA 2 1 School of Engineering, Osaka University 2-1 Yamadaoka, Suita, Osaka 565-871, Japan 2 Graduate School of Engineering, Osaka University 2-1 Yamadaoka, Suita, Osaka 565-871, Japan A motion design method for humanoid robots is proposed, which concerns dynamical constraints on the external forces and satisfies them even at the kinematic exceptions such as singular configurations and joint angle limits. It enables to mimic human s natural and vigorous motions, which leverage the limit of kinematics, on robots based on a robust numerical solver of the prioritized inverse kinematics. The idea of prioritization is to strictly constrain the least number of contact points to form the minimal support region and to unconstrain the other points rather than to keep foot-flat. It increases available degrees-of-freedom instead of degenerate components. Motion continuity in spite of the discontinuous change of a set of constraints is also taken into account. Knee-stretched walks are demonstrated in computer simulations as applications. Key Words : Humanoid, Motion Design, Inverse Kinematics 1. ( ),, (1) (3)., ( ) 1 565-871 2-1 kenya.tanaka@ams.eng.osaka-u.ac.jp 2 565-871 2-1 zhidao@ieee.org (4) 1(a) (b) (b) (c) 第 19 回ロボティクスシンポジア (214 年 3 月 13 日 -14 日 兵庫 ) - 485 -
(a)solvable case (b)unsolvable case without priority (c)unsolvable case with contact prioritized Fig. 1 Inverse kinematics in a sense of least-squared-error (d)unsolvable case with minimal support region (5) 2 (i) (ii) (i) 1(d) (ii) (1) 2. (a)maximal support region (b)minimal support region with foot flat with toe/heel contact Fig. 2 Minimal support region with respect to the desired (6) (7) (8) (1) 2-486 -
Fig. 3 Desired trajectory and (11) t = t = T x K x S x Z (t) = x S (x S x K )(t/t 1) N (1) Fig. 4 Delete the constraint N t T 3 4 3 4 3. 5 Fig. 5 Velocity constraint T k d p[k] p[k] ε p[k] d p[k] (p[k] p[k 1]) ( d p[k] d p[k 1]) > ε (2). d p [k] = p[k 1] + ( d p[k] d p[k 1]) + ε (3) d p [k]. p [k],. 4. (1) (1) d p z = ( d x z, d y z, d z z ) d n = ( d n x, d n y, d n z ) p = (x,y,z) f = ( f x, f y, f z ) - 487 -
d n = (p d p z ) f (4) m f x = mẍ f y = mÿ m(z d z)ẍ (x d x) f z = d n y (5) m(z d z)ÿ (y d y) f z = d n x (6) d n x d n y m(z d z)(ẍ + ẍ) (x + x d x) f z = (7) m(z d z)(ÿ + ÿ) (y + y d y) f z = (8) (7) (5) (8) (6) x y i ( ẍ i, ÿ i ) i(i = 1...S) ( x i, y i ) ( ẋ i, ẏ i ) ( 4S ) i = 2...S 1 i = 1,S 4S x y d n x d n y f = {( d n x ) 2 + ( d n y ) 2 }dt (9) (I) ( 6) (II) ( 7) (III) ( 8) (IV) ( 9) (V) IV ( 1) (I) (II) (V) (II) (III) (IV) 3 4 (III) (V) (IV) (I) 11 (V) 12 (V) 13. (1) (12) (13) (14) 5. - 488 -
1 2 3 (a) and Foot 1 2 3 (a) and Foot 1 2 3 (a) and Foot.5.4.5.4.5.4 1 2 3 (b) and Fig. 6 Case I 1 2 3 (b) and Fig. 7 Case II 1 2 3 (b) and Fig. 8 Case III.8.6.4.2 -.2 -.4 -.6 -.8 1 2 3 (c) and.8.6.4.2 -.2 -.4 -.6 -.8 1 2 3 (c) and.8.6.4.2 -.2 -.4 -.6 -.8 1 2 3 (c) and 1 2 3 (a) and Foot 1 2 3 (a) and Foot.5.4.5.4 1 2 3 (b) and Fig. 9 Case IV 1 2 3 (b) and Fig. 1 Case V.8.6.4.2 -.2 -.4 -.6 -.8 1 2 3 (c) and.8.6.4.2 -.2 -.4 -.6 -.8 1 2 3 (c) and - 489 -
.6[s] 1.[s] 1.2[s] 1.6[s] 2.[s] 2.2[s] Fig. 11 Conventional method.6[s] 1.[s] 1.2[s] 1.6[s] 2.[s] 2.2[s] Fig. 12 Proposed method.[s].4[s] 1.[s] 1.2[s] 1.6[s] 2.[s] Fig. 13 Side stepping(proposed method) (K23 XVI 355 ) (1),,, (1999). (2),,,, Vol.17, No.2(1999), pp.268 274. (3) K.Yamane et al, Dynamics Filter - Concept and Implementation if On-Line Motion Generator for Human Figures, Proc. of the 2 IEEE Int. Conf. on Robotics and Automation, (2), pp.688 695. (4) Levenberg-marquardt,, Vol.29, No.3(211), pp.269 227. (5), 19 ( ), (214). (6) M.Vukobratović and J.Stepanenko On the Stability of Anthropomorphic Systems, Mathematical Biosciences, 15(1),(1972), pp.1 37. (7), Vol.11, No.4(1993), pp.557 563. (8),,,,,, 6, (21), pp.113 118. (9),,,,,, 24, (26), 3H14. (1),,, 12, (27),pp.28 33. (11),,, 25, (27), 1G26. (12), 22, (24). (13), 24, (26), 2A1-D19. (14) 2, (C ), (24), pp147 153. - 49 -