1B4 SY0004/14/0000-0049 © 2014 SICE ྗֶม༰ʹͮ͘جೋ٭ϩϘοτͷࠨӈํҠಈ੍ޚ ా ༸࢙ ∗1 , ਿ ݪಓ ∗2 Sideway Locomotion Control of Biped Robots Based on Dynamics Morphing Hiroshi Atsuta∗1 and Tomomichi Sugihara∗2 ∗1∗2 Department of Adaptive Machine Systems, Graduate school of engineering, Osaka University 2-1 Yamadaoka, Suita-shi, Osaka 565-0871, Japan This paper presents a biped locomotion control to step sideways based on the framework of the dynamics morphing. Since the proposed controller doesn’t require detailed referential motion trajectories, it enables a robot to walk sideway at arbitrary velocity given at random timing. Sideway locomotion is realized by alternating the velocity following control and the self-excited oscillating control accompanying with the exchange of the supporting foot. Techniques to automatically update the referencial position of COM and phase for the consistent footstep are also proposed. Key Words : Biped robot, Sideway locomotion, COM-ZMP regulator, Dynamics morphing 1. ͡ Ί (8) ʹ ͨɽཱʹطҐ੍ ޚɼఆৗతͳ౿Έସ͑ ೋ٭ϩϘοτɼਓͷੜۭؒ׆ࡂݱͷΑ͏ͳ Έग़͠ (11) ɼલޙาߦ (12) (10) ɼᄉᅀͷ౿ Λ౷߹͢Δ੍ثޚΛఏҊͯ͠ ෳࡶͳ͍͓ͯʹڥߴ͍ҠಈೳྗΛൃ͢شΔ͜ͱ͕ ͍ΔɽຊߘͰ͜ΕΛ֦ு͠ɼཱҐ੍ޚલޙาߦͱ ظ͞Ε͍ͯΔɽϩϘοτͷஔ͔Εͨঢ়گͭʹܗ ౷߹ՄೳͳࠨӈํҠಈ੍ޚΛఏҊ͢Δɽ ͍ͯࣄલʹेͳใ͕ಘΒΕ͍ͯΕɼͦΕΒʹج ࠨӈํҠಈલޙาߦͷͨΊͷ੍ثޚΛͦͷ·· ͍ͮͯৄࡉͳࢀরيಓΛ࣌ࠁͷؔͱͯ͠ٻΊɼൺֱ ద༻͢Δ͜ͱͰ࣮͢ݱΔ͜ͱͰ͖ͳ͍ɽͳͥͳΒલ త༰қʹϩϘοτʹߴͳ࡞ۀΛߦΘͤΔ͜ͱ͕Ͱ͖ ͋ͰํޙΕৗʹඪͷํʹΛ౿Έग़͢͜ ɽ͔͠͠ෳࡶͳ͍͓ͯʹڥ࿏໘ܗঢ়͜ىΓ ͱ͕Ͱ͖Δ͕ɼࠨӈํͰͷަࠩΛݶ͍ͳ͞ڐΓ ͏Δ֎ཚʹ͍ͭͯࣄલʹΔ͜ͱ͘͠ɼෆ࣮֬͞ ৗʹඪͷํʹΛ౿Έग़͢͜ͱෆՄೳ͔ͩ ʹຬͪͨڥதΛҠಈ͢Δํ๏ͱͯ͠ݶք͕͋Δɽ ΒͰ͋Δɽ͕ͨͬͯ͠ɼࢦྩ͞Εͨํʹରͯ͠ (1) (2) Δ ࢀরيಓͷܭըΛϑΟʔυόοΫϧʔϓʹΈࠐΉ Ұݮͨ͠ޙɼ࠶ͼՃͯ͠ඪʹैͤ͞ͳ ͜ͱͰ֎ཚࢦྩͷมߋʹରԠ͢Δํ๏ɼ͜Ε· ͚ΕͳΒͳ͍ɽ͜ͷෳࡶͳ੍ޚΛࢀরيಓͷࣄલܭ (3)∼(5) Ͱͷͱ͜ΖՌΛ͍͋͛ͯΔ ɽ͔͜͠͠ΕΒͷํ ըʹཔΒͣঢ়ଶϑΟʔυόοΫͷΈʹΑ࣮ͬͯ͢ݱΔ ๏ɼݩͷࢀরيಓͱͷ͕ࠩͳΔ͘খ͘͞ͳΔΑ͏ ͜ͱ͕ຊߘͷతͰ͋Δɽ͜Ε (i) ࢧ࣋ঢ়ଶʹͮج ʹيಓΛमਖ਼͢ΔͷͰɼࢀরيಓΛେ͖͘֎Εͳ͚Ε ࣗ͘ྭৼಈ੍ͱثޚै੍ثޚͷΓସ͑ɼ(ii) ͳΒͳ͍Α͏ͳఆ֎ͷঢ়ʹگॊೈʹৼΔ͏͜ͱ ै੍ޚͷมʹܗಉͨ͠ظӡ੍ޚɼ(iii) ͕Ͱ͖ͳ͍ɽ ै੍͍͓ͯʹޚഁ͠ͳ͍ঢ੍߱ޚɼͷࡾऀʹ ࢀরيಓʹཔΒͳ੍͍ํޚ๏ఏҊ͞Ε͍ͯΔ (6) (7) ΑͬͯՄೳͱͳΔɽ ͕Λ৮ͱ͢Δͱ੍͍͕ͬͨ͋Δɽ͞Βʹਂࠁ 2. ྗֶม༰ʹثޚ੍ͮ͘جઃܭ ͳͷɼ͜ΕΒͷํ๏ࠓͷͱ͜ΖཱҐ੍ͱޚซ༻Ͱ ͖ͳ͍ɽSugihara (8)∼(12) ɼϩϘοτͷӡಈΛϩϘοτ ࣗͷμΠφϛΫεͱ੍ثޚͷͳ͢ྗֶͯ͠ͱܥઃ ͠ܭɼྡ͢ΔӡಈؒͷભҠΛ੍ثޚͷ࿈ଓతมʹܗ Α࣮ͬͯ͢ݱΔɼྗֶม༰ͱ͍͏ΈΛఏҊ͖ͯ͠ ∗1 ∗2 େࡕେֶେֶӃֶڀݚՊೳɾػೳֶઐ߈ʢ˟ 5650871 େࡕਧాࢢࢁాٰ 2-1ʣ[email protected] ಉ্ [email protected] ຊߘͰϩϘοτͷશ࣭ྔ͕ॏ৺ʹूதͨ͠Ϟσϧ Λ༻͍ΔɽFig. 1 ͷΑ͏ʹɼϩϘοτͷਐߦํʹ x ࣠ɼࠨํʹ y ࣠ɼԖ্ํʹ z ࣠ΛͱΔɽ؆୯ͷ ͨΊɼॏ৺·ΘΓʹൃੜ͢ΔτϧΫແࢹ͠ɼॏ৺ߴ ͞Ұఆͱ͢Δɽ͜ͷ؆୯Խఆੑతʹମ෦ʹ ൺͯ٭෦͕े͚ܰΕଥͰ͋Δɽॏ৺ҐஔΛ p = [x y z]T ɼZMP ҐஔΛ p Z = [xZ yZ zZ ]T ͱ͢Δͱɼॏ 第 19 回ロボティクスシンポジア（2014 年 3 月 13 日 -14 日・兵庫） － 49 － (a) (b) (c) (d) Fig. 3 Position of equilbrium point Fig. 1 COM-ZMP model Ͱ͋Γɼd x, d y ࢀরॏ৺Ґஔɼd vx લํޙͷࢀ ৺ͷӡಈํఔࣜ࣍ࣜͷΑ͏ʹද͞ΕΔɽ x¨ = ω 2 (x − xZ ) √ রɼఆ qx (≥ 0), qy (≥ 0), k(> 0), r(> 0) ͓Αͼ (1) y¨ = ω 2 (y − yZ ) (2) g/zɼॏྗՃ g = 9.8[m/s2 ] Ͱ͋Δɽ ͜ΕɼZMP Ґஔ p Z Λ੍ޚೖྗͱͯ͠ɼॏ৺Ґஔ p ͨͩ͠ɼω ≡ (13) Λ੍ޚՄೳͰ͋Δ͜ͱΛ͍ࣔͯ͠Δ ɽͨͩ͠ɼZMP Ґஔ p Z ࢧ࣋ྖҬ S ʹؔͯ࣍͠ͷ߆ଋ݅Λຬͨ͢ ඞཁ͕͋Δɽ pZ ∈ S (3) ͜ͷ߆ଋ݅Λຬͨͭͭ͠٭ҠಈΛ࣮͢ݱΔͨΊʹɼ ϩϘοτࢧ࣋ྖҬͷൣғͰ ZMP Λૢ࡞ͯ͠ॏ৺ Λ੍͢ޚΔ͜ͱͱࢧ࣋ྖҬΛෆ࿈ଓʹม͢ܗΔ͜ͱΛ ཱ྆ͤ͞ͳ͚ΕͳΒͳ͍ɽಛʹೋ٭ϩϘοτɼࢧ ࣋ྖҬͷมܗΛࠨӈҰରͷͷ౿Έସ͑ʹΑͬͯߦ͏ɽ Sugihara(12)ཱҐ੍ޚɼఆৗతͳ౿Έɼલޙਐา ρ (≥ 0) ઃܭύϥϝʔλͰ͋Δɽ·ͨࢧ࣋ྖҬ S x ํͷดྖҬ [xZmin , xZmax ] ͓Αͼ y ํͷดྖҬ [yZmin , yZmax ] Ͱද͞ΕΔͱ؆୯Խ͍ͯ͠Δɽ͜ͷ੍ޚ ࡍ࣮͕ͯͬͨ͠ʹثͷ ZMP ͕ඪ ZMP ʹΑ͘ै ͢ΔͳΒɼx, y ํͷॏ৺ӡಈͦΕͧΕ࣍ͷ۠ ࣗ͏ैʹܥɽ ω 2 x − ω 2 xZmax (T1) d 2 d x¨ = −ω (qx + 1)(x˙ − vx ) − ω qx (x − x) (T2)(10) ω 2 x − ω 2 x (T3) Zmin ω 2 y − ω 2 yZmax (S1) 2 d y¨ = −ω (qy + 1) f (ζ )y˙ − ω qy (y − y) (S2) (11) ω 2 y − ω 2 y (S3) Zmin ࣜ (10) ͓Αͼ (11) ʹै͏ॏ৺ͷৼΔ͍Λ૬ਤΛ ༻͍ͯઆ໌͢Δɽ·ͣঢ়໘ͷӡಈɼ͢ͳΘͪ x ํ ߦΛ౷߹͢Δ੍( ࣜͯ͠ͱثޚ1)(2) ͷඪ ZMP Ґஔ ͷӡಈʹ͍ͭͯड़ΔɽલʹํޙҠಈͤ͞ΔͨΊ xZ ͓Αͼ yZ Λ࣍ࣜͰఆΊΔ͜ͱΛఏҊͨ͠ɽ ) ( x˙ − d vx x˜Z = d x + (qx + 1) x − d x + ω x (T1 : x˜Z > xZmax ) Zmax xZ = x˜Z (T2 : xZmin ≤ x˜Z ≤ xZmax ) x (T3 : x˜Z < xZmin ) Zmin ʹॏ৺Λࢀর d vx ʹऩଋͤ͞ΔΑ͏ͳ੍ޚ ) y˙ y˜Z = y + (qy + 1) y − y + f (ζ ) ω yZmax (S1 : y˜Z > yZmax ) yZ = y˜Z (S2 : yZmin ≤ y˜Z ≤ yZmax ) y (S3 : y˜ < y ) d Zmin ͨͩ͠ ( Z { d (4) (5) + 1)2 ζ (qy f (ζ ) ≡ 1 − ρ exp k 1 − r2 √ y˙2 ζ ≡ (y − d y)2 + 2 ω qy (T2) ʹ͓͚Δॏ৺ӡಈ࣍ͷඍํఔࣜʹै͏ɽ x¨ = −ω (x˙ − d vx ) (12) ͜ͷͱ͖ฏߧଘࡏͤͣɼx˙ d vx ʹऩଋ͠Α͏ͱ ͢Δɽd vx > 0 ͳΒঢ়ଶ͍ͣΕ T1 ʹɼd vx < 0 ͳ (6) Βঢ়ଶ͍ͣΕ T3 ʹͦΕͧΕҠߦ͠ɼx˙ ൃࢄ͢ √ ΔɽFig. 2(a) ω = g/0.27, xZmin = −0.4, xZmax = (7) ͷΑ͏ʹยཱͪͷঢ়ଶͰ྆தԝΛฏߧͱ͢Δ 0.55, qx = 0 ʹର͢Δ૬ਤͰ͋ΔɽFig. 3(a) ʹࣔ͢Α͏ ʹঢ়໘ͷӡಈͰɼཱ྆ͪͷঢ়ଶͰ Fig. 3(b) ͜ͱ͕ՄೳͰ͋Δɽͦ͜Ͱ x˙ ͷൃࢄΛ͗ܧଓతʹӡ Zmin 2} ͕ثඞཁͰ͋Δɽࣜ (10) ʹ͓͍ͯ qx = 0 ͷͱ͖ঢ়ଶ (8) ಈΛߦ͏ͨΊʹɼཱҐՄ҆ఆ݅ Λৗʹຬ͢ΔΑ (12) ͏ͳӡ੍͕ޚఏҊ͞Ε͍ͯΔ (8) (9) dv x ɽ Fig. 4 ʹɼࢦྩ ʹΑཱͬͯҐ҆ఆԽ੍͔ثޚΒै੍ʹثޚ ࿈ଓతʹม༰͢Δ༷ࢠΛࣔ͢ɽ ࣍ʹલֹ໘ͷӡಈɼ͢ͳΘͪ y ํͷӡಈʹ √ ͍ͭͯड़ΔɽFig. 2(b) ω = g/0.27, yZmin = － 50 － S1 T1 T2 S1 S2 T3 (a) Velocity follower S2 S3 S3 (b) COM-ZMP regulator (c) Self-excited oscillator Fig. 2 Solution curves of the piecewoise linear autonomous system Fig. 4 Dynamics morphing along x axis Fig. 5 Dynamics morphing along y axis −0.7, yZmax = 0.7, qy = 0.5, d y = 0, k = 1, r = 0.5, ρ = 0 ʹର͢Δ૬ਤͰ͋Δɽͨͩ͠ɼԣ͕࣠ࠨӈํͷॏ ৺Ґஔ yɼॎ͕࣠ॏ৺ y˙ Ͱ͋Δɽρ = 0 ͷͱ͖ࣜ (8) ΑΓ f (ζ ) = 1 (const.) ͱͳΓɼ͜Ε Fig. 3(c) ͷΑ͏ ʹϩϘοτͷॏ৺ΛࢀরҐஔ d y ʹۙऩଋͤ͞Δ࠷ (8) ྑॏ৺-ZMP ϨΪϡϨʔλ ʹҰக͢Δɽ ੍ثޚΛॏ৺-ZMP ϨΪϡϨʔλʹͨ͠··Ͱ ౿ΈΛ͢Δ͜ͱͰ͖ͳ͍ɽͳͥͳΒ Fig. 3(c) ͷΑ͏ ʹॏ৺Λ྆தԝʹۙ҆ఆͤͨ͞ঢ়ଶͰɼยΛ্ ZMP Fig. 6 ͛Δ͜ͱͰ͖ͳ͍͔ΒͰ͋Δɽ͕ͨͬͯ͠ɼยΛ ZMP Difference of COM motion depending on supporting foot ্͛Δʹ Fig. 3(d) ͷΑ͏ʹࢧ࣋٭ଆʹॏ৺ΛҠಈ͞ ͕Βఆৗతʹ౿ΈΛߦ͏੍ʹط͕ޚఏҊ͞Ε͍ͯ ͤΔඞཁ͕͋Γɼఆৗతʹ྆౿ΈΛ͓͜ͳ͏ͨΊʹ Δ ɼॏ৺ʹࣗྭৼಈΛੜͤ͡͞Δඞཁ͕͋Δɽࣜ (8) ૬ਤͷมԽͷ༷ࢠΛ Fig. 5 ʹࣔ͢ɽ Ͱఆٛ͞ΕΔඇઢʹ߲ܗΑͬͯɼρ > e−1 (10) ͷͱ͖ঢ়ଶ (S2) ʹ͓͍ͯ࣍ͷϦϛοταΠΫϧΛൃ͢ݱΔɽ √ (1 + log k ρ r2 ) y˙2 (y − d y)2 + 2 = (13) ω qy (qy + 1)2 r 2π ɼप √ ظͷࣗ ಛʹɼρ = 1 ͷͱ͖ৼ෯ qy + 1 √ ω qy ྭৼಈͱͳΔɽFig. 2(c) ω = g/0.27, yZmin = −0.7, yZmax = 0.7, qy = 0.5, d y = 0, k = 1, r = 0.5, ρ = 1 ʹର͢Δ૬ਤͰ͋Δɽঢ়ଶ (S2) ʹ͓͍ͯϦϛοτα ΠΫϧ͕ܗ͞Ε͍ͯΔɽఆৗঢ়ଶͰɼZMP ৼ ෯ r Ͱॏ৺ͷࣗྭৼಈʹΕͳ͠Ͱಉ͢ظΔɽ͜ͷࣄ ࣮ʹ͍ͯͮجɼZMP ͷ߆ଋ݅Λࣗಈతʹຬ͠ͳ ɽͦͷͱ͖ʹࢧ࣋ྖҬ͕ෆ࿈ଓʹมԽ͢Δ༷ࢠͱ 3. ࠨӈํͷҠಈ੍ޚͷͨΊͷྗֶม༰ 3·1 ͷࢧ࣋ঢ়ଶʹΑΔैৼྭࣗͱܥಈܥ ͷΓସ͑ ॏ৺ͷӡಈํఔࣜ (1)(2) x, y ʹͭ ͍ͯରশͰ͋Δ͕ɼࠨӈํҠಈͷͨΊʹɼલઅͰड़ ͨ x ํͷै੍ثޚΛͦͷ·· y ํʹద༻ ͢Δ͜ͱͰ͖ͳ͍ɽͳͥͳΒલ͋ͰํޙΕৗʹ ඪͷํʹΛ౿Έग़͢͜ͱ͕Ͱ͖Δ͕ɼࠨӈ ํͰɼͷࢧ࣋ঢ়ଶʹΑͬͯͷަࠩΛͳ͞ڐ ͍ݶΓৗʹඪͷํʹΛ౿Έग़͢͜ͱ͕Ͱ͖ ͳ͍͔ΒͰ͋ΔɽFig. 6 ͷΑ͏ʹࢦྩͱͯ͠ d vy > 0 Λ༩͑ͨ߹ɼ͢ͳΘͪࠨํʹҠಈ͢Δ߹Λߟ͑ － 51 － (8) લํޙͷӡɼཱҐՄ҆ఆ݅ ʹ͍ͯͮج ඪ౿Έग़͠Ґஔ d xS Λܾఆ͢ΕΑ͔ͬͨɽ d xS = x + x˙ ω (16) ͔͠͠ࠨӈํʹ͍ͭͯɼॏ৺ৼಈ͍ͯ͠ΔͨΊɼ ཱҐՄ҆ఆ݅ʹͮ͘جඪ౿Έग़͠Ґஔ·ͨৼಈ ͯ͠͠·͏ɽͦ͜Ͱࠨӈํʹ͍ͭͯॏ৺ͷৼಈத ৺ɼͭ·ΓࢀরҐஔ d y ͕྆தԝʹҐஔ͢ΔΑ͏ʹ Fig. 7 Dynamics morphing along y axis with a desired velocity ܾΊΔɽ͢ͳΘͪɼࠨӈํͷඪ౿Έग़͠Ґஔ d yK Λ࣍ͷΑ͏ʹఆΊΔɽ Α͏ɽ͜ͷͱ͖ɼࠨ͕চ͍ͯ͠ΔͳΒै d y K = 2 d y − yP ੍ʹثޚม͢ܗΔ͜ͱͰࠨํʹҠಈ͢Δ͜ͱ͕Ͱ͖ ͨͩ͠ɼK ɼP ͦΕͧΕ༡ࠨ͕٭ͷͱ͖ LɼRɼ༡ Δɽ͔͠͠ɼࠨ͕ணচ͍ͯ͠Δঢ়ଶͰґવै ͕٭ӈͷͱ͖ RɼL Ͱ͋Δɽ (17) ੍͞ࢪ͕ޚΕͨ··Ͱ͋Δͱసͯ͠͠·͏ɽͦ͜Ͱɼ ͜ͷํ๏ʹΑͬͯ྆෯ΛมԽͤ͞Δ͜ͱʹͳΔ͕ɼ ࠨ͕ணচ͍ͯ͠Δͱ͖ɼࣗྭৼಈ੍ʹثޚҠߦ͠ɼ ͦΕʹͬͯࣗྭৼಈͷৼ෯ r Λௐઅ͢Δඞཁ͕͋Δɽ ॏ৺Λݮͤ͞Δඞཁ͕͋Δɽ͜ͷΑ͏ͳ੍ثޚ ͢ͳΘͪɼr ࣍ࣜʹΑͬͯௐઅ͢Δɽ Λ࣮͢ݱΔͨΊʹ y ํͷ੍ثޚΛࣜ (6) ͷΘΓʹ r= ࣍ࣜͰஔ͖͑Δɽ ) ( y˙ − d vy y˜Z = d y + (qy + 1) y − d y + f (ζ ) ω ͨͩ͠ɼ f (ζ ) ࣜ (8) Ͱఆٛ͞Εɼd vy (14) y ํͷࢀ রΛද͢ɽ͜Ε ρ = 0 ͔ͭ qy = 0 ͱͨ͠ͱ͖ 3·3 ঢ੍߱ޚ y pZ ≡ yZ − d y − z∗ ≡ ৼಈ੍ͳʹثޚΔɽͭ·Γɼै੍ৼྭࣗͱثޚ Fig. 7 ʹࢧ࣋٭ͷঢ়ଶʹΑͬͯै੍ࣗͱثޚ ྭৼಈ੍͕ࣗثޚಈతʹΓସΘΔ༷ࢠΛࣔ͢ɽ 3·2 ࢀরॏ৺Ґஔͷߋ৽ͱӡ੍ޚ ै ͷͨΊཱҐ੍ʹޚ෮͢ؼΔࡍɼҎલͷࢀরҐஔ͕ͬ ͍ͯͯ͋ͰݥةΔɽ͜ͷΑ͏ͳໃ६͕ੜ͡ͳ͍Α͏ ʹલํޙͷҠಈ੍ͰޚɼࢀরҐஔΛࡏݱͷॏ৺Ґ ஔɼ͢ͳΘͪ d x = x ͱ͢ΕΑ͔ͬͨɽ͔͠͠ࠨӈํ ʹ͍ͭͯɼࢀরҐஔ d y ॏ৺ͷࣗྭৼಈͷৼಈ த৺ͳͷͰɼࢀরҐஔΛࡏݱͷॏ৺ҐஔͰߋ৽͢Δ͜ = 0 ͱͨ͠ͱ͖ʹ͔ ʹఀࢭɼ·ͨͦͷͰ౿Έ͕ߦ͑ΔΑ͏ɼॏ৺ y˙ ͷਖ਼ෛ͕ d vy ͷਖ਼ෛͱҰக͢Δͱ͖ʹ √ √ 1 + log k ρ d r y = y± qy + 1 (19) 1 h|pZ | σ (ρ¯ )(1 − cos 2πϕ∗ ) 2 r (20) ͨͩ͠ ∗ ʹ L ·ͨ R ͕ೖΔɽϕL ɼϕR pZ ͕࣠ ྖҬʹ͋Δͱ͖ʹஞ࣍ਪఆ͞Εͨ૬ରҐஔ͔ (10) ·ΕͣɼϩϘοτࢀরҐஔͱແؔʹҠಈ͢Δɽͦ y (q¯y + 1)(y˙ − d vy ) i √ ω q¯y Βఆٛ͞ΕΔҐ૬Ͱ͋Δɽৄࡉจݙ ੍͍͓ͯʹޚࢀরҐஔ d x ͋Δ͍ d y ੍ؚ͕ʹثޚ ͱͰ͖ͳ͍ɽͦ͜Ͱɼd v ঢ੍߱ޚΛઆ໌͢Δɽy ํ ͓Αͼ্͛ߴ͞ z∗ ΛͦΕͧΕ࣍ͷΑ͏ʹఆٛ͢Δɽ = 0 ͔ͭ ρ = 1 ͱͨ͠ͱ͖ࣜ (6) ʹҰகͯࣗ͠ྭ ಈ੍ثޚΛΓସ͑ΒΕΔΑ͏ͳ੍ͳͱثޚΔɽ (18) ͷ ZMP Ґஔͱॏ৺ʹΑͬͯఆٛ͞ΕΔෳૉ pZ ࣜ (4) ͱಉ͡ߏΛͱͬͯै੍ͳʹثޚΓɼ dv y L − yR 2 Λࢀর͞Εͨ ͍ɽh ఆͰ͋Γɼσ (ρ¯ ) ࣍ͷΑ͏ʹఆٛ͞ΕΔɽ 1 (ρ¯ > 1) −1 ρ¯ − e (21) σ (ρ¯ ) ≡ (e−1 ≤ ρ¯ ≤ 1) 1 − e−1 0 (0 ≤ ρ¯ < e−1 ) ·ͨɼࣜ (19) ͓Αͼ (21) ͷ q¯y ͓Αͼ ρ¯ ͦΕͧΕ࣍ ͷΑ͏ʹఆٛ͞ΕΔɽ q (q ̸= 0) y y , q¯y ≡ ′ q (q = 0) y y ρ ρ¯ ≡ ρ ′ (ρ ̸= 0) (ρ = 0) (22) 3.1 અͰઆ໌ͨ͠Α͏ʹࢀরΛ༩͑ͯै (15) ͱ͢Δɽͨͩ͠ෳ߹ y˙ ͕ਖ਼ͷͱ͖ +ɼy˙ ͕ෛͷͱ͖ʹ − ͱ͢Δɽࣜ (15) ͷӈลୈ 2 ߲ॏ৺ͷৼ෯Λද͢ɽ ੍ʹثޚΓସ͑Δ߹ɼqy = 0, ρ = 0 ͱ͢ΕΑ͍ ′ ͕ɼͦͷࡍͰͷڍಈ͕߹͢ΔΑ͏ʹ qy ͓Αͼ ′ ρ ͦΕͧΕै੍ʹثޚΓସ͑Δલͷ qy ͓Αͼ ρ ͷϦβʔϒͱ͢Δɽ － 52 － Fig. 8 Snapshots of the movie of the simulation of the locomotion along y axis 4. γϛϡϨʔγϣϯ ΔͨΊͰ͋ΔɽલઅͰड़੍ͨߏ͕ثޚ͢Δྗֶܥ ਓܕϩϘοτ mighty Λఆ͠ɼγϛϡϨʔγϣϯ Λߦͬͨɽ؆୯ͷͨΊɼશ࣭ྔ͕ॏ৺ʹूத͍ͯ͠Δ ϞσϧΛ༻͍ɼॏ৺ߴ͞Λ z = 0.26[m]ɼͦͷଞͷύϥ ϝʔλΛ h = 0.02[m]ɼk = 1 ͱͨ͠ɽ྆ͱલํʹ 0.055[m]ɼ ʹํޙ0.04[m]ɼࠨӈʹͦΕͧΕ 0.035[m] ͷ෯ΛͭํͰܗද͞ݱΕ͍ͯΔɽॳظঢ়ଶʹ͓͍ ͯ྆ؒͷڑ 0.084[m] ͱͨ͠ɽ·֤͕ͨ༡٭ ͱͳΔؒɼඪ౿Έग़͠Ґஔʹै͢ΔΑ͏ͳೋ࣍ Ε੍ثޚΛઃͨ͠ܭɽ x¨∗ = K(d xS − x∗ ) −Cx˙∗ y¨∗ = K(d yK − y∗ ) −Cy˙∗ ʹ͓͍ͯɼॏ৺ϦϛοταΠΫϧͱࢦྩͷେ͖͞ ͷؔʹΑͬͯ૬ਤ্Ͱ Fig. 10 ͷΑ͏ͳيಓΛඳ͘ɽ Fig. 10(a) ͷΑ͏ʹࢦྩ d vy ͕ϦϛοταΠΫϧΑ Γେ͖͍߹ɼ੍ثޚॏ৺͕ै͋ʹܥΔͱ͖ ࢦྩʹऩଋͤ͞Α͏ͱ͢Δ͕ɼࣗྭৼಈʹܥΓ ସΘΔͱϦϛοταΠΫϧʹҾ͖ࠐ͏ͱ͢ΔͨΊɼ ࣮ࡍͷॏ৺ࢦྩʹ౸ୡ͢ΔલʹΛస͞ ͤͯ͠·͏ɽҰํɼFig. 10(b) ͷΑ͏ʹࢦྩ d vy ͕Ϧ ϛοταΠΫϧΑΓখ͍͞߹ಉ༷ʹɼ੍ثޚॏ (23) (24) ৺͕ै͋ʹܥΔͱ͖ʹࢦྩʹऩଋͤ͞Α͏ͱ ͢Δ͕ɼࣗྭৼಈʹܥΓସΘΔͱϦϛοταΠΫϧ ʹҾ͖ࠐ·ΕΔͨΊɼ࣮ࡍͷॏ৺ࢦྩΑΓ ͨͩ͠ ∗ ʹ L ·ͨ R ͕ೖΔɽK = 3000, C = 50 ͱ େ͖ͳΛग़ͯ͠͠·͏ɽ ඪ ZMP ͷࢧ࣋ྖҬ্࠷ۙΛͱΔ͜ͱͰସͨ͠ɽ ͭʹͳΔ͕ଟݟΒΕΔ͕ɼ͜Ε Fig. 10 ͷؙҹ ඍํఔࣜͷղ๏ʹ 4 ࣍ͷ Runge-Kutta ๏Λ༻͍ɼ Ͱࣔ͢Α͏ͳै੍͔ثޚΒࣗྭৼಈ੍ʹثޚ ੵ࣌ؒ෯ 0.01[s] ͱͨ͠ɽ Γସ͑ΔॠؒʹɼϦϛοταΠΫϧʹҾ͖ࠐ͏ͱ͠ ͨ͠ɽࣜ (5) (7) ʹ͓͚Δ؆୯Խ͞Εͨଇɼ ຊߘͰड़੍ͨثޚΛγϛϡϨʔγϣϯ্Ͱ࣮͠ ·ͨɼFig. 9(b) ʹ͓͍ͯ ZMP Ґஔ yZ ͷيಓ͕͍ͼ ͯʹܹٸΛมԽͤ͞ΔͨΊͰ͋Δɽ ͨɽͨͩ͠ϩϘοτͷঢ়ଶʹΑͬͯࢦྩͱؔͳ 5. ͓ ੍͘ޚύϥϝʔλΛ੍ڧతʹมߋ͢Δ͜ͱ͕͋ΔͷͰɼ Θ Γ ʹ ࢦྩ੍ޚύϥϝʔλͱผʹอ͓͖࣋ͯ͠ɼ੍ޚ ࣌ࠁͷؔͱͯ͠ఆٛͨ͠ࢀরيಓΛܭըͤͣʹཱ ͕ثඞཁͳλΠϛϯάͰࢦྩͷΛ੍ޚύϥϝʔλ Ґ੍͔ޚΒࠨӈํͷҠಈ͕ߦ͑Δ੍ثޚΛ։ൃ͠ ͱͯ͠өͤ͞ΔΑ͏ʹͨ͠ɽ ͨɽࠨӈํͷҠಈͷࢧ࣋ঢ়ଶʹΑͬͯॏ৺ ˙ y, y) ˙ = (0.0409, 0, 0, 0)ɼqx = 0.5ɼ ॳظঢ়ଶΛ (x, x, dv y ΛཹΊΔඞཁ͕͋Δ͕ɼै੍ৼྭࣗͱثޚಈ੍ qy = 0.5ɼρ = 0 ͱ͠ɼࢦྩ Λ −3.0 ∼ 3.0[m/s] ͷൣғͰແ࡞ҝʹมߋͨ͠ɽಈ࡞ͷ࠷͍͓ͯʹޙ ثޚΛΓସ͑Δ͜ͱʹΑΓ͜Ε͕Մೳͱͳͬͨɽ dv = 0[m/s] ͱ͠ɼཱҐ੍ʹޚͨ͠ɽಈըεφοϓ γϣοτΛ Fig. 8 ʹɼॏ৺ࢀরɾॏ৺Λϓϩο ߦ͑ΔΑ͏ʹͳͬͨɽࠨӈํࢀর͚ͩͰͳ͘લ τͨ͠άϥϑΛ Fig. 9(a)ɼॏ৺ҐஔɾZMP Ґஔɾઌ ટճΛؚΊͨશํҠಈʹ֦ு͢Δͷ͕ࠓޙͷ՝Ͱ Ґஔɾࢧ࣋ྖҬΛϓϩοτͨ͠άϥϑΛಉਤ (b) ʹͦ ͋Δɽ y ҙͷλΠϛϯάͰඪΛ༩͑ͯഁͤͣӡಈ͕ ࢀํޙর༩͑Δ͜ͱͰࣼΊาߦՄೳʹͳΔɽ ࢀ ߟ จ ݙ ΕͧΕࣔ͢ɽ Fig. 9(a) ʹࣔ͢Α͏ʹࢦྩ d vy Λແ࡞ҝʹมԽ͞ ͤͨʹؔΘΒͣɼಉਤ (b) ͷΑ͏ʹస͢Δ͜ͱͳ͘ ܧଓతʹӡಈΛߦ͍͑ͯΔ͜ͱ͕֬ೝͰ͖ΔɽFig. 9(a) ʹ͓͍ͯॏ৺ y˙ ͕ࢦྩ d vy ʹൺখ͘͞ͳΔ ͱେ͖͘ͳΔ͕͋Δ͕ɼ͜Εࣗྭৼಈ੍ࢦ͕ثޚ ྩ d vy ͱແؔʹϦϛοταΠΫϧ (13) Λܗ͢ (1) A. Takanishi, Y. Egusa, M. Tochizawa, T. Takeya, and I. Kato. Realization of Dynamic Walking Stabilized with Trunk Motion. In ROMANSY 7, pp. 68–79, 1988. (2) K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka. The development of Honda humanoid robot. In Proceedings of the 1998 IEEE International － 53 － reference velocity [m/s] COM limit cycle [s] switching point of the controllers (a) d vy , y˙ [m] supporting region COM (a) The case which d vy is large left foot limit cycle reference velocity ZMP right foot switching point of the controllers [s] (b) y, yZ , yL , yR (b) The case which d vy is small Fig. 9 Results of the simulation (3) (4) (5) (6) (7) (8) Conference on Robotics and Automation, Vol. 2, pp. 1321–1326, 1998. S. Kajita, M. Morisawa, K. Harada, K. Kaneko, F. Kanehiro, K. Fujiwara, and H. Hirukawa. Biped Walking Pattern Generator allowing Auxiliary ZMP Control. In Proceedings of the 2006 IEEE International Conference on Intelligent Robots and Systems, pp. 2993–2999, 2006. K. Nishiwaki and S. Kagami. Online Walking Control System for Humanoids with Short Cycle Pattern Generation. International Journal of Robotics Research, Vol. 28, No. 6, pp. 729–742, 2009. A. Herdt, N. Perrin, and P. B Wieber. Walking without thinking about it. In Proceedings of the 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 190–195, 2010. S.-H. Hyon and T. Emura. Symmetric Walking Control: Invariance and Global Stability. In Proceedings of the 2005 IEEE International Conference on Robotics & Automation, pp. 1455– 1462, 2005. C. Chevallereau, J.W. Grizzle, and Ching-Long Shih. Asymptotically Stable Walking of a FiveLink Underactuated 3-D Bipedal Robot. IEEE Transactions on Robotics, Vol. 25, No. 1, pp. 37– 50, 2009. T. Sugihara. Standing stabilizability and stepping maneuver in planar bipedalism based on the best Fig. 10 Loci of COM in sideway locomotion (9) (10) (11) (12) (13) － 54 － COM-ZMP regulator. In Proceedings of the 2009 IEEE International Conference on Robotics & Automation, pp. 1966–1971, 2009. T. Sugihara. Dynamics morphing from regulator to oscillator on bipedal control. In Proceedings of the 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2940–2945, 2009. T. Sugihara. Consistent biped step control with COM-ZMP oscillation based on successive phase estimation in dynamics morphing. In Proceedings of the 2010 IEEE International Conference on Robotics & Automation, pp. 4224–4229, 2010. T. Sugihara. Reflexive Step-out Control Superposed on Standing Stabilization of Biped Robots. In Proceedings of the 2012 IEEE-RAS International Conference on Humanoid Robots, pp. 741–746, 2012. T. Sugihara. Biped control to follow arbitrary referential longitudinal velocity based on dynamics morphing. In Proceedings of the 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 1892–1897, 2012. K. Mitobe, G. Capi, and Y. Nasu. Control of walking robots based on manipulation of the zero moment point. Robotica, Vol. 18, pp. 651–657, 2000.
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