z [m] Fig. 2 Bezier curve Controll Point x [m] Expression of individual ex) l = [ ] T innerfoot, outerfoot 3

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1 ( ) ( ) Gait Stabilization of Passive Dynamic Walker by Optimization of Foot Shape Toshihiko FUKUSHIMA (The University of Tokyo) and Kazuyuki HYODO (Fukuoka Institute of Technology) Passive dynamic walk is excellent at energy efficiency, but has some problems keeping gait stabilization in a real environment. There is one of problems that passive dynamic walker falter in some environment. To resolve this problem, it is experimentally known that specific foot shape confines pitch and roll swing to keep from faltering. In this paper, we report the method to optimize foot shape based on gait stabilization to improving gait performance. Key Words: Passive Dynamic Walker, Shape Optimization, Biped Walking, Constraint Mechanism, Genetic Algolithm 1. 1) 2 3 2) 2 3) Table 1 leg outerfoot Fig. 1 1 body innerfoot upperfoot 3D simulation model Composition of 3D simulation model Element Piece n Body body 1 Leg leg 2 upperfoot 2 Foot innerfoot 2 outerfoot OpenHRP Matlab OptimizationToolbox Body Leg Foot 3 Foot upperfoot,

2 z [m] Fig. 2 Bezier curve Controll Point x [m] Expression of individual ex) l = [ ] T innerfoot, outerfoot 3 (Fig.1, Table.1) 3 innerfoot outerfoot innerfoot outerfoot 9 y l (Fig.2) ) ) 1 6) 2 1 1) x x f 1) ϕ ϕ 1, ϕ 2, ϕ 3 4 ( ϕ 1 < ϕ 2 < ϕ 3 ) ϕ < ϕ ( ) ϕ 1 ϕ < ϕ 2 x f 1 ( ) ϕ 2 ϕ < ϕ 3 x f ( ) ϕ 3 ϕ x 7) (ϕ GS = ϕ 2 ϕ 1 ) Fig.3 Fig x sim x min x sim x min x min = 1.[m] Score = Score x f x x f x 1 x f

3 Start success = score = Update Slope Angle Simulate Angular velocity of hip joint [rad/s] Jumpping point x sim > x min YES Stable Gait? YES success = success + 1 NO NO score = success 1 + x sim Angle of hip joint [rad] Fig. 4 State trajectory: Sign means a point soon after jump Norm of x k Fig End Flowchart: Fitness function x 1 x f 1. k q k q + k (Fig.4) x k (Fig.) [ ] q + x + k k = 1 q + k 2. k x + k k + 1 x+ k 1 y k (Fig.6) y k = x + k x + k y k z k (Fig.7) Norm of y k k Fig. Norm of x k k Fig. 6 Norm of y k data ae bk z k = y k (x + k + x+ k 1 )/ z k z k z k ae bk (a, b )(Fig.7) z k ae bk 4. b < Norm of z k k Fig. 7 Norm of z k

4 .1 front foot_inner foot_outer.9 z [m] -. Foot Radius [rad] x [m] -.1 (a) Arc foot z [m] Slope Angle [rad] x [m] (b) Optimized foot Fig. 8 The relation of stability between foot radius and slope angle Fig. 1 Shape of innerfoot and outerfoot Stability Domain Supply Energy Surface High Energy 1 Best: -8 Mean: Mean fitness Best fitness -1 Consumption Energy Surface Fitness value Foot Radius Slope Angle Low Energy - -6 Fig. 9 The relation of stability between foot radius and slope angle Generation Fig.1 3 r ϕ r ϕ (Fig.8) Fig E sup. 2. E con 8). E sup E con Fig.9 Fig.9 r-ϕ Fig.8 Fig Fig. 11 Evaluation shift 3. 4 Fig.1-(a) (Fig.11) Fig.1-(b) ( ) l innerfoot: 3 l 4 outerfoot: 1 l ϕ GS l ϕ min =.3[rad] ϕ max =.12[rad] ϕ GS =.8[rad] ϕ GS =.[rad] 7% (Table 2)

5 Angular velocity of hip joint [rad/s] Arc foot Optimized foot -2 - Angle of hip joint [rad] Fig. 12 Table 2 Table 3 State trajectory: Arc foot VS Optimized foot: 1[s]- 6[s] Comperition between arc foot and optimized foot in stability domain Type ϕ GS Comparision Arc foot.[rad] 1[%] Optimized foot.8[rad] 17[%] Comperition between arc foot and optimized foot in state trajectory 4. 1 Type SD jump Comparision Arc foot.11 1[%] Optimized foot.13 13[%] Fig.12 SD jump =.13 SD jump =.11 87% (Table 3) innerfoot 2. outerfoot 3. outerfoot ϕ. 2 Fig.13 (Fig.13 ) 9) ϕ =.[rad].8[rad] ϕ R = T inn T inn +T out Fig.14 ϕ 3 2 (Fig.1) 9)

6 2 front 1 Ou te Angular Velocity [deg] 1 r fo e Inn ot r fo ot (a) Low slope angle - (b) High slope angle Toe Fig. 1 Virtual 2D foot shape outer foot -1 Inner foot 結 4 time [s] 論 (a) φ=. 遺伝的アルゴリズムを用いて歩行安定性を考慮した足形状最適化を行い受動歩行機の 3 次元空間における安定歩行 1 領域を拡大することに成功したまた定性的ではあるが歩 1 行機自身の持つ自己安定化の原理について考察したこの結果歩行機が環境の変化に対して自らの身体特徴を生かし運動を変化させることにより歩行を安定化しているという新 Angular Velocity [deg] 2 - たな示唆を与えた -1 本研究を通して歩行機の持つ形態の重要性を確認した -1 今後環境の変化に応じて形態を変化させる制御手法を検討 -2 する time [s] 参 (b) φ=.6 3 Angular Velocity [deg] time [s] (c) φ=.7 Fig. 13 Angular velocity.7.6 Outer foot Inner-Outer Rate.6 Inner foot Slope Angle [rad] Fig. 14 Time rate of inner foot shape and outer foot shape 考文献 1 T. McGeer. Passive dynamic walking. The International Journal of Robotics Research, Vol. 9, No. 2, pp , F. Asano and Z.W. Luo. Eﬃcient dynamic bipedal walking using eﬀects of semicircular feet. Robotica, Vol. 29, No. 3, pp , K. Hyodo, T. Oshimura, S. Mikami, and S. Suzuki. Stabilizing passive dynamic walk under wide range of environments by constraint mechanism ﬁtted to sole of foot. Journal of Robotics and Mechatronics, Vol. 21, No. 3, p. 43, Y. Sugimoto and K. Osuka. Stability analysis of passive-dynamic-walking focusing on the inner structure of poincare map. In Advanced Robotics, 2. ICAR. Proceedings., 12th International Conference on, pp IEEE, 2. M. Wisse, A.L. Schwab, R.Q. van der Linde, and F.C.T. van der Helm. How to keep from falling forward: elementary swing leg action for passive dynamic walkers. Robotics, IEEE Transactions on, Vol. 21, No. 3, pp , 2. 6 H. Poincare. Science and hypothesis. Science Press, Y. Ikemata, A. Sano, and H. Fujimoto. A physical principle of gait generation and its stabilization derived from mechanism of ﬁxed point. In Robotics and Automation, 26. ICRA 26. Proceedings 26 IEEE International Conference on, pp IEEE, 田崎勇一, 井村順一. 平面受動 2 足歩行における足形状の省エネルギー効果の考察. 日本ロボット学会誌 = Journal of Robotics Society of Japan, Vol. 23, No. 1, pp , F. Asano, M. Yamakita, and K. Furuta. Virtual passive dynamic walking and energy-based control laws. In Intelligent Robots and Systems, 2.(IROS 2). Proceedings. 2 IEEE/RSJ International Conference on, Vol. 2, pp IEEE, 2.

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