今後の予定

謝辞

本研究にあたり，熱心に指導していただいた浅野文彦准教授に心より深謝致します．ロ ボティクスの観点から鋭いご指摘を頂いた丁洛榮准教授に感謝致します．浅野文彦研究 室のメンバーであり，討論，論文作成，実験において貴重な意見・助言を頂いた肖軒氏，

安田芳樹氏，阿久津行裕氏，田村和希氏に感謝致します．学会活動中に鋭いご指摘を頂い た，東京工業大学山北昌毅准教授および広島大学原田祐志助教に心より感謝致します．学 外の研究室にかかわらず，数多くのご質問に回答してくださった東京工業大学山北研究室 の花澤雄太氏に心より感謝致します．また，円滑に研究活動を行えるよう数多くのサポー トをしてくださった吉高研究室の板宮吉宏氏に心より感謝致します．最後に，ここまで自 分を健康に育て支えてくれた両親に深く感謝致します．

参考文献

[1] T. McGeer:“Passive dynamic walking,” Int. J. of Robotics Research, vol. 9, no. 2, pp. 62–82, 1990.

[2] T. McGeer:“Passive walking with knees,” Proc. of the IEEE Int. Conf. on Robotics and Automation, vol. 3, pp. 1640–1645, 1990.

[3] A. Goswami, B. Thuilot and B. Espiau: “Compass-like biped robot Part I: Stability and bifurcation of passive gaits,” Technical report, INRIA, No. 2996, Oct 1996.

[4] H. Geyer and A. Seyfarth and R. Blickhan:“Compliant leg behaviour explains basic dynamics of walking and running,” Proc. of the Royal Society B, Vol. 273, No. 1603, pp. 2861–2867, 2006.

[5] F. Asano and J. Kawamoto, ”Passive dynamic walking of viscoelastic-legged rimless wheel,” Proc. of the 2012 IEEE International Conference on Robotics and Automa-tion, pp. 2331–2336, May 16, 2012.

[6] J. M. Font-Llagunes and J. Kovecses: “Dynamics and energetics of a class of bipedal walking systems,” Mechanism and Machine Theory Vol. 44, pp. 1999–2019, 2009.

[7] J. M. Bourgeot, C. Canudas-de-Wit and B. Brogliato: “Impactt shaping for double support walk: from the rocking block to the biped robot,” Proc. of the Int. Cof. on Climbing and walking Robots, pp. 509–516, 2005.

[8] F. Asano and Z.W. Luo:“Energy-efficient and high-speed dynamic biped locomotion based on principle of parametric excitation,” IEEE Trans. on Robotics, Vol. 24, No.

6, pp. 1289–1301, 2008.

[9] F. Asano:“Stability analysis of passive compass gait using linearized model,” Proc.

of the IEEE Int. Conf. on Robotics and Automation, pp. 557–562, 2011.

[10] J. C. Wall, D. B. Hogan, G.I. Turnbull and R. A. Fox:“The kinematics of idiopathic gait diisorder: A comparison with healthy young and elderly females,” Scand. J.

Med., Vol. 23, pp159–164, 1991.

[11] Y. Barak, R. C. Wagenaar and K. G. Holt:“Gait characteristics of elderly people with a history of falls: a dynamic approach,” Physical Therapy, Vol. 86, pp. 1501–1510, 2006.

[12] 山崎昌廣，佐藤陽彦：“ヒトの歩行–歩幅，歩調，速度，およびエネルギー代謝の観点 から–”，人類学雑誌，Vol. 98，No. 4，pp. 385–401，1990．

[13] 山海嘉之，鍋嶌厚太，河本浩明：“ロボットスーツHALの安全技術”， 日本ロボット 学会誌， Vol. 29，No. 9， pp. 780–782，2011．

付 録 A _{運動方程式の詳細}

第2章式(2.1)における行列M(q)，h(q,q)˙ の詳細な項について述べる．

M(q)について，

M(q) =









M₁₁ M₁₂ M₁₃ M₁₄ M₁₅ M₂₂ M₂₃ M₂₄ M₂₅ M₃₃ M₃₄ M₃₅ M₄₄ M₄₅

Sym. M₅₅









(A.1)

とおいたとき，各項の詳細は以下通りである．

M₁₁ = 8m+m_H M₁₂ = 0

M₁₃ = (am+L₁(7m+m_H)) cosθ₁+L₂(7m+m_H) cos(θ₁+θ₂)−m(L₂cos(θ₁+θ₂+α) +bcos(θ₁+θ₂+θ₃ +α) + 2(cos(α/2) + cos(3α/2) + cos(5α/2))(L₂cos(θ₁+θ₂ +9α/2) +bcos(θ₁+θ₂+ 9α/2 +β))

M14 = L2(7m+mH) cos(θ1 +θ2)−m(L2cos(θ1+θ2+α) +bcos(θ1+θ2+θ3+α) +2(cos(α/2) + cos(3α/2) + cos(5α/2))(L₂cos(θ₁+θ₂+ 9α/2) +bcos(θ₁+θ₂ +9α/2 +β)))

M₁₅ = −bmcos(θ₁+θ₂+θ₃+α) M₂₂ = 8m+m_H

M₂₃ = −(am+L₁(7m+m_H)) sinθ₁ −L₂(7m+m_H) sin(θ₁+θ₂) +m(L₂sin(θ₁+θ₂+α) +bsin(θ₁+θ₂+θ₃+α) + 2 cos(α/2 + cos(3α/2) + cos(5α/2))(L₂sin(θ₁+θ₂+α/2) +bsin(θ1+θ2+α/2 +β))

M₂₄ = −L₂(7m+m_H) sin(θ₁ +θ₂) +m(L₂sin(θ₁+θ₂+α) +bsin(θ₁+θ₂+θ₃+α) +2(cos(α/2) + cos(3α/2) + cos(5α/2)))(L₂sin(θ₁+θ₂+ 9α/2) +bsin(θ₁+θ₂ +9α/2 +β))

M₂₅ = bmsin(θ₁ +θ₂+θ₃+α)

M₃₃ = (a²+ 7(b²+L²₁+ 2L²₂))m+ (L²₁+L²₂)m_H + 2L₁L₂(7m+m_H) cosθ₂+ 2bL₂mcosθ₃

−2m(L²₂cosα+L²₂cos 2α+L²₂cos 3α+L²₂cos 4α+L²₂cos 5α+L²₂cos 6α

+L²₂cos 7α+L₁L₂cos(θ₂+α) +bL₂cos(θ₃+α) +bL₁cos(θ₂ +θ₃+α) +L₁L₂cos(θ₂ +2α) +L₁L₂cos(θ₂+ 3α) +L₁L₂cos(θ₂+ 4α) +L₁L₂cos(θ₂+ 5α) +L₁L₂cos(θ₂ +6α) +L₁L₂cos(θ₂+ 7α)−6bL₂cosβ+bL₂cos(2α+β) +bL₁cos(θ₂+ 2α+β) +bL₂

×cos(3α+β) +bL₁cos(θ₂+ 3α+β) +bL₂cos(4α+β) +bL₁cos(θ₂+ 4α+β) +bL₂

×cos(5α+β) +bL₁cos(θ₂+ 5α+β) +bL₂cos(6α+β) +bL₁cos(θ₂+ 6α+β) +bL₂

×cos(7α+β) +bL₁cos(θ₂+ 7α+β))

M34 = 7b²m+L²₂(14m+mH) +L1L2(7m+mH) cosθ2−m(−2bL2cosθ3 + 2L²₂cosα+ 2L²₂

×cos 2α+ 2L²₂cos 3α+ 2L²₂cos 4α+ 2L²₂cos 5α+ 2L²₂cos 6α+ 2L²₂cos 7α+L₁L₂

×cos(θ₂+α) + 2bL₂cos(θ₃+α) +bL₁cos(θ₂+θ₃+α) +L₁L₂cos(θ₂+ 2α) +L₁L₂

×cos(θ₂+ 3α) +L₁L₂cos(θ₂+ 4α) +L₁L₂cos(θ₂+ 5α) +L₁L₂cos(θ₂+ 6α) +L₁L₂

×cos(θ₂+ 7α)−12bL₂cosβ+ 2bL₂cos(2α+β) +bL₁cos(θ₂+ 2α+β) + 2bL₂cos(3α +β) +bL₁cos(θ₂+ 3α+β) + 2bL₂cos(4α+β) +bL₁cos(θ₂+ 4α+β) + 2bL₂cos(5α +β) +bL₁cos(θ₂+ 5α+β) + 2bL₂cos(6α+β) +bL₁cos(θ₂+ 6α+β) + 2bL₂cos(7α +β) +bL1cos(θ2+ 7α+β))

M₃₅ = bm(b+L₂cosθ₃−L₂cos(θ₃+α)−L₁cos(θ₂+θ₃+α))

M₄₄ = 7b²m+L²₂(14m+m_H)−2L₂m(L₂(cosα+ cos 2α+ cos 3α+ cos 4α+ cos 5α+ cos 6α + cos 7α) +b(−cosθ₃+ cos(θ₃+α)−6 cosβ))−4bL₂m(cos(α/2) + cos(3α/2)

+ cos(5α/2)) cos(9α/2 +β) M₄₅ = bm(b+L₂cosθ₃−L₂cos(θ₃+α)) M₅₅ = b²m

h(q,q)˙ についても，

h(q,q) =˙







 h₁ h₂ h3

h₄ h₅









(A.2)

とおいたとき，各項の詳細は以下通りである．

h₁ = −θ˙²₁(am+L₁(7m+m_H)) sinθ₁−( ˙θ₁+ ˙θ₂)²L₂(7m+m_H) sin(θ₁ +θ₂) +m(( ˙θ₁+ ˙θ₂)²

×L2sin(θ1+θ2+α) +b( ˙θ1+ ˙θ2+ ˙θ3)²sin(θ1+θ2+θ3 +α) + 2( ˙θ1+ ˙θ2)²

×(cos(α/2) + cos(3α/2) + cos(5α/2))(L₂sin(θ₁+θ₂+ 9α/2) +bsin(θ₁ +θ₂+ 9α/2 +β)) h₂ = 8gm+gm_H −θ˙₁²(am+L₁(7m+m_H)) cosθ₁−( ˙θ₁+ ˙θ₂)²L₂(7m+m_H) cos(θ₁+θ₂)

+m(( ˙θ₁+ ˙θ₂)²L₂cos(θ₁+θ₂+α) +b( ˙θ₁+ ˙θ₂+ ˙θ₃)²cos(θ₁+θ₂+θ₃+α) + 2( ˙θ₁ + ˙θ₂)²

×(cos(α/2) + cos(3α/2) + cos(5α/2))(L₂cos(θ₁+θ₂+ 9α/2) +bcos(θ₁+θ₂+ 9α/2 +β)) h₃ = −θ˙₂(2 ˙θ₁+ ˙θ₂)L₁L₂(7m+m_H) sinθ₂−gm_H(L₁sinθ₁ +L₂sin(θ₁+θ₂)) +m(−bθ˙₃(2( ˙θ₁

+ ˙θ₂) + ˙θ₃)L₂sinθ₃+ ˙θ₂(2 ˙θ₁+ ˙θ₂)L₁L₂sin(θ₂+α) + 2bθ˙₁θ˙₃L₂sin(θ₃+α) +2bθ˙2θ˙3L2sin(θ3+α) +bθ˙²₃L2sin(θ3+α) + 2bθ˙1θ˙2L1sin(θ2+θ3+α) +bθ˙₂²L₁sin(θ₂+θ₃+α) + 2bθ˙₁θ˙₃L₁sin(θ₂+θ₃+α) + 2bθ˙₂θ˙₃L₁sin(θ₂

+θ₃+α) +bθ˙²₃L₁sin(θ₂+θ₃+α) + 2 ˙θ₁θ˙₂L₁L₂sin(θ₂+ 2α) + ˙θ²₂L₁L₂sin(θ₂+ 2α) +2 ˙θ₁θ˙₂L₁L₂sin(θ₂ + 3α) + ˙θ²₂L₁L₂sin(θ₂+ 3α) + 2 ˙θ₁θ˙₂L₁L₂sin(θ₂+ 4α)

+ ˙θ²₂L₁L₂sin(θ₂+ 4α) + 2bθ˙₂(2 ˙θ₁ + ˙θ₂)L₁(cos(α/2) + cos(3α/2) + cos(5α/2)) cosβsin(θ₂ +9α/2) + 2 ˙θ₁θ˙₂L₁L₂sin(θ₂+ 5α) + ˙θ²₂L₁L₂sin(θ₂+ 5α) + 2 ˙θ₁θ˙₂L₁L₂sin(θ₂ + 6α)

+ ˙θ²₂L₁L₂sin(θ₂+ 6α) + 2 ˙θ₁θ˙₂L₁L₂sin(θ₂+ 7α) + ˙θ²₂L₁L₂sin(θ₂+ 7α) + 2bθ˙₂(2 ˙θ₁+ ˙θ₂)

×L1(cos(α/2) + cos(3α/2) + cos(5α/2) cos(θ2+ 9α/2) sinβ) +gm(−(a+ 7L1) sinθ1

−7L₂sin(θ₁+θ₂) +L₂(1 + 2 cosα+ 2 cos 2α+ 2 cos 3α) sin(θ₁+θ₂+ 4α) +b(sin(θ₁ +θ₂+θ₃+α) + 2(cos(α/2) + cos(3α/2) + cos(5α/2)) sin(θ₁+θ₂+ 9α/2 +β))) h₄ = ˙θ²₁L₁L₂(7m+m_H) sinθ₂−gL₂m_Hsin(θ₁+θ₂) +m(−b(−θ˙₃(2( ˙θ₁+ ˙θ₂) + ˙θ₃)L₂(−1

+ cosα) + ˙θ²₁L₁cos(θ₂+α)) sinθ₃+bcosθ₃( ˙θ₃(2( ˙θ₁+ ˙θ₂) + ˙θ₃)L₂sinα−θ˙²₁L₁sin(θ₂+α)) + ˙θ²₁L₁(−L₂(1 + 2 cosα+ 2 cos 2α+ 2 cos 3α) sin(θ₂+ 4α)−2b(cos(α/2) + cos(3α/2) + cos(5α/2)) sin(θ₂+ 9α/2 +β))) +gm(−7L₂sin(θ₁+θ₂) +bsin(θ₁+θ₂+θ₃+α) +L₂(1 +2 cosα+ 2 cos 2α+ 2 cos 3α) sin(θ1+θ2+ 4α) + 2b(cos(α/2) + cos(3α/2) + cos(5α/2)) sin(θ1+θ2+ 9α/2 +β))

h₅ = bm(( ˙θ₁ + ˙θ₂)²L₂(sinθ₃−sin(θ₃+α))−θ˙₁²L₁sin(θ₂+θ₃ +α) +gsin(θ₁+θ₂ +θ₃+α))

次に，第3章式(3.1)における行列M(q)，h(q,q)˙ の詳細な項について述べる．

M(q)について，

M(q) =













M₁₁ M₁₂ M₁₃ M₁₄ M₁₅ M₁₆ M₂₂ M₂₃ M₂₄ M₂₅ M₂₆ M₃₃ M₃₄ M₃₅ M₃₆ M₄₄ M₄₅ M₄₆ M55 M56

Sym. M₆₆













(A.3)

とおいたとき，各項の詳細は以下通りである．

M₁₁ = 2(m₁+m₂) +m_H M12 = 0

M13 = (a1m1+L1(m1+ 2m2+mH)) cosθ1

M₁₄ = (a₂m₂+L₂(m₁+m₂+m_H)) cosθ₂ M₁₅ =−(L₂m₁+b₂m₂) cosθ₃

M₁₆ =−b₁m₁cosθ₄ M21 = 0

M22 = 2(m1+m2) +mH

M23 =−(a1m1+L1(m1+ 2m2+mH)) sinθ1

M₂₄ =−(a₂m₂+L₂(m₁+m₂+m_H)) sinθ₂ M₂₅ = (L₂m₁+b₂m₂) sinθ₃

M26 =b1m1sinθ4

M33 =a²₁m1+L²₁(m1+ 2m2+mH)

M34 =L1(a2m2+L2(m1+m2+mH)) cos(θ1−θ2) M₃₅ =−L₁(L₂m₁+b₂m₂) cos(θ₁−θ₃)

M₃₆ =−b₁L₁m₁cos(θ₁−θ₄) M44 =a²₂m2+L²₂(m1+m2+m_H) M45 =−L2(L2m1+b2m2) cos(θ2−θ3) M46 =−b1L2m1cos(θ2−θ4)

M₅₅ =L²₂m₁+b²₂m₂ M₅₆ =b₁L₂m₁cos(θ₃−θ₄) M₆₆ =b²₁m₁

h(q,q)˙ についても，

h(q,q) =˙









 h1

h2

h₃ h4

h5

h₆











(A.4)

とおいたとき，各項の詳細は以下通りである．

h₁ = −θ˙₁²(a₁m₁+L₁(m₁+ 2m₂+m_H)) sinθ₁−θ˙₂²(a₂m₂+L₂(m₁+m₂+m_H)) sinθ₂ + ˙θ₃²(L₂m₁+b₂m₂) sinθ₃+b₁θ˙²₄m₁sinθ₄

h₂ = g(2(m₁+m₂) +m_H)−θ˙²₁(a₁m₁+L₁(m₁+ 2m₂+m_H)) cosθ₁−θ˙²₂(a₂m₂ +L2(m1+m2+mH)) cosθ2+ ˙θ²₃(L2m1+b2m2) cosθ3+b1θ˙₄²m1cosθ4

h3 = −g(a1m1+L1(m1+ 2m2+mH)) sinθ1+L1( ˙θ²₂(a2m2+L2(m1+m2+mH))

×sin(θ₁−θ₂)−θ˙₃²(L₂m₁+b₂m₂) sin(θ₁−θ₃)−b₁θ˙²₄m₁sin(θ₁−θ₄))

h₄ = −θ˙₁²L₁(a₂m₂+L₂(m₁+m₂+m_H)) sin(θ₁−θ₂)−g(a₂m₂+L₂(m₁+m₂+m_H)) sinθ₂−L₂( ˙θ²₃(L₂m₁+b₂m₂) sin(θ₂−θ₃) +b₁θ˙²₄m₁sin(θ₂−θ₄))

h5 = (L2m1+b2m2)( ˙θ²₁L1sin(θ1−θ3) + ˙θ₂²L2sin(θ2−θ3) +gsinθ3) +b1θ˙₄²L2m1

×sin(θ3−θ4)

h6 = b1m1( ˙θ₁²L1sin(θ1−θ4) + ˙θ²₂L2sin(θ2−θ4)−θ˙₃²L2sin(θ3−θ4) +gsinθ4)

最後に，第4章式(4.1)における行列M(q)，h(q,q)˙ の詳細な項について述べる．

M(q)について，

M(q) =













M₁₁ M₁₂ M₁₃ M₁₄ M₁₅ M₁₆ M₁₇ M22 M23 M24 M25 M26 M27

M33 M34 M35 M36 M37

M₄₄ M₄₅ M₄₆ M₄₇ M55 M56 M57

M66 M67

Sym. M₇₇













(A.5)

とおいたとき，各項の詳細は以下通りである．

M11 = 2m1+ 2m2+mt+mH

M₁₂ = 0

M₁₃ = (a₁m₁+L₁(m₁+ 2m₂+m_t+m_H)) cosθ₁ M₁₄ = (a₂m₂+L₂(m₁+m₂+m_t+m_H)) cosθ₂ M15 = −(L2m1+b2m2) cosθ3

M16 = −b1m1cosθ4

M17 = Ltmtcosθ5

M₂₂ = 2m₁+ 2m₂+m_t+m_H

M₂₃ = −(a₁m₁+L₁(m₁+ 2m₂+m_t+m_H)) sinθ₁ M24 = −(a2m2+L2(m1+m2+mt+m_H)) sinθ2

M25 = (L2m1+b2m2) sinθ3

M26 = b1m1sinθ4

M₂₇ = −L_tm_tsinθ₅

M₃₃ = a²₁m₁+L²₁(m₁+ 2m₂+m_t+m_H)

M34 = L1(a2m2+L2(m1+m2+mt+m_H)) cos(θ1−θ2) M35 = −L1(L2m1+b2m2) cos(θ1−θ3)

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