博士論文
令和元年度
マルチロータの状態変数モデリングと
飛行状態、モータ故障への応用
令和
2 年 3 月
湘南工科大学大学院
工学研究科電気情報工学専攻
磯貝 海斗
1 1 1.1 . . . 1 1.2 . . . 3 2 5 2.1 . . . 5 2.2 . . . 8 3 11 3.1 . . . 11 3.2 . . . 15 4 18 4.1 . . . 20 4.2 . . . 24 5 30 5.1 . . . 32 5.2 . . . 39 6 47 6.1 . . . 50 7 55 7.1 . . . 55 7.2 . . . 56 57 60
A . . . 60 B . . . 60
C . . . 63
64 65
1
1.1
1.1
[1]–[17]
1.1 ELEV-8 Quadcopter Kit
[1] 4 [1], [3]–[12] 6 8 ∗ ∗
• Mueller D’Andrea [18] 1 2
2 3
[19]
• Dongjie i), ii)
i) ii) PNPNPN PPNNPN P N [20] • Saied 4 1 [21] • Yu Dong [22] • Giribet 1 [23] [24], [25] ( [x, y, z, u, v, w]) Kalman [26] Kalman 6 [27] 1 ∗ ∗ “ ”
1950 [28] 1. Kalman 2. 3.
1.2
2 2 [29] 3 3 24 3 1 ∗ 5 6 7 ∗
2
[29] 32.1
[29] 2.1 2.2 R3 3 V ⊂ R3 A3 : a → a + v, a ∈ A3, v ∈ V, a + v ∈ A3. (2.1)Oc+span{E1,E2,E3}, Oc+span{e1,e2,e3}, O + span{e1,e2,e3} [30] (
[31]), ( [31]), ( [31])
w:O + span{e1,e2,e3} [30] O W:
Oc+span{E1,E2,E3} (or Oc+span{e1,e2,e3}) [30] Oc
1 w W 3 Euclid w W
([33], pp. 125–128)
T(t) : W → w, (2.2)
2.1 2 T(t) T(t) W w T(t) [29], p. 124 T(t) B(t) C(t) : T(t) = C(t)B(t), (2.3) C(t)q(t) = r(t) + q(t), (q(t), r(t) ∈ w). 3 T(t) B(t): Oc+span{E1,E2,E3} → Oc+span{e1,e2,e3} t B(t) = B(0) = B, T(t)Q = r(t) + BQ. 4 w W q ∈ w Q q(t) = r(t) + B(t)Q(t) (2.4) Q(t) 2.2 r(t), Ω(t) t 2.2 (w) (W) “ ”˙q Q B 2.4 t [ (2.5)] ˙ q = ˙r + ˙BQ + B ˙Q. (2.5)
5 2.3 ψ, θ, φ(ψ, θ, φ ∈ R) 0 ≤ ψ < 2π, −π2 < θ < π 2, − π 2 < φ < π 2. ψ 5 ψ ψ ψ(t) = (ψ(t) + π) mod 2π − π ei (i = 1, 2, 3) Ei (i = 1, 2, 3) 3 2.3 1. e3 ψ e3 e2 E(−2)2 Rψ 2.7 . 2. E(−2) 2 θ E(−2)2 E(−2)1 E(−1)1 Rθ 2.8 3. E(−1) 1 φ E(−1)1 E(−1)3 E3 Rφ 2.9 E(l)i , i = 1, 2, 3, l = −1, −2 ei(i = 1, 2, 3) Ei(i = 1, 2, 3) 3 e1 E1 e2 E2 e3 E3 ψ, θ, φ [27] 2.3 [29] B(ψ, θ, φ) ∈
SO(3): (ψ, θ, φ) ∈ R3(span{ε1,ε2,ε3}) → R3×3,B(ψ, θ, φ): Oc+span{E1,E2,E3} → Oc+span{e1,e2,e3}
B(ψ, θ, φ) = RψRθRφ =
cos ψ cos θ cos ψ sin θ sin φ − sin ψ cos φ cos ψ sin θ cos φ + sin ψ sin φsin ψ cos θ sin ψ sin θ sin φ + cos ψ cos φ sin ψ sin θ cos φ − cos ψ sin φ
− sin θ cos θ sin φ cos θ cos φ
Rψ=
cos ψ − sin ψ 0sin ψ cos ψ 0
0 0 1 , (2.7) Rθ= cos θ 0 sin θ 0 1 0 − sin θ 0 cos θ , (2.8) Rφ= 1 0 0 0 cos φ − sin φ 0 sin φ cos φ . (2.9) B(ψ, θ, φ) B(ψ, θ, φ)E1 B(ψ, θ, φ)E2 B(ψ, θ, φ)E3 = B(ψ,θ,φ)T e1 e2 e3 , (2.10) ( · )T ( · ) Q ='3 i=1 QiEi∈ W (or Oc+span{E1,E2,E3}), (2.11) B(ψ, θ, φ)Q ='3 i=1 QiB(ψ, θ, φ)Ei∈ Oc+span{e1,e2,e3}, (2.12) 3 ' i=1 QiB(ψ, θ, φ)Ei= 3 ' i=1 Qi( 3 ' j=1 (B(ψ, θ, φ)Ei,ej)ej) = 3 ' i=1 Qi( 3 ' j=1 (Ei,B(ψ, θ, φ)Tej)ej) = 3 ' j=1 ( 3 ' i=1 QiEi,B(ψ, θ, φ)Tej)ej= 3 ' j=1 (Q, B(ψ, θ, φ)Tej)ej= 3 ' j=1 (B(ψ, θ, φ)Q, ej)ej, (2.13) ( · , · ) (2.10)–(2.13) B [32] B (ψ, θ, φ)
2.2
W Q ˙Q = 0 W r = 0 q ˙ q = ˙r + ˙BQ = ˙r + B[Ω, Q] = ˙r + [BΩ, BQ], (2.14) [ · , · ] Ω∈ W Ω Ω =BTω. (2.15)ω∈ w w ω = ˙ψe3+ ˙θE(−2)2 + ˙φE(−1)1 . (2.16) ˙ψ, ˙θ, ˙φ Ei, i = 1, 2, 3 2.16 2.16 2.14 ˙ q = ˙r + ˙ψ ∂ ∂ψ(BQ) + ˙θ ∂ ∂θ(BQ) + ˙φ ∂ ∂φ(BQ). (2.17) ˙Q ! 0 (2.5) t ¨ q = ¨r + ¨BQ + 2 ˙B ˙Q + B ¨Q =¨r + B[Ω, [Ω, Q]] + B[ ˙Ω, Q] + 2B[Ω, ˙Q] + B ¨Q =¨r + [ω, [ω, BQ]] + [ ˙ω, BQ] + 2[ω, B ˙Q] + B ¨Q, (2.18) 2.18 3 “ ” B[ ˙Ω, Q] 2B[Ω, ˙Q] Coriolis B[Ω, [Ω, Q]] h ∈ w H ∈ W ˆI B h = ˆIω = BH ∈ w, (2.19) H = ˆIΩ ∈ W. (2.20) τ∈ w d dth = τ = d dtBH = BT = ˙BH + B ˙H = B[Ω, H] + B ˙H, B( ˙H + [Ω, H] − T) = ˙H + [Ω, H] − T = ˆI ˙Ω + [Ω, H] − T = 0. (2.21) 2.21 Ω ˙Ω
Ω =(− ˙ψ sin θ + ˙φ)E1+( ˙ψ cos θ sin φ + ˙θ cos φ)E2+( ˙ψ cos θ cos φ − ˙θ sin φ)E3, (2.22) ˙Ω = (− ¨ψ sin θ − ˙ψ˙θ cos θ + ¨φ)E1
+( ¨ψ cos θ sin φ − ˙ψ˙θ sin θ sin φ + ˙ψ ˙φ cos θ cos φ + ¨θ cos φ − ˙θ ˙φ sin φ)E2
2.21 τ τ =[r, f]. (2.24) r f H ='3 i=1 HiEi. (2.25) (H1,H2,H3)T = ˆI(Ω1, Ω2, Ω3)T. (2.26) ˆI ˆI = I11 I12 I13 I21 I22 I23 I31 I32 I33 , (2.27) ˆIEk=I1kE1+I2kE2+I3kE3. (2.28) 3.2 ˆI 0 Ikl=0, for k ! l
3
3.1
span{ε1,ε2,ε3} 3 R3 x = (ψ, θ, φ)T x = ( ˙ψ, ˙θ, ˙φ)˙ T (3.1) (3.2) x = ψε1+ θε2+ φε3= ψε1+η, (3.1) ˙ x = ˙ψε1+ ˙θε2+ ˙φε3 = ˙ψε1+˙η, (3.2) η = θε2+ φε3, (3.3) ˙η = ˙θε2+ ˙φε3. (3.4) Ω∈ W r ˙r L L = 1 2m(˙r, ˙r) + 1 2(ˆIΩ, Ω) − mg(r, e3), (3.5) m g ( · , · )3.1.1
B(x)Frot(u) [34]
(B(x)Frot(u), δω) = (B(x)Frot(u), ωx˙(ψ, θ)δ˙x)
=(ω˙x(ψ, θ)TB(x)Frot(u), δ˙x), (3.6) ω˙x(ψ, θ)TB(x)Frot(u) u ∈ Λ ⊂ Rρ, ρ ∈ N ρ [26] x ˙x L = 1 2m(˙r, ˙r) + 1 2(ˆIB(x)Tω(ψ, θ, ˙x), B(x)Tω(ψ, θ, ˙x)) − mg(r, e3) = 1 2m(˙r, ˙r) + 1 2(ˆIB(x)Tωx˙(ψ, θ) · ˙x, B(x)Tωx˙(ψ, θ) · ˙x) − mg(r, e3), (3.7) d dtL˙x(η, ˙x, ¨x) − Lx(η, ˙x) = ω˙x(ψ, θ)TB(x)Frot(u), (3.8) ω(ψ, θ, ˙x) = B(x)Ω(η, ˙x). (3.9) ω(ψ, θ, ˙x) ω˙x(ψ, θ)
ω(ψ, θ, ˙x) = (−˙θ sin ψ + ˙φ cos θ cos ψ)e1+(˙θ cos ψ + ˙φ cos θ sin ψ)e2+( ˙ψ − ˙φ sin θ)e3, (3.10)
ωx˙(ψ, θ) =
0 − sin ψ cos θ cos ψ0 cos ψ cos θ sin ψ
1 0 − sin θ . (3.11) B x ωx˙ ψ θ L˙xk = 1 2(ˆIBTω˙xεk,BTωx˙x) +˙ 1 2(ˆIBTω˙x˙x, BTω˙xεk), (3.12) d dtL˙xk = 1 2(ˆI ˙B T ωx˙εk+ ˆIBT˙ω˙xεk,BTωx˙x) +˙ 12(ˆIBTω˙xεk, ˙BTωx˙x + B˙ T˙ωx˙x + B˙ Tω˙xx)¨ +1 2(ˆI ˙B Tω ˙ xx + ˆIB˙ T˙ω˙x˙x + ˆIBTωx˙x, B¨ Tωx˙εk) + 1 2(ˆIBTωx˙x, ˙B˙ Tω ˙ xεk+BT˙ω˙xεk) =εTkωT˙x˙B ˆIBTωx˙x + ε˙ Tk˙ωT˙xB ˆIBTωx˙x˙
+εTkωTx˙B ˆI ˙BTωx˙x + ε˙ TkωTx˙B ˆIBT˙ωx˙x + ε˙ TkωxT˙B ˆIBTω˙xx,¨ (3.13)
Lxk = 1 2(ˆIBTxkωx˙x + ˆIB˙ Tω ˙ xxkx, B˙ Tω˙xx) +˙ 1 2(ˆIBTωx˙x, B˙ Txkω˙x˙x + B Tω ˙ xxk˙x) =x˙T(ωTx˙Bxk+ωTxx˙ kB) ˆIB Tω ˙ xx,˙ (3.14)
k = 1, 2, 3 ˙B(x, ˙x) ∈ R3×3 B(x) t ˙ω˙x(ψ, θ, ˙ψ, ˙θ) ∈ R3×3 ωx˙(ψ, θ) t [35] (p. 248) B ˙ωT ˙ xB = ΩTx (3.13) εTk˙ωT ˙ xB ˆIBTωx˙x˙ (3.14) Lxk εTk˙ωT ˙ xB ˆIBTω˙x˙x − ˙xT(ωTx˙Bxk +ωTxx˙ kB) ˆIB Tω ˙ xx = 0,˙ (3.15) k = 1, 2, 3.
ωTx˙˙B ˆIBTωx˙x + ω˙ xT˙B ˆI ˙BTω˙xx + ω˙ Tx˙B ˆIBT˙ω˙x˙x + ωT˙xB ˆIBTωx˙x = ω¨ T˙xBFrot(u). (3.16) (ωT ˙ x)−1 (3.16) (3.17) (3.16) (3.18) ˙BT ωx˙˙x = ˙BTω =−BT˙BBTω =−BTB[Ω, Ω] = (0, 0, 0)T, (3.17) ˙B ˆIBTω ˙
xx + B ˆIB˙ T˙ωx˙x + B ˆIB˙ Tωx˙x = BF¨ rot(u), (3.18) (B(x) ˆIB(x)T): x = (ψ, θ, φ) ∈ R3 → R3×3. ( ˙B(x, ˙x) ˆIB(x)T): (x, ˙x) = (ψ, θ, φ, ˙ψ, ˙θ, ˙φ) ∈ R6 → R3×3. 1 1 det(B ˆIBTω ˙ x) = −I11I22I33cos θ ! 0 (3.19) (x, ˙x)T∈ Σ rot⊂ R6 u ∈ Λ ⊂ Rρ d dt ( x ˙ x ) = ( ˙ x
Y(η, ˙x) + Z(η)Frot(u) ) , (3.20) Z(η) = (B(x) ˆIB(x)T· ω ˙ x(ψ, θ))−1· B(x), (3.21) Y(η, ˙x) = −(B(x) ˆIB(x)T· ω ˙ x(ψ, θ))−1( ˙B(x, ˙x) ˆIB(x)T· ωx˙(ψ, θ) +B(x) ˆIB(x)T· ˙ωx˙(ψ, θ, ˙ψ, ˙θ)) · ˙x, (3.22) t ∈ R x = (ψ, θ, φ)T=(ψ, η)T, η = (θ, φ)T, ˙x = ( ˙ψ, ˙θ, ˙φ)T =( ˙ψ, ˙η)T, ˙η = (˙θ, ˙φ)T, ρ∈ N. 1 3.18 3.23 ( I3×3 03×3 03×3 B ˆIBTωx˙ ) d dt ( x ˙ x ) = ( ˙ x
−( ˙B ˆIBTω˙x+B ˆIBT˙ωx˙)˙x + BFrot(u) )
I3×3 3 × 3 03×3 3 × 3 3.23 I3×3 03×3 03×3 B ˆIBTωx˙ 3.23 3.20 [36] det( I03×3 03×3 3×3 B ˆIBTωx˙ ) =det(B ˆIBTωx),˙ (3.24) det(B ˆIBTω ˙
x) = det(B)det( ˆI)det(BT)det(ωx)˙
=−I11I22I33cos θ. (3.25)
det(B) = det(BT) = 1, det( ˆI) = I
11I22I33, det(ωx) = − cos θ˙ 3.25 det( ˆI) ! 0 θ ! π/2 [rad] 3.20 3.21 ψ, ˙x 3.22 ψ
3.1.2
r, ˙r d dtLr(˙r) − Lr = B(x)F˙ tra(u), (3.26) B(x)Ftra(u) ¨r = −ge3+m1B(x)Ftra(u). (3.27)
d dt ( r ˙ r ) = ( ˙ r
−ge3+m1B(x)Ftra(u) ) . (3.28) 2 2 (x(t), ˙x(t))T 1 3.20 d dt ( r ˙ r ) = ( ˙ r
−ge3+m1B(φ1(t, (x0,x˙0)T,u))Ftra(u) ) ; (3.29) (t0,(x0,x˙0)T) ∈ R × Σrot ⊂ R × R6 u ∈ Λ (x(t), ˙x(t))T =(φ1(t, (x0,x˙0)T,u), φ2(t, (x0,x˙0)T,u))T=φ(t, (x0,x˙0)T,u) ∈ Σrot⊂ R6 2 3.28 3.28 B(x) x 3.20 φ1(t, (x0,x˙0)T,u) 3.28 3.29
3.2
[20] 3.1–3.3 3.1 3.2 3.3 3.1–3.3 Fi (i = 1, 2, . . . , 2p, p = 2, 3, 4) Mi (i = 1, 2, . . . , 2p, p = 2, 3, 4) ωi Fi(i = 1, 2, . . . , 2p, p = 2, 3, 4) Mi(i = 1, 2, . . . , 2p, p = 2, 3, 4) [12]2p, p = 2, 3, 4 p = 2: p = 3: p = 4: Fi=kFiω2Mi, i = 1, 2, . . . , 2p, p = 2, 3, 4, (3.30) Mi=kMiω2Mi, i = 1, 2, . . . , 2p, p = 2, 3, 4. (3.31) kF >0 kM >0 p p = 2 p = 3
p = 4 (3.20) Frot(u) (3.29) Ftra(u) Srot2pu2p
Stra2pu2p, p = 2, 3, 4 p = 2 Srot4= 0 −) · kF2 0 )· kF4 )· kF1 0 −) · kF3 0 −kM1 kM2 −kM3 kM4 , (3.32)
Srot4: span{εM1,εM2,εM3,εM4} → span{E1,E2,E3}, (3.33) u4=(ω2M1, ω2M2, ω2M3, ω2M4)T. (3.34) p = 3 Srot6= 0 − √ 3 2 )· kF2 − √ 3 2 )· kF3 0 √ 3 2 )· kF5 √ 3 2 )· kF6 )· kF1 0.5) · kF2 −0.5) · kF3 −) · kF4 −0.5) · kF5 0.5) · kF6 −kM1 kM2 −kM3 kM4 −kM5 kM6 , (3.35) Srot6: span{εM1,εM2, . . . ,εM6} → span{E1,E2,E3}, (3.36) u6=(ω2M1, ω2M2, . . . , ω2M6)T. (3.37) p = 4 Srot8= 0 −√22)· kF2 −) · kF3 − √2 2)· kF4 0 √2 2)· kF6 )· kF7 √2 2 )· kF8 )· kF1 √ 2 2 )· kF2 0 − √ 2 2)· kF4 −) · kF5 − √ 2 2)· kF6 0 √ 2 2 )· kF8 −kM1 kM2 −kM3 kM4 −kM5 kM6 −kM7 kM8 , (3.38)
Srot8: span{εM1,εM2, . . . ,εM8} → span{E1,E2,E3}, (3.39) u8=(ω2M1, ω2M2, . . . , ω2M8)T. (3.40) p = 2, 3, 4 Stra2p= 0 0 · · · 0 0 0 · · · 0 kF1 kF2 · · · kF2p , (3.41)
Stra2p: span{εM1,εM2, . . . ,εM2p} → span{E1,E2,E3}, (3.42) u2p=(ω2M1, ω2M2, . . . , ω2M2p)T, (3.43) span{εM1,εM2, . . . ,εM2p} 2p p = 2, 3, 4 d dt ( x ˙ x ) = ( ˙ x Y(η, ˙x) + Z(η)Srot2pu2p ) , (3.44) d dt ( r ˙ r ) = ( ˙ r −ge3+m1B(φ1(t, (x0,x˙0)T,u2p))Stra2pu2p ) , (3.45) u2p ∈ Λ2p ⊂ R2p
4
∗
4.1
5
4.1 4.1 (i-1) r3 dtd ˙r3>0 (i-2) r3 dtd ˙r3<0 (ii) r3 ˙r1 = ˙r2 = ˙r3 =0, dtd ˙r1 = d dt ˙r2= dtd ˙r3=0, ˙ψ = ˙θ = ˙φ = 0 1 (iii-1) ψ, 0 ≤ ψ < 2π, ˙ψ > 0 (iii-2) ψ, −2π < ψ ≤ 0, ˙ψ < 0 (iv-1) r1 dtd ˙r1>0, θ, 0 ≤ θ < π2, ˙θ >0 (iv-2) r1 dtd ˙r1<0, θ, −π2 < θ≤ 0, ˙θ <0 (v-1) r2 dtd ˙r2<0, φ, 0 ≤ φ < π2, ˙φ >0 (v-2) r2 dtd ˙r2>0, φ, −π2 < φ≤ 0, ˙φ < 0 x = (ψ, θ, φ)T, ˙x = ( ˙ψ, ˙θ, ˙φ)T r = (r1,r2,r3)T, ˙r = ( ˙r1,˙r2,˙r3)T 4.1 4.1 2p p = 2 p = 3 p = 4 u2p 4.1 [ (3.44)] d dt(x, ˙x)T = (03,03)T,03 = (0, 0, 0)T ψ θ φ
[ (3.45)] ¨r x(t) = φ1(t, (x0,x˙0)T,u) ri d dt˙ri =(ei,m1B(x)Stra2pu2p), i = 1, 2 r3 dtd ˙r3 = (e3,−ge3+m1B(x)Stra2pu2p) 6
4.1
6 (xop,x˙op)T∈ R3×R3,xop=(ψop,ηop)T, ηop=(θop, φop)T, (rop,˙rop)T∈ R3× R3 ( x˙ op Z(ηop)Srot2pu2p(op) ) = ( 0 3 03 ) , (4.1) ( ˙ rop(e3,−ge3+m1B(xop)Stra2pu2p(op)) ) = ( 03 c ) , (4.2) Z(ηop) = (B(xop) ˆIB(xop)T· ωx˙(ψop, θop))−1· B(xop), (4.3) ¨r1=(e1,m1B(xop)Stra2pu2p(op)), (4.4) ¨r2=(e2,m1B(xop)Stra2pu2p(op)), (4.5) ¨r3=c, (4.6) c ∈ R 03=(0, 0, 0)T c = 0 (xop,x˙op)T (rop,˙rop)T ¨r1=¨r2 =0 c = 0 (¨r3=0) (xe,˙xe)T (re,˙re)T 3, 4 3 H(u2p,x) u˜2p∈ R2p x ∈ R˜ 3 R3 C1 (4.1) (4.2) H(˜u2p,x) = c˜ H(u2p,x) = (ε i,Z(η)Srot2pu2p) (ej,−ge3+m1B(x)Stra2pu2p) (e3,−ge3+m1B(x)Stra2pu2p) , (4.7) Z(η) = (B(x) ˆIB(x)T· ω ˙ x(ψ, θ))−1· B(x), (4.8) 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, i, j ∈ N, c = (0, ¨rj,c)T ∈ R3, ¨rj,c ∈ R x = (ψ, η)T, η = (θ, φ)T εi ej Hx(˜u2p,x)˜ u2p ∈ U
U H(u2p,z) = c V z
z = x(u2p) x˜ V V x(U) x
(i)x 3 × 2p dx(u2p)
dx(u2p) = −[Hx(u2p,x(u2p))−1Hu2p(u2p,x(u2p))]. (4.9)
(ii)x(u2p),u2p∈ U (˜u2p,x)˜
x(u2p) ≈ ˜x + dx(˜u2p) · (u2p− ˜u2p)
=x − [H˜ x(˜u2p,x)˜ −1Hu2p(˜u2p,x)] · (u˜ 2p− ˜u2p). (4.10) (iii) ¨ψ ¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2pu2p). (4.11) 3 (4.2) 4 (e3,−ge3+m1B(x)Stra2pu2p) (4.8) ψ ∂ ∂x(Z(η)Srot2pu2p) 2 1 (3.44) ψ θ φ ¨ψ ψ ˙ψ (3.44)
¨ψ = (ε1,Y((θ, φ)T,( ˙ψ, ˙θ, ˙φ)T)+Z((θ, φ)T)Srot2pu2p) H(u2p,x) c H(u2p,x) ∈ R3,c ∈ R3,u2p∈ R2p, x ∈ R3 F(x, y) ∈ Rm,c ∈ Rm,x ∈ Rn, y ∈ Rm (iii) (3.44) 3 3 3 4.2 5 4.2 (m = 1.656 [kg], g = 9.80665 [m/s2], ) = 0.365 [m], I 11=0.01982 [kg · m2], I22=0.01954 [kg · m2], I33= 0.03221 [kg · m2], k F=1.79 × 10−7[N/rpm2], kM =4.38 × 10−9[Nm/rpm2]) ˜
x = ( ˜ψ [rad], ˜θ [rad], ˜φ [rad])T =(0 [rad], 0.5236 [rad], 0 [rad])T, u˜4=( ˜ωM1[rpm], ˜ωM2[rpm], ˜ωM3[rpm], ˜ωM4[rpm])T =(5118.6279 [rpm], 5117.6279 [rpm], 5117.6279 [rpm], 5118.6279 [rpm])T, x(u4),u4∈ U (˜u4,x)˜
x(u4) ≈ ˜x − −0.7946 × 10 −5 0.7946 × 10−5 −0.7946 × 10−5 0.7946 × 10−5 −0.1653 × 10−7 −0.1653 × 10−7 −0.1653 × 10−7 −0.1653 × 10−7 −0.3973 × 10−5 0.3973 × 10−5 −0.3973 × 10−5 0.3973 × 10−5 · (u4− ˜u4). (4.12) ¨ψ 3 (iii) 3 4 F(x, u2p) ˜x ∈ R3 u˜2p ∈ R2p R2p, p = 2, 3, 4 C1
(4.1) (4.2) Z(ηop)Srot2pu2p(op) (e3,−ge3+m1B(xop)Stra2pu2p(op)) F(˜x, ˜u2p) =02p=(0, 0, . . . , 0)T F(x, u2p) =A2p(x)u2p− b2p. (4.13) p = 2 A4(η) ∈ R4×4= ( Z(η)Srot4 eT 3m1B(x)Stra4 ) , (4.14) b4=( ¨ψ, ¨θ, ¨φ, c + g)T∈ R4 c ∈ R p = 3 4 A2p(x) ∈ R2p×2p=( A ! 4×2p(η) Q(2p−4)×2p ) , (4.15) A! 4×2p(η) ∈ R4×2p= ( Z(η)Srot2p eT 3m1B(x)Stra2p ) , (4.16) Q(2p−4)×2p∈ R(2p−4)×2p b2p=( ¨ψ, ¨θ, ¨φ, c + g,b! 2p−4)T∈ R2p.b!2p−4∈ R2p−4 ˜ x ∈ R3 F(˜x, ˜u 2p) =02p, det(A2p(˜x)) ! 0, ˜u2p∈ R2p u˜2p =A2p(˜x)−1b2p ¨ψ, ¨θ, ¨φ ¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2pu2p), ¨θ = (ε2,Y(η, ˙x) + Z(η)Srot2pu2p), (4.17) ¨φ = (ε3,Y(η, ˙x) + Z(η)Srot2pu2p). 4 p = 2 ˜η ∈ R2 A 4(˜η) 4 × 4 p = 3 4 x ∈ R˜ 3 A! 4×2p(˜x) 4 × 2p Q(2p−4)×2p (2p − 4) × 2p ˜
x ∈ R3, A2p(˜x)˜u2p− b2p =02p det(A2p(˜x)) ! 0, ˜u2p ∈ R2p ˜
u2p=A2p(˜x)−1b2p ¨ψ, ¨θ, ¨φ (3.44) 4–6 Y(η, ˙x) + Z(η)Srot2pu2p 4
R/C R/C
R/C
[16] Throttle, Rudder, Elevator, Ailerons
4
Throttle Rudder Ailerons Elevator
throttle, rudder, elevator, ailerons altitude ( ) yaw ( ) pitch ( ) roll ( )
[15]
3 x(u2p) u2p∈ U
x(u2p),u2p ∈ U (3.45)
¨
r = −ge3+m1B(x(u2p))Stra2pu2p (3.45) 3 u2p
4 ( θ, φ, ¨ψ, ¨θ, ¨φ,
c ( ¨r3)) u2p∈ U
3 4 throttle ( altitude),
rudder ( yaw), elevator ( pitch), ailerons ( roll)
4.2
i εMi Vi⊂ R2p [36], pp. 628–632 Vi Pi [36], pp. 628–632 v ∈ R2p Vi Piv Piv ∈ Vi (v − Piv, w) = 0 for all w ∈ Vi, (4.18) (Piv, εMi) = (v, εMi). (4.19) 7 i ui 2p−1=u2p− Piu2p = i−1 ' j=1 ω2M jεM j+ 2p ' j=i+1 ω2M jεM j. (4.20) Frot(ui2p−1) Ftra(ui2p−1)Frot(ui2p−1) =Srot2pu2p−1i =Sirot2p−1ui2p−1, (4.21) Ftra(ui2p−1) =Stra2pu2p−1i =Sitra2p−1ui2p−1, (4.22) Si ξ2p−1 Sξ2p i (ξ = rot, tra) 8 i1th, i2th, . . . , inth (1 ≤ i1 <i2 <· · · < in ≤ 2p, 1 ≤ n ≤ 2p − 2, p = 2, 3, 4, n ∈ N) ui1,i2,...,in 2p−n = i1−1 ' j=1 ω2M jεM j+ i2−1 ' j=i1+1 ω2M jεM j+· · · + 2p ' j=in+1 ω2M jεM j. (4.23) Srot2pui2p−n1,i2,...,in Stra2pui2p−n1,i2,...,in, p = 2, 3, 4
Srot2pui2p−n1,i2,...,in=Srot2p−ni1,i2,...,inui2p−n1,i2,...,in, (4.24) Stra2pui2p−n1,i2,...,in =Stra2p−ni1,i2,...,inui2p−n1,i2,...,in, (4.25) Si1,i2,...,in
1 1 3 5 4.3 ui1,i2,...,in 2p−n =u1,3,53 = ω2M2εM2+ ω2M4εM4+ ω2M6εM6, (4.26) i1=1, i2=3, i3=5, n = 3. Srot3u1,3,53 Stra3u1,3,53 Srot3u1,3,53 =S1,3,5rot3u1,3,53 , (4.27) Stra3u1,3,53 =S1,3,5tra3 u1,3,53 , (4.28) S1,3,5 rot3u1,3,53 = − √ 3 2 )· kF2 0 √ 3 2 )· kF6 0.5) · kF2 −) · kF4 0.5) · kF6 kM2 kM4 kM6 ω2M2 ω2M4 ω2M6 , (4.29) S1,3,5 tra3 u1,3,53 = 0 0 0 0 0 0 kF2 kF4 kF6 ω2M2 ω2M4 ω2M6 , (4.30) S1,3,5 ξ3 Sξ6 1 3 5 (ξ = rot, tra) 4.3 1 3 5 [ (3.44)] [ (3.45)] (p = 2, 3, 4) d dt ( x ˙ x ) = ( ˙x Y(η, ˙x) + Z(η)Si1,i2,...,in rot2p−nui2p−n1,i2,...,in ) , (4.31) d dt ( r ˙ r ) = ( ˙r
−ge3+m1B(x)Stra2p−ni1,i2,...,inui2p−n1,i2,...,in )
. (4.32)
4.2 9 4.2 5 6 9 2 4.2 4.2 2 4.1 (I) No Problem*1 ( 4.4)
(II) Admissible Problem*2 ( 4.5).
4.4 4.1 4.2 (I) 4.5 4.1 4.2 (II) p = 3, 3.2 2 4 2 (n = 4) 4.2 (I) *1 (I) 4.1 *2 (II) 4.1
(I) (II) (I) (II) 5 4.42
4 1 ≤ i1<i2<· · · < in≤ 2p, 1 ≤ n ≤ 2p − 3, p = 3, n ∈ N 9 4.2 (xop,x˙op)T ∈ R3×R3,xop =(ψop,ηop)T, ηop =(θop, φop)T, (rop,˙rop)T∈ R3×R3 ( x˙ op Z(ηop)Si1,i2,...,in
rot2p−nui2p−n(op)1,i2,...,in ) = ( 03 03 ) , (4.33) ( ˙r 3(op)
(e3,−ge3+m1B(xop)Stra2p−ni1,i2,...,inui2p−n(op)1,i2,...,in) ) = ( 0 c ) , (4.34) Z(ηop) = (B(xop) ˆIB(xop)T· ωx˙(ψop, θop))−1· B(xop), (4.35) ¨r1=(e1,m1B(xop)S i1,i2,...,in
tra2p−nui2p−n(op)1,i2,...,in), (4.36) ¨r2=(e2,m1B(xop)Stra2p−ni1,i2,...,inui2p−n(op)1,i2,...,in), (4.37)
¨r3=c, (4.38) c ∈ R 03=(0, 0, 0)T c = 0 (xop,x˙op)T (rop,˙rop)T ¨r1 = ¨r2 =0 c = 0 (¨r3 =0) (xe,x˙e)T (re,˙re)T 4.2 (II) (4.33) 1 ˙ψ 4 ¨ψ (4.33) ˙ηop
(ε2,Z(ηop)Srot2p−ni1,i2,...,inui2p−n(op)1,i2,...,in) (ε3,Z(ηop)Srot2p−ni1,i2,...,inui2p−n(op)1,i2,...,in)
= 02 0 0 , (4.39)
¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in), (4.40)
˙ψ = cψt + ˙ψ(0), (4.41) cψ∈ R 4.1 (ii) 5 4.2 (I) F(x, ui1,i2,...,in 2p−n ), p = 2, 3, 4 x ∈ R˜ 3 ˜ ui1,i2,...,in 2p−n ∈ R2p−n R2p−n C1 (4.33) (4.34) Z(ηop)Si1,i2,...,in
F(˜x, ˜ui1,i2,...,in
2p−n ) =02p−n=(0, 0, . . . , 0)T F(x, ui1,i2,...,in
2p−n ) = Ai2p−n1,i2,...,in(x)u2p−ni1,i2,...,in− b2p−n. (4.42) p = 2 Ai1,i2,...,in 4−n (η) ∈ R(4−n)×(4−n)= ( Z(η)Si1,i2,...,in rot4−n eT 3m1B(x)Sitra4−n1,i2,...,in ) , (4.43) b4−n=( ¨ψ, ¨θ, ¨φ, c + g)T∈ R4−n c ∈ R p = 3 4 Ai1,i2,...,in 2p−n (x) ∈ R(2p−n)×(2p−n)= A!i1,i2,...,in 4×(2p−n)(η) Qi1,i2,...,in (2p−4)×(2p−n) , (4.44) A!i1,i2,...,in 4×(2p−n)(η) ∈ R4×(2p−n)= Z(η)Si1,i2,...,in rot2p−n eT 3m1B(x)Sitra2p−n1,i2,...,in , (4.45) Qi1,i2,...,in (2p−4)×(2p−n)∈ R(2p−4)×(2p−n) b2p−n =( ¨ψ, ¨θ, ¨φ, c + g,b!2p−4)T∈ R2p−n. b! 2p−4∈ R2p−4 ˜ x ∈ R3 F(˜x, ˜ui1,i2,...,in
2p−n ) = 02p−n, det(Ai2p−n1,i2,...,in(˜x)) ! 0, ˜u2p−ni1,i2,...,in ∈ R2p−n u˜i2p−n1,i2,...,in = Ai1,i2,...,in
2p−n (˜x)−1b2p−n ¨ψ, ¨θ, ¨φ
¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in),
¨θ = (ε2,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in), (4.46) ¨φ = (ε3,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in).
˜ ui1,i2,...,in 2p−n ∈ R2p−n 4.2 (I) 5 p = 2 ˜η ∈ R2 Ai1,i2,...,in 4−n (˜η) (4 −n)×(4−n) p = 3 4 x ∈ R˜ 3(˜x = ( ˜ψ, ˜η)T) A!i1,i2,...,in 4×(2p−n)(˜η) 4 × (2p − n) Qi(2p−4)×(2p−n)1,i2,...,in (2p − 4) × (2p − n) ˜ x ∈ R3, Ai1,i2,...,in
2p−n (˜x)˜ui2p−n1,i2,...,in− b2p−n =02p−n det(A2p−ni1,i2,...,in(˜x)) ! 0, ˜ui2p−n1,i2,...,in ∈ R2p−n u˜i1,i2,...,in 2p−n =A2p−ni1,i2,...,in(˜x)−1b2p−n ¨ψ, ¨θ, ¨φ (4.31) 4–6 Y(η, ˙x) + Z(η)Si1,i2,...,in rot2p−nui2p−n1,i2,...,in 5 6 4.2 (II) F(x, ui1,i2,...,in 2p−n ), p = 2, 3, 4 x ∈ R˜ 3 ˜ ui1,i2,...,in 2p−n ∈ R2p−n R2p−n C1 (4.39)
(4.34) Z(ηop)Sirot2p−n1,i2,...,inu2p−n(op)i1,i2,...,in (e3,−ge3+m1B(xop)Stra2p−ni1,i2,...,inui2p−n(op)1,i2,...,in) F(˜x, ˜ui1,i2,...,in
2p−n ) =02p−n=(0, 0, . . . , 0)T F(x, ui1,i2,...,in
p = 2 Ai1,i2,...,in 4−n (η) ∈ R(4−n)×(4−n)= εT2Z(η)Si1,i2,...,in rot2p−n εT3Z(η)Si1,i2,...,in rot2p−n eT 3m1B(x)Sitra2p−n1,i2,...,in , (4.48) b4−n=(¨θ, ¨φ, c + g)T∈ R4−n c ∈ R p = 3 p = 4 Ai1,i2,...,in 2p−n (x) ∈ R(2p−n)×(2p−n)= A!!i1,i2,...,in 3×(2p−n)(η) Qi1,i2,...,in (2p−4)×(2p−n) , (4.49) A!!i1,i2,...,in 3×(2p−n)(η) ∈ R3×(2p−n)= εT2Z(η)Si1,i2,...,in rot2p−n εT3Z(η)Si1,i2,...,in rot2p−n eT 3m1B(x)Sitra2p−n1,i2,...,in , (4.50) Qi1,i2,...,in (2p−4)×(2p−n) ∈ R(2p−4)×(2p−n) b2p−n =(¨θ, ¨φ, c + g,b!2p−4)T ∈ R2p−n. b! 2p−4∈ R2p−4 ˜ x ∈ R3 F(˜x, ˜ui1,i2,...,in
2p−n ) = 02p−n, det(Ai2p−n1,i2,...,in(˜x)) ! 0, ˜u2p−ni1,i2,...,in ∈ R2p−n u˜i2p−n1,i2,...,in = Ai1,i2,...,in
2p−n (˜x)−1b2p−n ¨ψ, ¨θ, ¨φ
¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in),
¨θ = (ε2,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in), (4.51) ¨φ = (ε3,Y(η, ˙x) + Z(η)Srot2p−ni1,i2,...,inui2p−n1,i2,...,in).
˜ ui1,i2,...,in 2p−n ∈ R2p−n 4.2 (II) 6 p = 2 ˜η ∈ R2 Ai1,i2,...,in 4−n (˜η) (4 −n)×(4−n) p = 3 4 x ∈ R˜ 3(˜x = ( ˜ψ, ˜η)T) A!!i1,i2,...,in 3×(2p−n)(˜η) 3 × (2p − n) Qi(2p−4)×(2p−n)1,i2,...,in (2p − 4) × (2p − n) ˜ x ∈ R3, Ai1,i2,...,in
2p−n (˜x)˜ui2p−n1,i2,...,in− b2p−n =02p−n det(A2p−ni1,i2,...,in(˜x)) ! 0, ˜ui2p−n1,i2,...,in ∈ R2p−n u˜i1,i2,...,in 2p−n =A2p−ni1,i2,...,in(˜x)−1b2p−n ¨ψ, ¨θ, ¨φ (4.31) 4–6 Y(η, ˙x) + Z(η)Si1,i2,...,in rot2p−nui2p−n1,i2,...,in 6 ˜ ui1,i2,...,in 2p−n ∈ R2p−n 4.2 n Si1,i2,...,in rot2p−n ∈ R3×(2p−n) Si1,i2,...,in tra2p−n ∈ R3×(2p−n) ui2p−n1,i2,...,in∈ R2p−n Ai1,i2,...,in 2p−n ∈ R(2p−n)×(2p−n) 4.2 (II) (4.48) (4.50) εT1Z(η)Si1,i2,...,in rot2p−n
5
4.1 4.2 2 4–6 4–6 (p = 2) (p = 3) (p = 4) 5.1, 5.2, 5.3 m, I11,I22,I33, ),kF, kM, ELEV-8 Quadcopter Kit [15], [16]
[37] 5.1 p = 2 m 1.656 [kg] I11 0.01982 [kg · m2] I22 0.01954 [kg · m2] I33 0.03221 [kg · m2] g 9.80665 [m/s2] kF 1.79 × 10−7[N/rpm2] kM 4.38 × 10−9[Nm/rpm2] ) 0.365 [m]
5.2 p = 3 m 2 [kg] I11 0.02973 [kg · m2] I22 0.02931 [kg · m2] I33 0.048315 [kg · m2] g 9.80665 [m/s2] kF 1.79 × 10−7[N/rpm2] kM 4.38 × 10−9[Nm/rpm2] ) 0.365 [m] 5.3 p = 4 m 3.5 [kg] I11 0.03964 [kg · m2] I22 0.03908 [kg · m2] I33 0.06442 [kg · m2] g 9.80665 [m/s2] kF 1.79 × 10−7[N/rpm2] kM 4.38 × 10−9[Nm/rpm2] ) 0.365 [m] 4–6 Q(2p−4)×2p ∈ R(2p−4)×2p, p = 3 4 p = 3 Q2×6= ( 0 1 −1 0 0 0 0 0 0 1 −1 0 ) , (5.1) p = 4 Q4×8= 0 1 −1 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 −1 1 . (5.2)
5.1
4.1
4 [
(3.44)], [ (3.45)] Maple MATLAB ode45
[38] MATLAB (3.45)
ψ(t), θ(t), φ(t) (3.44)
(3.45) [39] ψ(t), θ(t),
φ(t)
(a) 5.1 5.2 4.1 (ii) ˜θ = 0 [rad], ˜φ =
0 [rad], ¨ψ = 0 [rad/s2], c = 0 [m/s2] , 4 ˜ωM1= ˜ωM2 = ˜ωM3= ˜ωM4=4762.4891 [rpm]. 0 1 2 3 4 5 −1 −0.5 0 0.5 1
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.1 (a) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.2 (a) r3 =3 [m] 5.1 ψ(t), θ(t), φ(t) 5.2 r1(t) r2(t) (a) (ψ(t), θ(t), φ(t)) (r1(t), r2(t), r3(t)) 4.1 (ii) (b) 5.3 5.4 4.1 (iii-1)
˜ωM1= ˜ωM3=4359.3992 [rpm], ˜ωM2= ˜ωM4 =5134.0279 [rpm]. 0 1 2 3 4 5 0 5 10 15 20 25
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.3 (b) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.4 (b) r3= 3 [m] 5.3 θ(t) φ(t) 5.4 r1(t) r2(t) (b) (θ(t), φ(t)) (r1(t), r2(t), r3(t)) 4.1 (iii-1) (c) 5.5 5.6 4.1 (iv-1)
θ ˜θ = 0.5236 [rad], ˜φ = 0 [rad], ¨ψ = 0 [rad/s2], c = 0 [m/s2] , 4 ˜ωM1= ˜ωM2 = ˜ωM3= ˜ωM4=5117.6279 [rpm]. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.5 (c) 0 1 2 3 4 5 0 10 20 30 40 50 60 70 80
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.6 (c) r3=3 [m]
5.5 ψ(t) φ(t) (c)
θ r1
(ψ(t), φ(t)) (r2(t), r3(t))
4.1 (iv-1) θ
(d) 5.7 5.8 4.1 (v-1)
φ ˜θ = 0 [rad], ˜φ = 0.3491 [rad], ¨ψ = 0 [rad/s2], c = 0 [m/s2] , 4 ˜ωM1= ˜ωM2 = ˜ωM3= ˜ωM4=4912.9359 [rpm]. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.7 (d) 0 1 2 3 4 5 −50 −40 −30 −20 −10 0 10 20
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.8 (d) r3=3 [m] 5.7 ψ(t) θ(t) (d) φ r2 (ψ(t), θ(t)) (r1(t), r3(t)) 4.1 (v-1) φ (e) 5.9 5.10 4.1 (i-1)
˜θ = 0 [rad], ˜φ = 0 [rad], ¨ψ = 0 [rad/s2], c = 3 [m/s2] , 4 ˜ωM1= ˜ωM2 = ˜ωM3= ˜ωM4=5442.4121 [rpm].
0 1 2 3 4 5 −1 −0.5 0 0.5 1
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.9 (e) 0 1 2 3 4 5 0 10 20 30 40 50
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.10 (e) r3=3 [m] 5.10 ψ(t), θ(t), φ(t) 5.10 r1(t) r2(t) (e) r3 (ψ(t), θ(t), φ(t)) (r1(t), r2(t)) 4.1 (i-1) ( f ) 5.11 5.12 4.1 (i-2)
˜θ = 0 [rad], ˜φ = 0 [rad], ¨ψ = 0 [rad/s2], c = −0.1 [m/s2] , 4 ˜ωM1= ˜ωM2 = ˜ωM3= ˜ωM4=4738.1449 [rpm]. 0 1 2 3 4 5 −1 −0.5 0 0.5 1
Rotational behavior of the quadrotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.11 ( f ) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the quadrotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.12 ( f ) r3=3 [m] 5.12 ψ(t), θ(t), φ(t) 5.12 r1(t) r2(t) ( f ) r3 (ψ(t), θ(t), φ(t)) (r1(t), r2(t))
4.1 (i-2)
(g) 5.13 5.14 4.1 (iii-1)
˜θ = 0 [rad], ˜φ = 0 [rad], ¨ψ = 1 [rad/s2], c = 0 [m/s2] , 4 ˜ωM1=3570.2251 [rpm], ˜ωM2= ˜ωM3=4483.3457 [rpm], ˜ωM4= ˜ωM5=4052.5853 [rpm], ˜ωM6 =4876.2003 [rpm]. 0 1 2 3 4 5 −2 0 2 4 6 8 10 12 14
Rotational behavior of the hexarotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.13 (g) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the hexarotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.14 (g) r3= 3 [m] 5.13 θ(t), φ(t) 5.14 r1(t), r2(t) (g) (θ(t), φ(t)) (r1(t), r2(t), r3(t)) 4.1 (iii-1) (h) 5.15 5.16 4.1 (iv-1)
θ ˜θ = 0.01745 [rad], ˜φ = 0 [rad], ¨ψ = 0 [rad/s2], c = 0 [m/s2] , 4
0 1 2 3 4 5 0 0.005 0.01 0.015 0.02 0.025 0.03
Rotational behavior of the hexarotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.15 (h) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the hexarotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.16 (h) r3=3 [m] 5.15 ψ(t), φ(t) (h) θ r1 (ψ(t), φ(t)) (r2(t), r3(t)) 4.1 (iv-1) θ (i) 5.17 5.18 4.1 (iii-2)
˜θ = 0 [rad], ˜φ = 0 [rad], ¨ψ = −1 [rad/s2], c = 0 [m/s2] , 4 ˜ωM1=3489.4736 [rpm], ˜ωM2= ˜ωM3= ˜ωM4=5184.9965 [rpm], ˜ωM5= ˜ωM6=4419.3107 [rpm], ˜ωM7 = ˜ωM8=5470.8561 [rpm]. 0 1 2 3 4 5 −15 −10 −5 0 5 10
Rotational behavior of the octorotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.17 (i) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the octorotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.18 (i) r3= 3 [m] 5.17 θ(t), φ(t) 5.18 r1(t), r2(t) (i)
(θ(t), φ(t)) (r1(t), r2(t), r3(t))
4.1 (iii-2)
( j) 5.19 5.20 4.1 (iv-2)
θ ˜θ = 0.01745 [rad], ˜φ = 0 [rad], ¨ψ = 0 [rad/s2], c = 0 [m/s2] , 4
˜ωM1= ˜ωM2= ˜ωM3= ˜ωM4 = ˜ωM5= ˜ωM6= ˜ωM7= ˜ωM8 =4896.1635 [rpm]. 0 1 2 3 4 5 0 0.005 0.01 0.015 0.02 0.025 0.03
Rotational behavior of the octorotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 5.19 ( j) 0 1 2 3 4 5 0 1 2 3 4 5
Translational behavior of the octorotor
Time [s] Movement distance [m] r1(t) r2(t) r3(t) 5.20 ( j) r3=3 [m] 5.19 ψ(t), φ(t) ( j) θ r1 (ψ(t), φ(t)) (r2(t), r3(t)) 4.1 (iv-2) θ
5.2
4.2 2 5 6 [ (4.31)], [ (4.32)] Maple MATLAB ode45 [38] MATLAB (4.32) ψ(t), θ(t), φ(t) (4.31) (4.32) [39] ψ(t), θ(t), φ(t) (A) 3 ( 6.1) t[s] ∈ [0 5], θ = φ =0 [rad], ¨θ = ¨φ = 0 [rad/s2], kF1=kF2=kF4=kF5=kF6=1.79 × 10(−7)[N/rpm2], kM1=kM2= kM4 =kM5 =kM6=4.38 × 10(−9)[Nm/rpm2], c (or ¨r3) = 0 [m/s2] 5.22 5.23 ( 4.2 (I) 4.1 (ii)) 5 ωM1= ωM2= ωM4 = ωM5=5233.820507 [rpm], ωM6=0 [rpm]. 5.21 (A) 35.22 3 (A) 5.23 3 (A) r3=3 [m] 5.22 ψ(t), θ(t), φ(t) 5.23 r1(t) r2(t) (A) 3 3 6 (ψ(t), θ(t), φ(t)) (r1(t), r2(t), r3(t)) ( 4.2 (I) 4.1 (ii)) 5 F(x, ui1,i2,...,in 2p−n ) F(x, u3 5) =A35(x)u35− b5, (5.3) A3 5(x) ∈ R5×5=( A ! 4×5(η) εTQ2Q32×5 ) , (5.4) A! 4×5(η) ∈ R4×5= ( Z(η)S3 rot5 eT 3m1B(x)S3tra5 ) , (5.5) b5=( ¨ψ, ¨θ, ¨φ, c + g,b!1)T ∈ R5, (5.6) i1 =3 n = 1. span{εQ1,εQ2} 2 R2 Q3 2×5∈ R2×5 Q2×6 3 b!1 ∈ R S3 ξ5 Sξ6 3 (ξ = rot, tra) (B) 2 5 8 5.24 t[s] ∈ [0 5],
θ = 0.0872665 [rad], φ = 0 [rad], ¨θ = ¨φ = 0 [rad/s2], kF1 = kF3 = kF4 = kF6 = kF7 = 1.79 × 10(−7)[N/rpm2], kM1 =kM3 =kM4=kM6=kM7=4.38 × 10(−9)[N/rpm2], c (or ¨r3) = 0 [m/s2] 5.25
5.26 4.2 (I) 4.1
ωM1= ωM6=8249.4166 [rpm], ωM3= ωM4=5309.2771 [rpm], ωM7 =0 [rpm]. 5.24 (B) 2, 5, 8 5.25 2 5 8 (B) 5.26 2 5 8 (B) r3=3 [m] 5.25 ψ(t) φ(t) (B) 2 5 8 3 7 θ r1 (ψ(t), φ(t)) (r2(t), r3(t)) 4.2 (I) 4.1 (iv-1) θ ) 5 F(x, ui1,i2,...,in 2p−n ) F(x, u2,5,8 5 ) =A2,5,85 (x)u2,5,85 − b5, (5.7) A2,5,8 5 (x) ∈ R5×5= ( A!2,5,8 4×5(η) εTQ2Q2,5,8 4×5 ) , (5.8)
A!2,5,8 4×5(η) ∈ R4×5= ( Z(η)S2,5,8 rot5 eT 3m1B(x)S2,5,8tra5 ) , (5.9) b5=( ¨ψ, ¨θ, ¨φ, c + g,b!1)T ∈ R5, (5.10) i1 =2, i2 =5, i3=8, n = 3. span{εQ1,εQ2,εQ3,εQ4} 4 R4 Q2,5,8 4×5 ∈ R4×5 Q4×8 2 5 8 b!1 ∈ R S2,5,8 ξ5 Sξ8 2 5 8 (ξ = rot, tra) (C) 2 t[s] ∈ [0 5], θ = φ = 0 [rad], ¨θ = ¨φ = 0 [rad/s2], k F1=kF3=kF4=1.79 × 10(−7)[N/rpm2], kM1=kM3=kM4=4.38 × 10(−9)[Nm/rpm2], c (or ¨r3) = 0 [m/s2] 5.28 5.29 ( 5.27 4.2
(II) 4.1 (i-1), (i-2) r3 ¨ψ = −12.3371 [rad/s2]
) 6 ωM1= ωM3=6735.1766 [rpm], ωM4=0 [rpm]. 5.27 (C) 2 5.28 2 (C) 5.29 2 (C) r3=3 [m]
5.28 θ(t) φ(t) 5.29 r1(t) r2(t)
(C) 2
2 4
( 5.27 4.2 (II) 4.1 (i-1), (i-2)
r3 ¨ψ = −12.3371 [rad/s2] ) (θ(t), φ(t)) (r1(t), r2(t), r3(t)) 6 F(x, ui1,i2,...,in 2p−n ) F(η, u2 3) =A23(η)u23− b3, (5.11) A2 3(η) ∈ R3×3= εT2Z(η)S2 rot3 εT3Z(η)S2rot3 eT 3m1B(x)S2tra3 , (5.12) b3=(¨θ, ¨φ, c + g)T∈ R3. (5.13) i1=2 n = 1. S2ξ3 Sξ4 2 (ξ = rot, tra). (D) 1 3 t[s] ∈ [0 0.1], θ = φ = 0 [rad], ¨θ = 0 [rad/s2], ¨φ = 17.4533 [rad/s2], k F2 = kF4 = 1.79 × 10(−7) [N/rpm2], kM2 = kM4 = 4.38 × 10(−9)[Nm/rpm2], c (or ¨r 3) = 0 [m/s2] 5.31 5.32 ( 5.30) 4.2 (II) 4.1 (v-1) ¨ψ = 12.3371 [rad/s2] ) 6 ωM2=6535.6936 [rpm], ωM4=6928.9189 [rpm]. 5.30 (D) 1 3
5.31 1 3 (D) 5.32 1 3 (D) r3=3 [m] 5.32 r1(t) r2(t) t[s] ∈ [0 0.1] (D) 1 3 ( 4.2 (II) 4.1 (v-1) ¨ψ = 12.3371 [rad/s2] ) 6 F(x, ui1,i2,...,in 2p−n ) F(η, u1,3 2 ) =A1,32 (η)u1,32 − b2, (5.14) A1,3 2 (η) ∈ R2×2= ( εT3Z(η)S1,3 rot2 eT 3m1B(x)S1,3tra2 ) , (5.15) b2=( ¨φ, c + g)T ∈ R2. (5.16) i1 = 1, i2 = 3, n = 2. S1,3ξ2 Sξ4 1 3 (ξ = rot, tra).
(E) 1 3 4 6 8 t[s] ∈ [0 5], θ = φ = 0 [rad], ¨θ = ¨φ = 0 [rad/s2], k
F2 = kF5 = kF7 = 1.79 × 10(−7)[N/rpm2], kM2 = kM5 = kM7 = 4.38 × 10(−9)[Nm/rpm2], c (or ¨r
3) = −0.1 [m/s2] 5.34 5.35
( Fig. 5.33 4.2 (II) 4.1 (i-2) ¨ψ = −2.2140 [rad/s2]
) 6 ωM2=8866.5435 [rpm], ωM5= ωM7=7455.8446 [rpm]. 5.33 (E) 1 3 4 6 8 5.34 1 3 4 6 8 (E) 5.35 1 3 4 6 8 (E) r3=3 [m] 5.34 θ(t) φ(t) 5.35 r1(t) r2(t) (E) 1 3 4 6 8 4.2 (II) 4.1 (i-2) ¨ψ = −2.2140 [rad/s2] ) (θ(t), φ(t)) (r1(t), r2(t))
6 F(x, ui1,i2,...,in 2p−n ) F(η, u1,3,4,6,8 3 ) =A1,3,4,6,83 (η)u1,3,4,6,83 − b3, (5.17) A1,3,4,6,8 3 (η) ∈ R(3)×(3)= εT2Z(η)S1,3,4,6,8 rot3 εT3Z(η)S1,3,4,6,8 rot3 eT 3m1B(x)S1,3,4,6,8tra3 , (5.18) b3=(¨θ, ¨φ, c + g)T∈ R3. (5.19) i1=1, i2 =3, i3=4, i4=6, i5 =8, n = 5. S1,3,4,6,8ξ3 Sξ8 1 3 4 6 8 (ξ = rot, tra).
6
4.2 (I) 4.2 (I) (1 ≤ i1 <· · · < in ≤ 6, 1 ≤ n ≤ 2, n ∈ N) (p = 3) [ (4.31) ] 4.2 (I) (xop,x˙op)T (rop,˙rop)T ddtδXrot=Frot6−n(op)δXrot+Grot6−n(op)δU i1,...,in 6−n , (6.1) δXrot=(δx, δ˙x)T, (δx, δ˙x)T =(x − x (op),x − ˙x˙ (op))T, Frot6−n(op)= ( 03×3 I 3×3 frotx(ηop,xop,˙ ui1,...,in
6−n(op)) frot˙x(ηop,˙xop)
)
Grot6−n(op)= ( 0 3×(6−n) frotui1,...,in 6−n (ηop) ) , (6.3)
frotζ(ηop,x˙op,u6−n(op)i1,...,in ) =Yζ(ηop,x˙op) +
* ∂ ∂ζ +
Z(η)Si1,...,in rot6−nui6−n(op)1,...,in
,-.... ..η=η op , (6.4) frot˙ζ(ηop,x˙op) =Y˙ζ(ηop,x˙op), (6.5) frotui1,...,in 6−n (ηop) =Z(ηop)S i1,...,in rot6−n, (6.6) ζ =x ∈ R3, η∈ R2, ˙ζ = ˙x ∈ R3, ˙η ∈ R2, 0 3×(6−n) 3 × (6 − n) .Yx Y˙x Y x Y x˙ frotx, frotx˙, frotui1,...,in
6−n frot x , frot x˙ , frot u
i1,...,in
6−n
7 4.2 (I) (1 ≤ i1 < · · · < in ≤ 6,
1 ≤ n ≤ 2, n ∈ N) CMrot6−n(op)∈ R6×(6×(6−n)) 6 × (6 × (6 − n)) (6.7)
CMrot6−n(op)=[Grot6−n(op),Frot6−n(op)Grot6−n(op), F2
rot6−n(op)Grot6−n(op), . . . , F5
rot6−n(op)Grot6−n(op)], (6.7)
CMrot6−n(op) 6 (Frot6−n(op),Grot6−n(op))
(6.8) 06 = (0, 0, . . . , 0)T ∈ R6
(6 − n) × 6 K6−n(op)
(6.9) δ¨r 03=(0, 0, 0)T ∈ R3
d
dtδXrot=(Frot6−n(op)− Grot6−n(op)K6−n(op))δXrot, (6.8)
δ¨r = 1 m * ∂ ∂x + B(x)Si1,...,in
tra6−nui6−n(op)1,...,in ,-....
..x=xopδx − 1
mB(xop)S i1,...,in
tra6−nK6−n(op)δXrot, (6.9) δXrot=(δx, δ˙x)T, δ˙r = ˙r − ˙rop .
7 4.2 (I) (1 ≤ i1 < · · · < in ≤ 6,
1 ≤ n ≤ 2, n ∈ N) (6.1) δUi1,...,in
[43] (Frot6−n(op),Grot6−n(op)) Frot6−n(op)−Grot6−n(op)K6−n(op) K6−n(op) (Frot6−n(op),Grot6−n(op)) (6.8) 06 (6.9) δXrot→ 06, ( δx → 03) δ¨r → 03 7 8 4.2 (I) (1 ≤ i1 < · · · < in ≤ 6,
1 ≤ n ≤ 2, n ∈ N) (i) (6.7)CMrot6−n(op) 6 , δui6−n1,...,in
=MδUi1,...,in
6−n =−K6−n(op)(x − x(op),x − ˙x˙ (op))T [ (4.31)] [ 4.32]
[ (6.10)] [ (6.12)]
(ii) (Frot6−n(op)− Grot6−n(op)K6−n(op))
(xop,x˙op)T (t0,(x0,x˙0)T) (6.10) (ϕ(t, (x0,x˙0)T), ˙ϕ(t, (x0,x˙0)T)) (xop,x˙op)T (6.12) ¨r(t) ¨r(op) d dt ( x ˙ x ) = ( ˙ x frot(x, ˙x) ) , (6.10) M =+ 03×3 I3×3 ,,
frot(x, ˙x) = Y(TLx, ˙x) + Z(TLx)Sirot6−n1,...,in(u6−n(op)i1,...,in + δui6−n1,...,in), (6.11)
TL= ( 0 1 0 0 0 1 ) , d dt ( r ˙ r ) = ( ˙ r ftra(ϕ(t, (x0,x˙0)T), ˙ϕ(t, (x0,x˙0)T)) ) , (6.12)
ftra(x, ˙x) = −!e3+m1B(x)Sitra6−n1,...,in(ui6−n(op)1,...,in + δui6−n1,...,in), (6.13) K6−n(op) 7 , (δx, δ˙x)T = (x − x(op),x − ˙x˙ (op))T, (x(t), ˙x(t))T = (ϕ(t, (x0,x˙0)T), ˙ϕ(t, (x0,˙x0)T))T =(φ(t, (x0,x˙0)T,u6−n(op)i1,...,in + δui1,...,in
6−n ), ˙φ(t, (x0,x˙0)T,u6−n(op)i1,...,in + δui6−n1,...,in))T ∈ Σrot⊂ R6, (t0,(x0,x˙0)T) ∈ R × U, u6−n(op)i1,...,in + δui6−n1,...,in ∈ Λ, U ,(x0− xop,x˙0−
˙
xop)T, < 1 (xop,˙xop)T∈ Σrot⊂ R6
8 4.2 (I) (1 ≤ i1 < · · · < in ≤ 6,
δui1,...,in
6−n (x, ˙x) = −K6−n(op)(x−x(op),x− ˙x˙ (op))T
[ (4.31)] [ 4.32] [ (6.10)] [ (6.12)]
(6.7)CMrot6−n(op) 6 (6.10)Frot6−n(op)− Grot6−n(op)K6−n(op)
K6−n(op) 7
(Frot6−n(op)− Grot6−n(op)K6−n(op)) (xop,x˙op)T
(t0,(x0,˙x0)T) (6.10) (ϕ(t, (x0,x˙0)T), ˙ϕ(t, (x0,˙x0)T)) (Dϕ(t, (xop,x˙op)T), D ˙ϕ(t, (xop,x˙op)T)) (x0− x(op),x˙0− ˙x(op))T (xop,x˙op)T (t0,(x0,x˙0)T) ∈ R × U (ϕ(t, (x0,x˙0)T), ˙ϕ(t, (x0,x˙0)T)) (6.12) ¨r(t) ¨r(op) 8
6.1
4.1 (ii) 4.2 (I) 1 ( 6.1) 5: ψ(op) = θ(op) = φ(op)=0 [rad], ˙ψ(op)= ˙θ(op)= ˙φ(op) =0 [rad/s2], r1(op) =r2(op) = 0[m], r3(op) =3[m], ˙r1(op) =˙r2(op) =˙r3(op) =0[m/s], c (or ¨r3) = 0 [m/s2],
: ωM2(op)= ωM3(op)= ωM5(op)= ωM6(op)=5233.8205 [rpm], ωM4(op)=0 [rpm].
6.1 1 4.2 (I) 5 (a) (a) 1 ( 6.1) t[s] ∈ [0 1], θ = φ = 0 [rad], ¨θ = ¨φ = 0 [rad/s2], k F2 =kF3=kF4 =kF5 =kF6=1.79 × 10(−7)[N/rpm2], kM2 =kM3 =kM4 = kM5 =kM6 =4.38 × 10(−9)[Nm/rpm2], c (or ¨r3) = 0 [m/s2] ( 4.2 (I) 4.1 (ii) ) 6.2 6.3
0 0.2 0.4 0.6 0.8 1 −1 −0.5 0 0.5 1
Rotational behavior of the hexarotor
Time [s]
Tait
−
Bryan angles [rad]
ψ(t) θ(t) φ(t) 6.2 1 ( (a)). 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5
Translational behavior of the hexarotor
Time [s] Movement distance [m] r1(t) r 2(t) r 3(t) 6.3 1 (a) r3=3 [m] 6.2 6.3 ( 4.2 (I) 4.1 (ii) ) 5 F(η, ui1 5) = Ai51(η)ui51− b5, (6.14) Ai1 5(η) ∈ R5×5= ( A! 4×5(η) εTQkQi1 2×5 ) , (6.15) A! 4×5(η) ∈ R4×5= ( Z(η)Si1 rot5 eT 3m1B(x)Sitra51 ) , (6.16) b5=( ¨ψ, ¨θ, ¨φ, c + g,b!1)T ∈ R5, (6.17) i1=1, k = 1,Qi1 2×5∈ R2×5, b!1∈ R . span{εQ1,εQ2} 2 R2 Qi1 2×5 Q2×6 1 Siξ51 Sξ6 1 (ξ = rot, tra) (a) 5 ( 4.2 (I) 4.1 (ii) ) 7 (b) . (b) 6.1 7 1 K5(op)∈ R5×6 K5(op)∈ R5×6=1011· 0.8273 −0.0336 −0.0591 0.0083 −0.0003 −0.0006 −0.8273 0.0336 −0.0591 −0.0083 0.0003 −0.0006 1.6546 0.1346 0 0.0165 0.0013 0 −0.8273 0.0336 0.0591 −0.0083 0.0003 0.0006 0.8273 −0.0336 0.0591 0.0083 −0.0003 0.0006 , (6.18)
(6.8)Frot5(op)− Grot5(op)K5(op) −300, −300, −300, −150, −150,
−150 (6.8)
• δψ(0) = 0.00174533 [rad], δθ(0) = 0 [rad], δφ(0) = 0 [rad], δ ˙ψ(0) = 0 [rad/s], δ˙θ(0) = 0 [rad/s], δ ˙φ(0) = 0 [rad/s].
• θop = 0, [rad], φop = 0, [rad], ˙ψop = 0 [rad/s], ˙θop = 0 [rad/s], ˙φop = 0 [rad/s], ωM2(op) = ωM3(op)= ωM5(op)= ωM6(op)=5233.8205 [rpm], ωM4(op)=0 [rpm].
(6.8) 6.4 6.5 (6.9) δ¨r
6.6 6.7
6.8 6.4 δθ(t) δφ(t) 6.5 d/dt δθ(t)
d/dt δφ(t) 6.6 δr1(t) δr2(t)
6.7 d/dt δr1(t) d/dt δr2(t) 6.8 δu2(t) δu6(t), δu3(t) δu5(t)
0 0.02 0.04 0.06 0.08 0.1 −0.5 0 0.5 1 1.5 2 2.5 3x 10
−3Rotational behavior of the hexarotor
Time [s]
Tait
−
Bryan angles [rad]
δψ(t) δθ(t) δφ(t) 6.4 1 ( (b)) 0 0.02 0.04 0.06 0.08 0.1 −0.1 −0.05 0 0.05 0.1
Rotational behavior of the hexarotor
Time [s]
Angular velocities of Tait
−
Bryan angles [rad/s]
d/dt δψ(t) d/dt δθ(t) d/dt δφ(t) 6.5 1 ( (b)) 0 0.02 0.04 0.06 0.08 0.1 −5 −4 −3 −2 −1 0 1 2 3 4x 10
−4Translational behavior of the hexarotor
Time [s] Movement distance [m] δr 1(t) δr 2(t) δr 3(t) 6.6 1 ( (b)) 0 0.02 0.04 0.06 0.08 0.1 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
Translational behavior of the hexarotor
Time [s] Movement velocity [m/s] d/dt δr1(t) d/dt δr 2(t) d/dt δr 3(t) 6.7 1 ( (b))
0 0.02 0.04 0.06 0.08 0.1 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5x 10
8 Motor speed control signals of the hexarotor
Time [s]
Motor speed control signals [rpm
2] δu 2(t) δu 3(t) δu 4(t) δu 5(t) δu 6(t) 6.8 1 ( (b)) (b) 7 ( 4.2 (I) ) 8 (c) (6.18) K5(op) (c) 6.1 (t [s] ∈ [0 0.1]):
• δψ(0) = 0.00174533 [rad], δθ(0) = 0 [rad], δφ(0) = 0 [rad], δ ˙ψ(0) = 0 [rad/s], δ˙θ(0) = 0 [rad/s], δ ˙φ(0) = 0 [rad/s].
• θop = 0, [rad], φop = 0, [rad], ˙ψop = 0 [rad/s], ˙θop = 0 [rad/s], ˙φop = 0 [rad/s], ωM2(op) = ωM3(op)= ωM5(op)= ωM6(op)=5233.8205 [rpm], ωM4(op)=0 [rpm].
[ (6.10)] 6.9 6.10 [ (6.12)] 6.11 6.12 6.13 6.9 δθ(t) δφ(t) 6.10 d/dt δθ(t) d/dt δφ(t) 6.11 δr1(t) δr2(t) 6.12 d/dt δr1(t)
d/dt δr2(t) 6.13 δu2(t) δu6(t), δu3(t) δu5(t)
(c) 8
0 0.02 0.04 0.06 0.08 0.1 −0.5 0 0.5 1 1.5 2 2.5 3x 10
−3Rotational behavior of the hexarotor
Time [s]
Tait
−
Bryan angles [rad]
δψ(t) δθ(t) δφ(t) 6.9 1 ( (c)) 0 0.02 0.04 0.06 0.08 0.1 −0.1 −0.05 0 0.05 0.1
Rotational behavior of the hexarotor
Time [s]
Angular velocity [rad/s]
d/dt δψ(t) d/dt δθ(t) d/dt δφ(t) 6.10 1 ( (c)) 0 0.02 0.04 0.06 0.08 0.1 −5 −4 −3 −2 −1 0 1 2 3 4x 10
−4Translational behavior of the hexarotor
Time [s] Movement distance [m] δr 1(t) δr 2(t) δr 3(t) 6.11 1 ( (c)) 0 0.02 0.04 0.06 0.08 0.1 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
Translational behavior of the hexarotor
Time [s] Movement velocity [m/s] d/dt δr 1(t) d/dt δr2(t) d/dt δr 3(t) 6.12 1 ( (c)) 0 0.02 0.04 0.06 0.08 0.1 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5x 10
8 Motor speed control signals of the hexarotor
Time [s]
Motor speed control signals [rpm
2] δu 2(t) δu 3(t) δu 4(t) δu 5(t) δu 6(t) 6.13 1 ( (c))
7
7.1
1. B 3 2 w W Kalman ei∈ w, Ei∈ W (i = 1, 2, 3) 1 2 1 3.20 ψ 2. Kalman 1 2 4.1 3 4 3 4throttle ( altitude), rudder ( yaw), elevator ( pitch), ailerons ( roll)
“ ” 3. 2 4.2 2 4.2 (I) (II) 5 6 4. 5 6
2
4.2
2 5 6 1
Electric Speed Controller
4 5 6 Q(2p−4)×2p Q(2p−4)×2p ( (5.1), (5.2)). 5. 4.2 (I) 7 8
7.2
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A
Implicit Function Theorem: theorem 13.1.1 [40] F(x, y) x ∈ R˜ n ˜y ∈ Rm
R3 C1 F(˜x, ˜y) = c
Fy(˜x, ˜y) x ∈ U y(˜x) = ˜y F(x, y(x)) = c
C1 y : U → Rm x˜ U F(x, z) = c V
z z = y(x) x˜ V(V y(U)
) y
(i)y , m × n dy(x)
dy(x) = −[Fy(x, y(x))−1Fx(x, y(x))]. (A.1) (ii)y(x), x ∈ U (˜x, ˜y)
y(x) ≈ ˜y + dy(˜x) · (x − ˜x)
=y − [F˜ y(˜x, ˜y)−1Fx(˜x, ˜y)] · (x − ˜x). (A.2)
(i) [40] theorem 13.1.1 Strichartz (ii)
theorem 13.1.1 (ii)
B
F(x, y) x ∈ U+ ⊂ U ⊂ Rn y ∈ Vδ⊂ V ⊂ Rm Rm C1 (B.3) F(x, y) = c. (B.3) (B.3) F(x, y) = c y Fy(x, y) (x, y) = (˜x, ˜y) 0 F(x, y) = c : ˜y s.t. F(˜x, ˜y) ≈ c : +, δ : κ1<1, κ2<1 (κ1+ κ2<1) (i) U+ ={x | ,x − ˜x, ≤ +} ⊂ U, (ii) Vδ={y | ,y − ˜y, ≤ δ} ⊂ V, (iii) ,Fy(˜x, ˜y) − Fy(˜x, y), ≤ κ1/M, for anyy ∈ Vδ,
(iv) ,Fy(˜x, y) − Fy(x, y), ≤ κ2/M, for anyy ∈ Vδ,and for anyx ∈ U+, (v) M(r+M1−(κ1+κ-2+)) < δ,
(B.4)
r, M, M- (> 0) 3
(i) ,c − F(˜x, ˜y), ≤ r and ,Fy(˜x, ˜y)−1, ≤ M,
(ii) ,Fx(x, ˜y), ≤ M-for anyx ∈ U+, (B.5)
(B.3)F(x, y) = c x ∈ U+ F(x, y(x)) = c C1
: y(x) ∈ Vδ
,y(x) − ˜y, ≤ 1 − (κM(r + M-+)
1+ κ2). (B.6)
(i) x ∈ U+ y , m × n dy(x)
dy(x) = −[Fy(x, y(x))−1Fx(x, y(x))]. (B.7) (ii)y(x), x ∈ U+ F(x-,y(x-)) =c, x-(! x) ∈ U+ (x-,y(x-))
y(x) ≈ y(x-) + dy(x-) · (x − x-)
=y(x-) − [Fy(x-,y(x-))−1Fx(x-,y(x-))] · (x − x-). (B.8)
[40] [41] [42] [42] (i) y (ii) 9 3 H(u2p,x) u2p ∈ U+ ⊂ R2p x ∈ Vδ ⊂ R3 R3 C1 (4.1) (4.2) H(u 2p,x) = c H(u2p,x) = (ε i,Z(η)Srot2pu2p) (ej,−ge3+m1B(x)Stra2pu2p) (e3,−ge3+m1B(x)Stra2pu2p) = 0, (B.9) Z(η) = (B(x) ˆIB(x)T· ω ˙ x(ψ, θ))−1· B(x), (B.10) x = (ψ, η)T, η = (θ, φ)T. 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, i, j ∈ N. c = (0, ¨r j,c)T∈ R3, ¨rj,c ∈ R
(B.9) H(u2p,x) = c x Hx(u2p,x) (u2p,x) = (˜u2p,x)˜ 0 H(u2p,x) = c : ˜x s.t. H(˜u2p,x) ≈ c˜
+, δ κ1<1, κ2<1 (κ1+ κ2<1) (i) U+ = 4 u2p| ,u2p− ˜u2p, ≤ +5⊂ R2p, (ii) Vδ={x | ,x − ˜x, ≤ δ} ⊂ R3, (iii) ,Hx(˜u2p,x) − H˜ x(˜u2p,x), ≤ κ1/M, for anyx ∈ Vδ, (iv) ,Hx(˜u2p,x) − Hx(u2p,x), ≤ κ2/M, for anyx ∈ Vδ,and for anyu2p∈ U+, (v) M(r+M1−(κ1+κ-+)2) < δ,
(B.11)
r, M, M- (> 0) 3
(i) ,c − H(˜u2p,x), ≤ r and ,H˜ x(˜u2p,x)˜ −1, ≤ M, (ii) ,Hu2p(u2p,x), ≤ M˜ -for anyu2p∈ U+,
(B.12) (B.9)H(u2p,x) = c u2p∈ U+ H(u2p,x(u2p)) =c
C1 : x(u2p) ∈ V δ ,x(u2p) − ˜x, ≤ M(r + M -+) 1 − (κ1+ κ2). (B.13) (i) u2p∈ U+ x , 3 × 2p dx(u2p)
dx(u2p) = −[Hx(u2p,x(u2p))−1Hu2p(u2p,x(u2p))]. (B.14)
(ii)x(u2p),u2p∈ U+ H(u-2p,x(u-2p)) = c, u-2p(! u2p) ∈ U+ (u-2p,x(u-2p)) x(u2p) ≈ x(u-2p) + dx(u-2p) · (u2p− u-2p)
=x(u-2p) − [Hx(u-2p,x(u-2p))−1Hu2p(u-2p,x(u-2p))] · (u2p− u-2p). (B.15)
(iii) ¨ψ
¨ψ = (ε1,Y(η, ˙x) + Z(η)Srot2pu2p). (B.16) 3
C
f: U ⊂ Rn→ Rn dx dt = f(x), x(t0) =x0, (C.17) t ∈ R: ,x ∈ U ⊂ Rn: t n x 0: , n ∈ N. : (C.17) t = 0, t ∈ (−c, c), c ∈ R ϕt : Rn → Rn ϕt(x0) =x(t, x0) x,¯ f (¯x) = 0, (C.17) x¯ Dϕt(¯x)ξ x¯ ϕt Dϕt(¯x)ξ = exp(tD f(¯x))ξ. (C.18) ¯ x (C.17) Dϕt(¯x)ξ (C.19) dξ dt =D f(¯x)ξ, x(t0) =x0, (C.19) D f =6∂fi ∂xj 7 f = ( f1(x1, . . . ,xn), f2(x1, . . . ,xn), . . . , fn(x1, . . . ,xn))T, (T ) 1 x = ¯x + ξ, |ξ| ≤ 1. (C.17) x¯ ϕt (C.19) Dϕt(¯x)ξ Dϕt(¯x)ξ x¯ ϕt Theorem 1. 3. 1 (Hartman–Grobman) [44], p. 13 D f(¯x) 0 (C.17) ϕt (C.19) exp(tD f(¯x)) ¯ x ∈ Rn U h t1. K. Isogai and H. Okazaki, “CONDITIONS TO REALIZE HEXAROTOR STABLE FLIGHTS TO AVOID A CRASH UNDER COMPLETE PROPELLOR MOTOR FAILURES.,” Int. J. Adv. Res. 8(01), 2020, pp. 261–278. 6
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2. H. Okazaki, Siyuan Yin, K. Isogai, and H. Nakano, “Motor speed control signals for multirotor flightsin the presence of complete propeller motor failures,” 2018 IEEE 61st International Midwest Symposium on Circuits and Systems (The IEEE MWSCAS 2018), pp. 384–387, 5–8 August 2018, Windor, Canada. 4.2
3. K. Isogai, R. Inohara, H. Nakano, and H. Okazaki, “Modeling and Simulation of Motion of a Quad-copter,” 2016 International Symposium on Nonlinear Theory and Its Applications (NOLTA2016), pp. 165–168, 27–30 November 2016, Yugawara, Japan. 2
4. H. Okazaki, K. Isogai, and H. Nakano, “Modeling and Simulation of Motion of a Quadcopter in a Light Wind,” 2016 IEEE 59th International Midwest Symposium on Circuits and Systems (The IEEE MWSCAS 2016), pp. 117–120, 16–19 October 2016, Abu Dhabi, UAE. 2
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10 . 4.1 A B
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, vol. 117, no. 225, CAS2017-29, pp. 29–33, 2017 10 . 4.2
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