5D1 SY0004/14/ SICE 1, 2 Dynamically Consistent Motion Design of Humanoid Robots even at the Limit of Kinematics Kenya TANAKA 1 and Tomo

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1 5D1 SY4/14/ 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 , Japan 2 Graduate School of Engineering, Osaka University 2-1 Yamadaoka, Suita, Osaka , 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)., ( ) kenya.tanaka@ams.eng.osaka-u.ac.jp 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 )

2 (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)

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).

3 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 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 )

4 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 1 2 3 (a) and Foot (a) and Foot (a) and Foot (b) and Fig. 6 Case I (b) and Fig. 7 Case II (b) and Fig. 8 Case III (c) and (c) and (c) and (a) and Foot (a) and Foot (b) and Fig. 9 Case IV (b) and Fig. 1 Case V (c) and (c) and

.6[s] 1.[s] 1.2[s] 1.6[s] 2.[s] 2.2[s] Fig.

6 .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 (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 (4) Levenberg-marquardt,, Vol.29, No.3(211), pp (5), 19 ( ), (214). (6) M.Vukobratović and J.Stepanenko On the Stability of Anthropomorphic Systems, Mathematical Biosciences, 15(1),(1972), pp (7), Vol.11, No.4(1993), pp (8),,,,,, 6, (21), pp (9),,,,,, 24, (26), 3H14. (1),,, 12, (27),pp (11),,, 25, (27), 1G26. (12), 22, (24). (13), 24, (26), 2A1-D19. (14) 2, (C ), (24), pp

Nominal trunk path feasible region of the step )ϕ L (s) ϕ R (s) [ ] [ ] [ ][ x L (s) x B (s) cosθ B (s) sinθ B (s) = + y L (s) y B (s) sinθ B (s) cosθ

Nominal trunk path feasible region of the step )ϕ L (s) ϕ R (s) [ ] [ ] [ ][ x L (s) x B (s) cosθ B (s) sinθ B (s) = + y L (s) y B (s) sinθ B (s) cosθ 3B2 SY5/17/-177 217 SICE 1 1 Semi-autonomous Navigation of a Biped Robot Over Unlevel Terrain Based on the Combination of Discreate-Continuous Control Takanobu Yamamoto 1 and Tomomichi Sugihara 1 1 Department