MEMS INS/GPS 2004/11
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- あおし さどひら
- 5 years ago
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1 MEMS INS/GPS 4/11
2 INS/GPS Kalman Filter Kalman Filter Kalman Filter UD Kalman Filter System Design INS/GPS i
3 A Quaternion Algebra 44 A.1 Quaternion A. Quaternion A.3 Quaternion A.4 Quaternion A.5 Quaternion A.6 Quaternion A.7 Quaternion A.8 Quaternion A.9 Quaternion Direction Cosine Matrix B Coordinate Systems 51 B.1 Earth-Centered Inertial Frame i-frame B. Earth-Centered, Earth-Fixed Frame e-frame B.3 Local Geodetic Frame g-frame B.4 Navigation Frame Wander Azimuth Frame n-frame B.5 Body Frame b-frame B B.7 i-frame e-frame B.8 e-frame g-frame B.9 g-frame n-frame B.1 e-frame n-frame B.11 n-frame b-frame C Earth ModelWGS C C C C.4 WGS D Allan Variance 63 D.1 Allan Variance E 64 F 65 F F ii
4 F iii
5 Update Correct ADXRS15 Allan Variance Allan Variance Allan Variance B.1 e-frame g-frame B. g-frame n-frame B.3 n-frame b-frame C iv
6 A B C.1 WGS v
7 1 1.1 Unmanned Aero Vehicle UAV UAV g cm UAV R/C Radio Contorl UAV UAV 1 Inertial Navigation System INS kg UAV UAV UAV UAV 1
8 INS INS GPS DGPS GPS GPS GPS GPS m cm 1Hz 1Hz Hz Hz Hz 1. Inatial Navigation System INS Inatial Navigation System INS INS INS INS
9 INS INS INS INS.1 /hr Ring Laser GyroRLG RLG / Micro Electro Mechanical SystemsMEMS 1 RLG. GPS Global Positioning SystemGPS GPS GPS GPS m Hz 3
10 GPS DGPS GPS DGPS Differntial GPSDGPS cm GPS GPS 3. UAV 1.. UAV m m UAV 1m 1g W 1 D 1 H 1 cm 3 1 4
11 INS MEMS INS MEMS GPS INS GPS GPS 1.3 INS/GPS UAV INS GPS Kalman Filter INS/GPS Kalman Filter INS GPS Kalman Filter Loose Coupling INS GPS INS Loose Coupling GPS Tight Coupling INS GPS Tight Coupling Loose Coupling GPS GPS Loose Coupling
12 Kalman Filter INS/GPS INS GPS Kalman Filter Kalman Filter Kalman Filter.1 x z x z z = Hx + v.1.1 v R E[v] =, E [ vv T] = R.1. x z J LS J LS = z Hx T z Hx.1.3 x J LS x J LS = z T z z T Hx x T H T z + x T H T Hx.1.4 = T z T H H T z T + H T Hx T + x T H T H = z T H x T H T H.1.5 6
13 x J LS x = H T H.1.6 J LS x J LS x = x J LS x ˆx.1.5 z T H = ˆx T H T H.1.7 H T H ˆx = H T H 1 H T z.1.8 ˆx 1. ε ε x ˆx = x H T H 1 H T z = x H T H 1 H T Hx + v.1.9 = H T H 1 H T v. ν ν z ẑ = H x ˆx + v = Hε + v = I H H T H 1 H T v P [ E[ε] = E HH T ] 1 H T v = HH T 1 H T E[v] = P E [ εε T] = E [x ˆxx ˆx T] H = E[ T H 1 H H v T T H ] 1 T H T v = H T H 1 H T E [ vv T] H H T H 1 T = H T H 1 H H T RH T H 1 T
14 ..1.1 x z J WLS W W T = W J WLS = z Hx T W z Hx..1 x J LWS = z T Wz x T H T Wz x T WHx + x T H T WHx.. J x = zt WH z T WH + x T H T WH + x T H T WH = z T WH x T H T WH..3 J WLS x ˆx z T WH = ˆx T H T WH..4 ˆx = H T WH 1 H T Wz..5 W R 1 1. ε ε x ˆx = x H T R 1 H 1 H T R 1 z = x H T R 1 H 1 H T R 1 Hx + v..6 = H T R 1 H 1 H T R 1 v. P [ H P E T R 1 H 1 H H T R v 1 T R 1 H ] 1 T H T R 1 v [ H = E T R 1 H 1 H T R 1 v v T R 1 H H T R 1 H ] 1 = H T R 1 H 1 H T R 1 E [ vv T] R 1 H H T R 1 H 1 = H T R 1 H 1 H T R 1 RR 1 H H T R 1 H 1 = H T R 1 H 1 H T R 1 H H T R 1 H 1 = H T R 1 H
15 z 1 h 11 h 1... h 1n x 1 v 1 z.. = h 1 h... h n x v. z m h m1 h m... h mn x m v m.3.1 ˆx m..5 W R 1 P m..7 ˆx m = H T R 1 H 1 H T R 1 z.3. P m = H T R 1 H P m ˆx m ˆx m = P m H T R 1 z.3.4 z m+1 z = z m+1 [ H h T ] v x + v m ˆx m+1 ˆx m ˆx m+1 = ˆx m + x.3.6 ˆx m+1..1 J RWLS z J RWLS = z m+1 T [ ] H R 1 z h T x m+1 T r 1 z m+1 H x h T m [ ] R 1 T r 1..5 ˆx m+1 ˆx m+1 = H h T T [ R 1 T r 1 ] H h T 1 T [ ] H R 1 z h T T r 1 z m+1 = H T R 1 H + hr 1 h T 1 H T R 1 z + hr 1 z m
16 ..7 P m+1 P m+1 = H h T T [ R 1 ] 1 H T r 1 h T = H T R 1 H + hr 1 h T 1 = P m + hr 1 h T = P m P m h h T P m h + r 1 h T P m A 1 = B 1 +C T D 1 C D +CBC T 1 A = B BC T D +CBC T 1 CB.3.1 k m P m h h T P m h + r k m P m+1 = I k m h T P m ˆx m+1 = P m+1 H T R 1 z + hr 1 z m+1 = I k m h T P m H T R 1 z + hr 1 z m+1 = I k m h T ˆx m + I k m h T P m hr 1 z m+1 = I k m h T ˆx m + I P m h h T P m h + r 1 h T k m h T P m h + r r 1 z m+1 = I k m h T h T P m h + r I P m hh T ˆx m + h T k m h T P m h + r r 1 z m+1 P m h + r h T P m h + r.3.13 = I k m h T ˆx m + rk m r 1 z m+1 = I k m h T ˆx m + k m z m+1 = ˆx m + k m zm+1 h Tˆx m.3.6 ˆx z m+1 z m+1 h H.4 Kalman Filter x x k+1 = Φ k+1,k x k + Γ k w k.4.1 1
17 w k ˆx x ˆx x ˆε k x ε k+1 E[ ε k+1 ] x k+1 Γ k+1,kˆx k.4. ˆε k = ˆx k x k.4.3 ε k+1 = ˆx k+1 x k+1 = Φ k+1,kˆx k Φ k+1,k x k + Γ k w k = Φ k+1,k ˆε k Γ k w k.4.4 E[ ε k+1 ] = E[ x k+1 x k+1 ] = Φ k+1,k E[ˆε k ] Γ k E[w k ] =.4.5 P k+1 P k+1 E[ ε k+1 ε T k+1 ] = E[ Φ k+1,k ˆε k Γ k w k Φk+1,k ˆε k Γ k w k T] = Φ k+1,k E[ˆε k ˆε T k ]ΦT k+1,k Γ ke[w k ˆε T k ]ΦT k+1,k.4.6 Φ k+1,k E[ˆε k w T k ]ΓT k + Γ ke[w k w T k ]ΓT k w k ˆε k 1 w E[w k ε T k ] = E[w k Φk,k 1 ˆε k 1 Γ k 1 w k 1 T] = E[w k ˆε T k 1 ]ΦT k,k 1 E[w k wt k 1 ]ΓT k = P k+1 P k+1 = Φ k+1,k ˆP k Φ T k+1,k + Γ kq k Γ T k.4.8 x k+1 z k+1 x k+1 ˆ x k+1 ˆx k+1 = x k+1 + K k+1 [ zk+1 H k+1 x k+1 ].4.9 K k+1 = P k+1 H T k+1 Hk+1 P k+1 H T k+1 + R k
18 ˆ x k+1 x k+1 ˆε k+1 ˆx k+1 = x k+1 + ˆε k x k+1 + ˆε k+1 = Φ k+1,k x k + ˆε k + K k+1 [ zk+1 H k+1 Φ k+1,k x k + ˆε k ] = Φ k+1,k x k + Φ k+1,k ˆε k + K k+1 [ Hk+1 xk+1 Φ k+1,k x k + vk+1 H k+1 Φ k+1,k ˆε k ] ˆP k+1 ˆε k+1 = [I K k+1 H k+1 ]Φ k+1,k ˆε k [I K k+1 H k+1 ]Γ k w k + K k+1 v k ˆP k+1 = E[ˆε k+1 ˆε T k+1 ] = [I K k+1 H k+1 ]Φ k+1,k ˆP k Φ T k+1,k [I K k+1h k+1 ] T.4.14 [I K k+1 H k+1 ]Γ k Q k Γ T k [I K k+1h k+1 ] T + K k+1 R k+1 K T k ˆP k+1 = [I K k+1 H k+1 ] P k+1 [I K k+1 H k+1 ] T + K k+1 R k+1 Kk+1 T ˆP = [I KH] P[I KH] T + KRK T = P KH P PH T K T + KH PH T K T + KRK T = P PH T HPH T + R 1 HP PH T 1 HP + PH T 1 HPH T 1 HP + PH T 1 R 1 HP = P PH T 1 HP PH T 1 HP + PH T 1 1 HP.4.16 = P PH T HPH T + R 1 HP = [I KH] P Kalman Filter update x k+1 = Φ k+1,k x k + Γ k w k.4.1 P k+1 = Φ k+1,k ˆP k Φ T k+1,k + Γ kq k Γ T k.4.8 correct ˆx k+1 = x k+1 + K k+1 [ zk+1 H k+1 x k+1 ].4.9 K k+1 = P k+1 H T k+1 Hk+1 P k+1 H T k+1 + R k ˆP k+1 = [I K k+1 H k+1 ] P k
19 .5 Kalman Filter Kalman Filter.4.1 d x ẋ = Ax + Bw.5.1 dt t x x x A t x + B t w.5. x + x = I + A tx + B t w Φ k+1,k = I + A t.5.4 Γ k = B t.5.5 Kalman Filter.6 UD Kalman Filter.4.1,.4.17,.4.9,.4.17 Kalman Filter Kalman Filter Filter.4.9 Kalman Gain K Filter Filter 1 P, Q, R UD Kalman Filter.6.1 UD 13
20 3 System Design INS/GPS INS/GPS Kalman Filter INS/GPS Quaternion q q q 1 q q 3 3 x x 1 x x 3 A b-frame B 3.1 INS e-frame i-frame r e r i e-frame i-frame Quaternion { { = q ei q r e ri e i Quaternion q = 1 { q 3.1. q = q = 1 { q = 1 { q
21 3.1.1 { r e e = d { q ei q dt ri e i { { { = q e i q e r i + q e i i r i q e i + q e i q e i r i i = 1 { { { qe i e/i i q e r i + q e i i r i q e i + 1 { { i qe i r i e/i i q e i { { = q e i r i q e i q e i i e/i i r q e i i { { = r i e q e i e/i i r q e i i r 1 1-Frame -Frame 3 1/ -Frame 1-Frame 3-Frame e/i { r e e = d { q e i dt r i i { = q e i = q e i { = { r e i r i i r i i { q e i q e i e/i i r q e i i { q e i q e i e/i i r i i q e i { { q e i q e i e/i i r i i q e i + q e i e/i i { + q e i q e i { q e i e/i i r i i {{ { { q e i e/i i r i i q e i q e i e/i i r i e/i i e/i i r q e i i {{ e-frame { { r e n = q n e r e e q e i e/i i e/i i r q e i i q n e { r e n = d { q n e dt r e q n e e { { = q n e r e q n e q n e e n/e e r e e q n e
22 3.1.5 { r n e = q n e { = { r n i r e i { { q e i e/i i r i i q e i + q e i { q ne { e/i e r i e { r e i = { e/i i e/i i r i { + e/i e e/i e r e r e e = { r e e + q e i { + q e i e n/e r e e q e i { q n e q n e n/e e q n e r e e q n e e/i i r i { e e/i r e { { e/i e r i e = e/i{ e r i e { = e/i[{ e { = e/i e r e e r e e { + + ] e/i e r e { e/i e e/i e r e { { { r e n = r i n q ne e/i e r e e { = { r n i q ne { + e/i e e/i e r e { e/i e r e e + { e/i e e/i e r e { + e n/e r e e q n e e/i e e/i e r e { + e n/e r e e q n e a b i-frame b-frame { { { a b = q b n r n q b n i g b {{ { { r i n = q n b a b = q n b { a b + { g b q n b q n b + { g n
23 n-frame { { d dt r e n r e n = q n b { a b { q n b + g n { q n e { { = q n b a b q n b + g n { e/i e e/i n { { = q n b a b q n b + g n { r n e r e e e/i n + n n/e { + { + { e/i e e/i e r + e n n/e r n e q ne { e n/e r e e q n e e/i e e/i e r q n e e { r e n q ne e/i e e/i e r q n e e e/i n { { e/i n = q n e e/i e q n e ω e/i Ω e/i e/i i = e e/i = Ω e/i n/e n { { n/e n = q n g g q n g n/e g n/e φ λ d dt φ v N φ = R meridian + h d dt λ λ = v E β = v E R normal + hcosφ R meridian R normal h v N v E WGS-84 C R meridian R normal R meridian = r e 1 ε 1 ε sin φ
24 r e R normal = 1 ε sin φ v N v E v D v N v E v D n/e { r g e { = q g n r n q g n 3.1. e g n/e east = φ g n/e down = n n/e down = λ sinφ e n/e z = λ { g n/e = g n/e east + q g e q g e g n/e down n/e e z e/i e e/i e r e 3.1 λ cosφ g n/e = φ = v E R normal v +h N R meridian +h e/i e e/i e r e = Ω e/i β cosλ sinλ = Ω e/i R normal + hcosφ cosλ sinλ B q n e e/i e e/i e r e = Ω e/i R normal + h qn e q n e + q n e 1 q n e 3 q n e 3 q n e q n e 1 q n e
25 Ze β re ϖe/i ϖe/i * ϖe/i * re λ φ Ye Xe φ λ h q n e h q n e Quaternion d dt qn e q n e = 1 { n/e e q n e = 1 { qe n n/e n q e n q n e { = 1 qn e n n/e n n/e h d dt h = v D ṙ n e Z q b n Quaternion d dt qb n q b n = 1 { b/n n q b n
26 n b/n = n b/i n n/i { = n b/i n e/i + n n/e n b/i { = q n b b/i b q n b e/i n n n/e b b/i i-frame b-frame b-frame q b n = 1 = 1 [ q n b { b b/i [ { q b n b b/i { q n b { n e/i n e/i { + { + n n/e n n/e ] q b n q b n ] INS/GPS INS GPS Kalman Filtering { d dt r e n = d { { dt r e n + r e n r e n [ { = q b n + q b { n a b + a b q b n + q n b + g n + g n { e/i n + n e/i + n/e n + n n/e r n e + r e n { ] q n e + q n e e/i e e/i e r q n e + r e e + q n e [ { { q b n a b q b n + g n { e/i n + n n/e r e n { q n e ] e/i e e/i e r q n e e 3..1
27 { { d dt r e n = q b n a b q b n + q b n { e/i n + n n/e r e n + { q n e e/i e e/i e r e { q n e e/i e e/i e r e { { { a b q b n + q b n a b q b n + g n e/i n + n n/e r e n q n e q n e q n e { e/i e e/i e r q n e e 3.. n e/i e/i { n e/i { { = q n e + q n e e/i i q n e + q n e q n e e/i i q n e 3..3 n/e n n n/e { n n/e = q n g { g n/e q n g v E = q n g R normal +h v N q n g q n g R meridian +h = 1 r e + h qn g q g n ṙn e Y ṙ e n X q g n q n g Z g Z n = 1 r e + h ṙn e Y ṙ e n X 1 r e +h v E v N n/e n = 1 ṙn e Y + ṙ n e Y ṙ n r e + h + h e X ṙe n X 1 ṙn e Y ṙ n r e + h e X = 1 ṙn e Y ṙ n r e + h e X 1 ṙn e Y r e + h ṙe n X h q n g
28 e/i e e/i e r e e/i e e/i e r e = Ω e/i R normal + h + h q + q q + q + q 1 + q 1 q 3 + q 3 q 3 + q 3 q + q q 1 + q 1 q + q Ω e/i R normal + h q q + q 1 q 3 q 3 q q 1 q q n e = Ω e/i q q + q 1 q 3 q 3 q q 1 q h q n e +R normal + h q q 3 q q 1 q 1 q q 3 q q n e q n e q n e d dt qn e = 1 { qn e + q n e n/e n + 1 { n n/e qn e n/e n = 1 { qn e + 1 { 3..7 qn e n n/e 3..3 n n/e d dt h = ṙn e Z + ṙ n e Z ṙ n e Z = ṙ n e Z d dt qb n = 1 [ { q b n + q b n = 1 1 [ q b n b/i b + b b/i { { e/i n + + n e/i n/e n + n n/e [ { { { ] q b n + q b n { b b/i b b/i { n e/i { + q b n { + n e/i b b/i n n/e n n/e ] q b n + q b n { { ] q b n e/i n + n/e n q b n 3..9
29 d dt x = A x + B w 3..1 x w A B ṙ n e X ṙe n Y ṙ n e Z q n e q n e 1 x = q n e q n e 3 h q b n q b n 1 q b n q b n 3 a b X a b Y a b Z w = ωb/i b X ω b b/i Y ωb/i b Z g
30 A = n e/i + n n/e n e/i + n n/e Y Z e/i n + n n/e e/i n + n n/e ṙ e n Z re+h n e/i + n n/e X ṙ e n Z re+h Z ṙn e X Y re+h e/i n + n n/e ṙn e Y X re+h q n e re+h qn e1 re+h q n e3 q n e re+h re+h qn e q n e3 re+h re+h qn e1 qn re+h e re+h 1 q b n1 re+h qb n re+h q b n3 re+h qb n re+h q b n q b n3 re+h re+h qb n1 re+h qb n re+h 4Ω e/i q n e1ṙ ez n q n eṙ ey n 4Ω e/i q n eṙ ez n + q n e1ṙ ey n 4Ω e/i q n e3ṙ ez n + q n eṙ ey n 4Ω e/i q n eṙ ez n q n e3ṙ ey n 4Ω e/i q n eṙ ex n + q n eṙ ez n 4Ω e/i q n e1ṙ ex n q n e3ṙ ez n 4Ω e/i q n eṙ ex n + q n eṙ ez n 4Ω e/i q n e3ṙ ex n q n e1ṙ ez n 4Ω e/i q n eṙ ey n q n e1ṙ ex n 4Ω e/i q n e3ṙ ey n q n eṙ ex n 4Ω e/i q n eṙ ey n q n e3ṙ ex n 4Ω e/i q n e1ṙ ey n q n eṙ ex n / ω n/e e / X / ω e n/e Y / / ω e n/e Z / ω n/e e X / / ω e n/e Z ω e n/e Y / ω n/e e Y / ω e n/e Z / / ω e n/e X ω n/e e Z ω e n/e Y ω e n/e X { Ω e/i q b n 1q n e q b nq n e1 q b n3q n { e Ω e/i q b n 1q n e3 q b nq n e + q b n3q n { e1 Ω e/i q b n 1q n e q b nq n e3 + q b n3q n { e Ω e/i q b { n 1q n e1 q b nq n e q b n3q n e3 Ω e/i q b n q n e + q b n3q n e1 q b nq n { e Ω e/i q b n q n e3 + q b n3q n e + q b nq n { e1 Ω e/i q b n q n e + q b n3q n e3 + q b nq n { e Ωe/i +q b { n q n e1 + q b n3q n e q b nq n e3 Ω e/i q b n 3q n e + q b nq n e1 + q b n1q n { e Ω e/i q b n 3q n e3 + q b nq n e q b n1q n { e1 Ω e/i q b n 3q n e + q b nq n e3 q b n1q n { e Ω e/i q b { n 3q n e1 + q b nq n e + q b n1q n e3 Ω e/i q b n q n e q b n1q n e1 + q b nq n { e Ω e/i q b n q n e3 q b n1q n e q b nq n { e1 Ω e/i q b n q n e q b n1q n e3 q b nq n { e Ω e/i q b n q n e1 q b n1q n e + q b nq n e3 4 { a b Xq b n1 + a b Y q b n + a b Zq b n3 { a b Xq b n + a b Y q b n1 + a b Zq b n { a b Xq b n3 a b Y q b n + a b Zq b n1 ṙn e X ṙ e n Z { a b Xq b re+h n a b Y q b n3 + a b Zq b n { a b Xq b n a b Y q b n3 + a b Zq b n { a b Xq b re+h n3 + a b Y q b n a b Zq b n1 { a b Xq b n a b Y q b n1 a b Zq b n { a b Xq b n1 + a b Y q b n + a b Zq b n3 ṙ e n X +ṙn e Y { a b Xq b re+h n + a b Y q b n1 + a b Zq b { n a b Xq b n3 + a b Y q b n a b Zq b { n1 a b Xq b n + a b Y q b n3 a b Zq b n ṙn e Y ṙ n e Z { a b Xq b n1 + a b Y q b n + a b Zq b n3 q n eṙ n e X +q n e1ṙ n e Y re+h q n e3ṙ n e X q n eṙ n e Y re+h q n eṙ n e X q n e3ṙ n e Y re+h q n e1ṙ n e X +q n eṙ n e Y re+h { / { / / { / { { ω b/i b Z + ω e/i n + ωn n/e Z ω b/i b X + ω e/i n + ωn n/e X ω b/i b Y + ω e/i n + ωn n/e Y / { / ω ω b/i b Y ω e/i n + ωn n/e Y b/i b Z + ω e/i n + ωn n/e Z / { / / ω b/i b X + ω e/i n + ωn n/e X b n + ωn ω b/i b Z ω e/i n + ωn n/e Z / { / { ω b/i b Y + ω e/i n + ωn n/e Y / ω b/i b X ω e/i n + ωn n/e X q b nṙ e n X q b n1ṙ e n Y re+h q b n3ṙ e n X +q b nṙ e n Y re+h {ω b b/i X ω n e/i + ωn n/e X q b nṙ n e X q b n3ṙ n e Y re+h {ω b b/i Y ω n e/i + ωn n/e Y q b n1ṙ n e X +q b nṙ n e Y re+h {ω b b/i Z ω n e/i + ωn n/e Z
31 3..14 q b q b n1q b q b nq b q b n1q b + q b nq b n + q b n1 q b n q b n3 n n3 q b n1q b n + q b nq b n3 q b n q b n1 + q b n q b n3 n3 n q b nq b n3 q b nq b n1 q b n1q b n3 q b nq b n q b nq b n3 + q b nq b n1 q b n q b n1 q b n + q b n3 1 / q b n1 b n b n 3 / q / q / / / n q b n 3 / q b / q b n / n3 q b n / q b / q b n 1 n q b n 1 / q b q b n B = 5
32 3.3 INS/GPS GPS INS GPS φ λ h 3..1 INS GPS INS GPS GPS φ λ INS Azimuth α GPS q n egps INS q n eins 1 GPS h GPS INS h INS GPS INS Azimuth α n-frame ṙe n GPS X ṙe n GPS Y ṙe n INS X ṙe n INS Y 3 z = Hx + v x 3..4 v z ṙe n GPS X ṙ n e INS X ṙe n GPS Y ṙe n INS Y q n egps q n eins z = q n egps 1 q n eins 1 q n egps q n eins q n egps 3 q n eins 3 h GPS h INS 3.3. H H =
33 3.4 INS/GPS INS/GPS Initialize Flight Flight Flight INS GPS Update Correct Initialize Initialize INS r e n q n e q b n r e n = r e n GPS GPS m 3 1 Initialize Filight GPS INS/GPS Kalman Filter INS P 3.4. Flight :Update INS Update INS r n e q n e q b n INS/GPS Kalman Filter P INS r n e qb n qb n INS INS P INS.4.8 P Φ Γ
34 n g h Strap-down a r b & r b ϖ b/ i q ~b * n n r &r e n q ~ e h r b ϖ b/ n r b ϖ n/ i ΦPΦ +ΓQ Γ T T Pk k k P k Update Flight :Correct GPS GPS Kalman Filter Correct GPS INS P INS r n e q n e q b n P P.4.17 GPS INS 3.3 INS r e n qb n qb n GPS INS P.4.9 INS 8
35 r &r n e q ~ n e h b q ~ n r &r n e q ~ n e h b q ~ n z K P k+1 ˆ +1 P k H K 3.3 Correct 9
36 4 4.1 INS MEMS GPS 1m 1 1 1[m] π/1 [sec] 5[m/s] UAV INS Flight Initialize GPS GPS 1Hz m 3m 1m/s 5 3
37 1 INS/GPS Kalman Filter Q R INS/GPS Kalman Filter Q R GPS 1 INS INS/GPS GPS : Kalman Filter σ WN σ WN = σ f 1Hz σ 1Hz 1Hz f Analog Device MEMS ADXL13 MEMS ADXRS15 σ 1Hz ADXL13 = 11[µg] 1 3 [m/s ] σ 1Hz ADXRS15 =.5[ /s] 1 3 [rad/s] 1 31
38 : 1 x drift d dt x drift = βx drift t + wt 4.1. β wt E[wtwτ] = Nδt τ β x driftk+1 = βx drift t + Nu k+1 t u k MEMS Analog Devices MEMS ADXRS15 β N Hz. Allan Variance 3. β N 1Hz 3
39 Allan Variance 5. 4 Allan Variance 3 4 Allan Variance D β Gyro N Gyro β Gyro =.16 NGyro =.8 / SF ADXRS15 [rad/s] SF ADXRS15 = 1.5 [mv/deg/s] SF ADXRS15 ADXRS15 Scale Factor ADXRS15 Allan Variance Allan Variance Low Pass Filter Allan Variance 4. 33
40 ADXRS15 Allan STD DEV ADEV Line Fit Lower Bound Upper Bound.1 στ τ Produced by AlaVar ADXRS15 Allan Variance ADXRS_SIM Allan STD DEV ADEV Line Fit Lower Bound Upper Bound.1 στ τ Produced by AlaVar Allan Variance 34
41 ADXRS_SIM1 Allan STD DEV ADEV Line Fit Lower Bound Upper Bound.1 στ.1 1E τ Produced by AlaVar Allan Variance MEMS INS/GPS 1 INS/GPS 3/4 GPS 3m INS/GPS 35
42 3m ] [m 度 高 - -4 高度 [m] 真値 高度 [m]ins のみ モデル 1 高度 [m]ins/gps モデル 1 高度 [m]ins のみ モデル 高度 [m]ins/gps モデル GPS 高度 時間 [ 秒 ]
43 8 6 4 s ] / [m - 度速向 -4 方北 北方向速度 [m/s] 真値 北方向速度 [m/s]ins のみ モデル 1 北方向速度 [m/s]ins/gps モデル 1 北方向速度 [m/s]ins のみ モデル 北方向速度 [m/s]ins/gps モデル 時間 [ 秒 ] s ] / [m 度速向 方東 東方向速度 [m/s] 真値 東方向速度 [m/s]ins のみ モデル 1 東方向速度 [m/s]ins/gps モデル 1 東方向速度 [m/s]ins のみ モデル 東方向速度 [m/s]ins/gps モデル 時間 [ 秒 ]
44 s ] / [m 度.6.4 速. 向方力 重 重力方向速度 [m/s] 真値 重力方向速度 [m/s]ins のみ モデル 1 重力方向速度 [m/s]ins/gps モデル 1 重力方向速度 [m/s]ins のみ モデル 重力方向速度 [m/s]ins/gps モデル 時間 [ 秒 ] INS/GPS INS 1 INS/GPS Kalman Filter INS/GPS 1 INS/GPS 1/4 Kalman Filter 38
45 ] Ψ[deg] 真値 e g re Ψ[deg]INSのみ モデル1 e Ψ[deg]INS/GPS モデル1 [d ー Ψ[deg]INSのみ モデル ヨ -5 Ψ[deg]INS/GPS モデル 時間 [ 秒 ] ] -1 e Θ[deg] 真値 g re Θ[deg]INSのみ モデル1 e [d - Θ[deg]INS/GPS モデル1 チッ Θ[deg]INSのみ モデル ピ -3 Θ[deg]INS/GPS モデル 時間 [ 秒 ]
46 ] e -1 Φ[deg] 真値 g re Φ[deg]INSのみ モデル1 e [d - Φ[deg]INS/GPS モデル1 ルー Φ[deg]INSのみ モデル ロ-3 Φ[deg]INS/GPS モデル 時間 [ 秒 ] INS/GPS Kalman Filter INS/GPS MEMS INS/GPS 4
47 ] 秒 557 [3 緯北 真値 INS のみ モデル INS/GPS モデル 東経 [139 秒 ] ] e g re e [d ル ーロ - Φ[deg] 真値 Φ[deg]INS のみ モデル Φ[deg]INS/GPS モデル 時間 [ 秒 ]
48
49 6 INS/GPS MEMS MEMS 43
50 A Quaternion Algebra 3 1. Euler. Direction Cosine MatrixDCM 3. Quaternion A.1 1. Euler. DCM 3. Quaternion ?? Kalman Filtering Quaternion Quaternion 44
51 Quaternion q x x 1 x x 3 q q 1 q q 3 3 q p 1 p p Quaternion p p 1 p p 3 3 { q p 3 A.1 Quaternion Quaternion 1 1, i, j, k q q q 1 1 q q + i q 1 + j q + k q 3 q 3 A.1.1 1, i, j, k 1 1 = 1 A.1. i i = j j = k k = 1 A.1.3 i j = j i = k A.1.4 j k = k j = i A.1.5 k i = i k = j A.1.6 A. Quaternion Quaternion q q q q 1 q q 3 = q q 1 q q 3 A..1 45
52 A.3 Quaternion Quarternion Quarternion q q q 1 q q 3 a q a q q a a q 1 a A.3.1 q a q 3 a A.4 Quaternion Quaternion Quaternion q a q q a a1, q q b a q a3 q b q b1 q b q b3 A.4.1 q a + q b = 1 q a + i q a1 + j q a + k q a3 + 1 q b + i q b1 + j q b + k q b3 = 1 q a + q b + i q a1 + q b1 + j q a + q b + k q a3 + q b3 q a + q b q a1 + q b1 q a + q b q a3 + q b3 A.4. q b + q a = q a + q b A.4.3 A.5 Quaternion Quaternion Quaternion q a q q a a1, q q b a q a3 q b q b1 q b q b3 A
53 q a q b = 1 q a + i q a1 + j q a + k q a3 1 q b + i q b1 + j q b + k q b3 = 1 q a q b q a1 q b1 q a q b q a3 q b3 + i q a q b1 + q a1 q b + q a q b3 q a3 q b + j q a q b q a1 q b + q a q b + q a3 q b1 + k q a q b3 + q a1 q b q a q b1 + q a3 q b q a q b q a1 q b1 q a q b q a3 q b3 q a q b1 + q a1 q b + q a q b3 q a3 q b q a q b q a1 q b + q a q b + q a3 q b1 q a q b3 + q a1 q b q a q b1 + q a3 q b A.5. 3 Quaternion q a q a q b = q q b a1 q a q b1 q b q a3 q b3 q a q b q a1 q a q b1 q b q a3 q b3 q a q b1 q b + q b q a1 q a + q a1 q a q b1 q b q b3 q a3 q a3 q b3 A.5.3 q b q a q b1 q b q b1 q b q q b q a = b3 q b3 q b q a1 q a + q a q b1 q b + q b1 q b q a1 q a q a3 q b3 q b3 q a3 q a q b q a1 q a q b1 q b q = a3 q b3 q a q b1 q b + q b q a1 q a q a1 q a q b1 q b q b3 q a3 q a3 q b3 A.5.4 q a q b A.6 Quaternion 3 Quaternion Quaternion 3 47
54 3 p 3 r r r = 1 θ 3 p p = cosθ p + sinθ q + p A.6.1 p p p r p p p, p r p r A.6. q p r 9 q p r A.6.3 p p r p { p p = cosθ p + sinθ q + p q { cos θ sin θ r = cosθ p + sinθ p r + p = cosθ p p + sinθ p p r + p = cosθ p + 1 cosθ r p p + sinθ p r A.6.4 { { { q cos θ p q cos θ sin θ r p sin θ { r sin θ { = r p cos θ cos θ p sin θ r p sin θ { r sin θ = cos θ r p cos θ p sin θ r p sin θ r sin θ r p r + cos θ cos θ p sin θ r p cos θ p + sin θ r p sin θ { r = sin θ r p r + cos θ p sin θ cos θ r p sin θ r p r r p r = { = sin θ r p r + cos θ p sin θ cos θ r p sin θ r r p r p r r p r = r r p r p r { = sin θ r p r + cos θ sin θ p sin θ cos θ { r p = cosθ p + 1 cosθ r p r + sinθ p r { p A
55 Quaternion p q p p 3 Quaternion q { cos θ sin θ r r r = 1 A.6.6 Quaternion q Quaternion Quaternion Quaternion q Quarternion q q { cos θ sin θ r { cos θ = sin θ { r cos θ = sin θ r A.6.7 A.7 Quaternion A.6 Quaternion 3 Quaternion { Quaternion q 1 q q 1 p { q p { { p = q q 1 q p 1 q { = q 1 q q p 1 q { = q 1 q q p 1 q { q 1 q p 1 q 1 q 1 q q 1 q 1 q Quaternion A
56 A.8 Quaternion Quaternion q d dt q q = 1 ω 1 ω q ω 3 ω 1 ω ω 3 3 A.8.1 A.9 Quaternion Direction Cosine Matrix A.6 Quaternion 3 Direction Cosine MatrixDCM 3 Quaternion q q q 1 q q 3 DCM C q + q 1 q q 3 q 1 q + q q 3 q 1 q 3 q q C = q 1 q q q 3 q q 1 + q q 3 q q 3 + q q 1 q 1 q 3 + q q q q 3 q q 1 q q 1 q + q 3 A.9.1 { { p = q q p A.9. p = C p A.9.3 5
57 B Coordinate Systems 5 1. Earth-Centered Inetial Frame i-frame. Earth-Centered, Earth-Fixed Frame e-frame 3. Body Frame b-frame 4. Local Geodetic Frame g-frame 5. Navigation Frame Wander Azimuth Frame n-frame?? X - Y - Z X i i-frame X B.1 Earth-Centered Inertial Frame i-frame Newton Z i Earth-Centered Inertial Framei-Frame B. Earth-Centered, Earth-Fixed Frame e-frame X e Z e Eeath-Centered, Earth-Fixed Framee-Frame B.3 Local Geodetic Frame g-frame X g Z g Local Geodetic Frameg-Frame Y g X g - Y g - Z g North-East-Down 51
58 B.1 e-frame g-frame N - E - D N E B.4 Navigation Frame Wander Azimuth Frame n-frame Z n Navigation Frameg- Frame g-frame Z g D α rad n-frame Azimuth α = λ sinφ B.4.1 λ φ B.5 Body Frame b-frame X b Z b Body Frameb-Frame b-frame X b Y b Z b 5
59 B. g-frame n-frame B.3 n-frame b-frame B.6 A Quaternion B.1 A.6.7 Quaternion q e i = q i e B.6.1 A.7.1 q b i = q e i q b e = q e i q n e q b n B.6. 53
60 B.1 i-frame e-frame g-frame n-frame b-frame i-frame q e i q g i q n i q b i e-frame q i e q g e q n e q b e g-frame q i g q e g q n g q b g n-frame q i n q e n q g n q b n b-frame q i b q g b q n b q e b B.7 i-frame e-frame i-frame e-frame e-frame i-frame Z i Earth Rate Ω e/i A.8.1 q e i = 1 Ω i/e q n i B.7.1 i-frame e-frame Z i Z e i/e e = i i/e = Ω i/e B.7. B.8 e-frame g-frame e-frame g-frame e-frame g-frame B.1 1. n-frame λ rad λ rad Z e X e - Y e - Z e X e - Y e - Z e X e - Y e - Z e. g-frame φ rad φ rad Y e X e - Y e - Z e X e - Y e - Z e X e - Y e - Z e 3. Y e X e Z e 9 X e - Y e - Z e N - E - D 54
61 Quaternion q g e 1 1 Quaternion cos λ q g e 1 = sin λ 1 cos φ q g e = sin φ q g e 3 = 1 cos 9 sin 9 cos λ = sin λ cos φ = sin φ 1 = A.7.1 Quaternion q g e 1 1 B.8.1 B.8. B.8.3 q g e = q g e 1 q g e q g e 3 cos λ cos φ = sin φ sin λ 1 1 = 1 cos λ cos φ φ + sin sin λ cos φ φ sin cos λ cos φ φ sin sin λ cos φ φ + sin B.8.4 B.9 g-frame n-frame g-frame D α rad n-frame g-frame n-frame B. cos α q n g = sin α B
62 B.1 e-frame n-frame e-frame n-frame q n e = q g e q n g cos λ cos φ φ + sin = 1 sin λ cos φ cos φ sin α cos λ cos φ φ sin sin λ cos φ sin φ + sin α cos λ+α cos φ φ + sin = 1 sin λ α cos φ φ sin cos λ α cos φ φ sin sin λ+α cos φ φ + sin φ = arcsin { q n e + q n e 1 + q n e q n e 3 = arcsin { 1 q n e + q n e 3 B.1.1 B.1. λ = arctan qn e 3 q n e arctan qn e 1 q n e B.1.3 α = arctan qn e 3 q n e + arctan qn e 1 q n e B.1.4 q n e φ λ Azimuth α φ sinφ = 1 q n e + q n e 3 B.1.5 cos φ = 1 sin φ = 1 1 q n e + q n e 3 = 4 q n e + q n e 3 4 q n e + q n e 3 = 4 q n e + q n e 3 1 q n e + q n e 3 = 4 q n e + q n e 3 q n e 1 + q n e B.1.6 cosφ = q n e + q n e 3 q n e 1 + q n e B
63 λ tanλ = tan arctan qn e 3 q n arctan qn e 1 e q n e tan arctan qn e 3 q = n tan arctan qn e 1 e q n e 1 + tan arctan qn e 3 q tan n arctan qn e 1 e q n e = q n e 3 q n e qn e 1 q n e 1 + qn e 3 q n e q n e 1 q n e = qn e 3 q n e q n e 1 q n e q n e q n e + q n e 3 q n e 1 B.1.8 cos λ = = tan λ {qn e 3 q n e q n e 1 q n e {q n e q n e +q n e 3 q n e 1 {q n = e q n e + q n e 3 q n e 1 {q n e q n e + q n e 3 q n e 1 + {q n e 3 q n e q n e 1 q n e {q n = e q n e + q n e 3 q n e 1 {q n e q n e + {q n e 3 q n e 1 + {q n e 3 q n e + {q n e 1 q n e {q n = e q n e + q n e 3 q n e 1 {q n e + q n e 3 {q n e 1 + q n e q n cosλ = e q n e + q n e 3 q n e 1 q n e + q n e 3 q n e 1 + q n e B.1.9 B.1.1 sin λ = tan λ 1 + tan λ {q n = e 3 q n e q n e 1 q n e {q n e + q n e 3 {q n e 1 + q n e sinλ = q n e q n e 1 q n e q n e 3 q n e + q n e 3 q n e 1 + q n e B.1.11 B.1.1 α tanα = tan arctan qn e 3 q n + arctan qn e 1 e q n e = cosα = q n e 3 q n e + qn e 1 q n e 1 qn e 3 q n e q n e 1 q n e = qn e q n e 1 + q n e q n e 3 q n e q n e q n e 1 q n e 3 q n e 1 q n e 3 q n e q n e q n e + q n e 3 q n e 1 + q n e B.1.13 B
64 q n sinα = e q n e 1 + q n e q n e 3 q n e + q n e 3 q n e 1 + q n e B.1.15 B.11 n-frame b-frame n-frame b-frame n-frame yawing Ψ pitching Θ rolling Φ b-frame B.3 cos Ψ q b n = sin Ψ cos Θ sin Θ cos Φ 1 sin Φ 1 1 cos Ψ cos Θ cos Φ + sin Ψ sin Θ sin Φ cos Ψ = cos Θ sin Φ sin Ψ sin Θ cos Φ cos Ψ sin Θ cos Φ + sin Ψ cos Θ sin Φ sin Ψ cos Θ cos Φ cos Ψ sin Θ sin Φ B.11.1 q b n q b Ψ = arctan n 1 q b n + q b n q b n 3 q b n + q b n 1 q b n q b B.11. n 3 Θ = arcsin q b n q b n q b n 1 q b n 3 B.11.3 q b Φ = arctan n q b n 3 + q b n q b n 1 q b n q b n 1 q b n + q b B.11.4 n 3 58
65 C Earth ModelWGS-84 INS WGS-84 WGS-84 C.1 C.1 β r e + z r = 1 C.1.1 p 59
66 βdβ re + zdz r = C.1. p φ dz dβ dz dβ = br p zr e dz π dβ = tan + φ = 1 tanφ 1 tanφ = βr p zr e C.1.3 C.1.4 C.1.5 ε ε = 1 r p r e 1 C.1.6 r p r e = ε 1 C.1.7 z β = 1 ε tanφ C.1.8 C.1.1 z β = r e cosφ 1 ε sin φ 1 C.1.9 β z = r e 1 ε sinφ 1 ε sin φ 1 C.1.1 C dz dβ R meridian = d z dβ C..1 C.1.3 β d z dβ = r4 p r ez 3 6 C..
67 C.1.3 C.1.8 C ε r e R meridian = 1 ε sin φ 3 C..3 C.1.9 R normal = β cosφ r e R normal = 1 ε sin φ 1 C..4 C..5 C.3 Earth RateΩ i/e i-frame G i 1 µ + 3 R J rer 1 5 z xr R + G i = 1 µ + 3 R J rer 1 5 z y R R + 1 µ + 3 R J rer 3 5 z y R R + C.3.1 R = x + y + z i-frame g CF g i g CF = Ω e/i Ω e/i r i g i = G i Ω e/i Ω e/i r i C.3. C.3.3 C.4 WGS-84 WGS-84 C.1 WGS-84 C..3 C..5 WGS-84 g = g WGS 1 + gwgs1 sin φ 1 ε sin φ 1 C
68 C.1 WGS-84 r e m Earth Rate Ω e/i rad/s µ m 3 /s J f r p m ε g WGS m/s g WGS
69 D Allan Variance Allan Variance Allan Variance D.1 Allan Variance Allan Variance MEMS MEMS Allan Variance AVARτ = 1 n 1 n forward y backward y τ D
70 E 64
71 F F.1 F. F.3 65
72 [1] Robert M. Rogers. Applied Mathematics in Integrated Navigation System, Second Edition. AIAA Education Series, 3. ISBN [], 3. ISBN [3] CQ, ISBN
73 MEMS 4/11 67
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
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64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
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医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
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数学の基礎訓練I
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The Physics of Atmospheres CAPTER :
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量子力学 問題
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液晶の物理１：連続体理論（弾性，粘性）
The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )
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II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
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127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
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9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (
9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)
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() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
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Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
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