重力方向に基づくコントローラの向き決定方法

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1 ( ) 2/Sep 09 1 ( ) ( ) 3 2 X w, Y w, Z w +X w = +Y w = +Z w = 1 X c, Y c, Z c X c, Y c, Z c X w, Y w, Z w Y c Z c X c 1: X c, Y c, Z c Kentaro 1

2 M M v 0, v 1, v 2 v 0 v 0x v 0y v 0x v 0y v 0z M v 1x v 1y v 1z (1) v 2x v 2y v 2z, v 1 v 1x v 1y, v 2 v 2x v 2y (2) v 0z v 1z v 2z M 3 ( ) M v T 0 v T 1 v T 2 (3) M ( ) M 1 = M T M 1 M 1 3 v 0x v 1x v 2x ] M 1 v 0y v 1y v 2y [v 0 v 1 v 2 v 0z v 1z v 2z e 0, e 1, e e 0 0, e 1 1, e 2 0 (5) 0 0 M 1 e 0 = v 0 M 1 e 1 = v 1 M 1 e 2 = v 2 (6) M v 0 = e 0 M v 1 = e 1 M v 2 = e 2 (7) v 0, v 1, v 2 e 0, e 1, e 2 ( +X w, +Y w, +Z w ) 1 (4) v u v u = 1 v u v u u x u y u z (8) 2

3 ( ) a = [a x a y a z ] T a v u = a/ a 3.2 v u M v u M v u = e 1 (9) v u = M 1 e 1 = v 1 (10) M 1 v 1 v u M (v 0 v 2 ) 3.3 v u M ( ) +X c (X c X c 0 ) X w Y w X w 0 M v 1 = v u (11) v 2x = 0, v 2y = u z, v 2z = u 2 y + u 2 z u y u 2 y + u 2 z (12) v 0 = v 1 v 2 (13) 3

4 X c Z w Y w Y c Z c Z w v u ±X c +X c 4.2 ( ) +Z c Y w Z w Z w 0 M v 1 = v u (14) v 0x = u y u 2 x + u 2 y, v 0y = u x, v 0z = 0 (15) u 2 x + u 2 y v 2 = v 0 v 1 (16) Z c X w Y w X c Y c X w v u ±Z c +Z c 5 2 : M : M 6 2 v u v u X w, Y w, Z w Y w Y w v u X w, Z w v u v u 3 v u = 1 v u 1 S X w, Z w v u S 4

5 Y c S Y c Y w X w X w X w Z w v u X w Z w v u Z w Z w Z c X c Z c X c 2: v u X w, Y w, Z w 3: X w, Z w v u X w, Y w, Z w S X w, Z w v u X w, Z w 4 S X w, Z w S Y c X w Z w Z c X c 4: S 1 ( ) 5 X c +X c S +X w +Z w 2 (+X c X c ) 5

6 2 ( ) 6 +Z c Z c 2 5: 1 ( ) 7 ( 7 ) v a, v b v a v b v a, v b v a v b v a v b ( v a = v b ) ( ) v a v b ( v a = v b ) 6

7 6: 2 ( ) v b v a 7: v a v b 7

8 v a v b M SR (v a, v b ) n 2 xt + cos φ n x n y t n z sin φ n z n x t + n y sin φ M SR (v a, v b ) = n x n y t + n z sin φ n 2 yt + cos φ n y n z t n x sin φ (17) n z n x t n y sin φ n y n z t + n x sin φ n 2 zt + cos φ [n x n y n z ] T φ [n x n y n z ] T = v a v b sin φ (18) sin φ = v a v b (19) cos φ = v a v b (20) t = 1 cos φ (21) v a v b v b v a M SR (v a, v b ) 1 = M SR (v b, v a ) (22) v a, v b 1 k, l M SR (k v a, l v b ) = M SR (v a, v b ) (23) (Fixed point theorem, Hairy ball theorem) f : S 2 S 2 f(p ) = P P S 2 P ( ) P f(p ) f(p ) = P S X w, Z w X w S X w X w S 2 1 8

9 8.2 ( ) M 1. ( ) 2. v u +Y w +Y w e 1 M ( A ) M = M SR (v u, e 1 ) (24) M 1 M 1 = M SR (v u, e 1 ) 1 (25) = M SR (e 1, v u ) (26) e 1 v u 8 Y c 1 v u Y c Y c 8: - ( ) 9

10 8.3 ( ) (8.2) ( ) 1 Y c v r v t = v r (target vector) M 1. ( ) 2. v u v t 3. v t +Y w M ( A ) M = M SR (v t, e 1 ) M SR (v u, v t ) (27) M 1 = M SR (v u, v t ) 1 M SR (v t, e 1 ) 1 (28) = M SR (v t, v u ) M SR (e 1, v t ) (29) ( ) M SR (e 1, v t ) v r 9 Y c +Z c θ = v r v u v r v r Y c θ Z c v r X c 9: v r 9 v p M p 10

11 10: - ( ) v u v u M 3 S X w, Z w X w, Z w v p v u M 1 = M SR (v p, v u ) M 1 p (30) M = M p M SR (v p, v u ) 1 (31) = M p M SR (v u, v p ) (32) 10 11

12 S X w ( Z w ) ( ) X w ( Z w ) 1. X w, Z w 2. (a) X w, Z w (b) X w, Z w 2 ( ) X w, Z w S X w, Z w 10.1 ( 1) ( ) 6 +Z c, Z c 11 S ( ) +Z w +Z w (+Z c ) λ ( ) ( 11 λ = π/3 ) v u X c Y c Z c +Z w +Z c v u ±Z c Z c Z w v u M ( ) M d M d M d +X w, +Z w v d, v e M d v T d v T u v T e (33) 12

13 11: ( 1) u z sin λ S M = M d ( ) M g M g v g, v h +X w, +Z w M g v T g v T u v T h (34) ρ ( ) ρ = π sin 1 u z (35) λ 12 +Z w (+Z w ) v h ±ρ ρ M d v e v h cos ρ M = M d v e 13 ρ, ρ v e v g 0 ρ = ρ 13

14 v g v h ρ +Z w N G v g ρ 0 OK v h ρ ρ N G OK 12: +Z w 13: ρ = ρ v e v h v u ρ M r cos ρ 0 sin ρ M r = (36) sin ρ 0 cos ρ M M 1 = M 1 g M r (37) M = M 1 r M g (38) 10.2 ( 2) ( ) 8 Y c 14 S ( ) +Z w S +Y c P 1 P 1 +Z w (+Z c ) P 1 ( S ) λ ( ) ( 14 λ = π/2 ) v u +Y c +Y c +Z w +Z c v u Y c +Y c Z w v u M ( ) M d M d (33) u y cos λ S M = M d 14

15 14: ( 2) ( ) M g M g (34) ρ ( ) ρ = π λ cos 1 u y (39) +Z w 10.1 ( 1) 12 v e v h cos ρ M = M d v e ( 1)

16 2 1 ( ) A A.1 M SR (v u, e 1 ) 8.2 M SR (v u, e 1 ) K 1 K 1 u 2 z + u y u x K 1 u x u z M SR (v u, e 1 ) = u x u y u z (40) K 1 u x u z u z K 1 u 2 x + u y K 1 = 1 u y u 2 x + u 2 z (41) u y 1 K 1 = 1 u y u 2 x + u 2 z = 1 u y 1 u 2 y = u y (42) A.2 M SR (v t, e 1 ) M SR (v u, v t ) 8.3 v r 9 v t = [0 cos θ sin θ] T M SR (v t, e 1 ) M SR (v u, v t ) K 2 u u 0 K 2 u 1 u s u c K 2 u 1 u c + u s M SR (v t, e 1 ) M SR (v u, v t ) = u x u y u z (43) K 2 u 1 u x K 2 u x u s u z K 2 u x u c + u y 16

17 u 0, u 1, u c, u s, K 2 u 0 = u y cos θ u z sin θ (44) u 1 = u y sin θ + u z cos θ (45) u c = u x cos θ (46) u s = u x sin θ (47) K 2 = 1 u 0 u 2 x + u 2 1 (48) u 0 1 K 2 = 1 u 0 u 2 x + u 2 = 1 u u 2 = 1 (49) u 0 [1] ( ) ( ) 17

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