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a
a
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2nd
4th
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m
k= sen(x)
m k= sen(x)
m
k= cos(x)
m
k= cos(x)
m
k= cos(x)
m
k= sen(x)
m1
6
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m1
4
m2
m2
10
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a
m
k= cos(x)
m
k=cos(x)
a
k= cos(x)
m
k= cos(x)
m
k= cos(x)
m
k= sen(x)
m
k=sen(x)
a
k= sen(x)
m
k= sen(x)
m k= sen(x)
a
m1 F1
m2 F2
a
m1 x1
m2 x2
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m1 v1
m2 v2
BZ
m
k= cos(x)
m
k= sen(x)
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a
JId JV al
JId JV al
a
JId JV al
m
k= cos(x)
JId JV al
a
JId JV al
JId JV al
JId
JId
JId JV al
a
JId JV al
JId JV al
JId JV al
m
k =
sen(x)
m
k= sen(x)
m
k =
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cos(x)
m
k= cos(x)
m
k= cos(x)
m
k= sen(x)
m1 x1
m1
v1
m2 x2
m2
v2
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y(t)
u(t)
yk k
uk k
J
JId
JV al
G(j) G(s)
N(, s)
D(, s)
L
JS() JS()
< A, B >
a >> b
a
b
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yp
ye
J=||yp ye||
Gp(j)
Ge(j)
J=||Gp(j) Ge(j)||
Gs(j)
= [w1w2 wm]
Gne (, s) =
nk=0
kPk(s) =N(, s)
D(, s) =0 s
n +1 sn1 + +n1 s+n
sn +1 sn1 + +n1 s+n
Pk(s) = snk
0 sn +1 sn1 + +n1 s+n
= [0 1 n]T
= [1 2 n]T
= [T T]T
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Gne (, s)
= [T T]T
JS() =||D(, j) G(j) N(, j)||22
D(, j) G(j) =
D(, j1)G(j1)
D(, j2)G(j2)
D(, jm)G(jm)
Gne (s) D(s) N(s)
JS() =m
i=1
|D(, ji) G(ji) N(, ji)|2
JS() =m
i=1
(ji)(ji)
(ji) =D(, ji)G(ji) N(, ji)
(ji) = (ji)
= [TT]T
JS()
JS() =
JS0
JS1
JSn
JS1
JS2
JSn
T= 0(2n+1) 1
t k k = 0, , n
l
l= 1, , n
JSt
=m
i=1
(ji)
t(ji) + (ji)
(ji)
t
Rn(ji) (ji)
n
(ji)
k l
(ji)
k= [ji]
nk =Rnk(ji)
(ji)
l = [ji]nl
G(ji) =Rnl(ji)G(ji)
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JSk
=m
i=1
[Rnk(ji)(ji) (ji)R
nk(ji)]
=
mi=1
2Re[Rnk(ji)
(ji)]
=m
i=1
2Re{Rnk(ji)[D(, ji)G
(ji) N(, ji)]}
=m
i=1
2Re[Rnk(ji)N(, ji) Rnk(ji)G
(ji)D(, ji)]
l
JS
l
=m
i=1 [Rnl(ji)G(ji)(ji) + (ji)R
nl(ji)G(ji)]
=m
i=1
2Re[Rnl(ji)G(ji)(ji)]
=m
i=1
2Re{Rnl(ji)G(ji)[D(, ji)G
(ji) N(, ji)]}
=m
i=1
2Re[Rnl(ji)(ji)D(, ji) Rnl(ji)G(ji)N
(, ji)]
(ji) =G(ji)G
(ji) Re[]
JSk
= 0 JSl =
0
n
k
l
Q = Y
Q1 Q2Q3 Q4
= Y1
Y2
Rn = Rn(ji) G = G(ji)
= (ji) =G(ji)G(ji) Q1, Q2, Q3, Q4 Y1 Y2
Q1=m
i=1
Re
RnR
n RnR
n1 RnR
0
Rn1R
n Rn1R
n1 Rn1R
0
R0Rn R0Rn1 R0R0
R(n+1) (n+1)
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Q2 =m
i=1Re
RnR
n1G RnR
n2G RnR
0G
Rn1R
n1G Rn1R
n2G Rn1R
0G
R0R
n1G R0R
n2G R0R
0G
R(n+1) n
Q3=m
i=1
Re
Rn1R
nG Rn1R
n1G Rn1R
0G
Rn2R
nG Rn2R
n1G Rn2R
0G
R0R
nG R0R
n1G R0R
0G
R
n (n+1)
Q4=m
i=1
Re
Rn1Rn1 Rn1Rn2 Rn1R0
Rn2R
n1 Rn2R
n2 Rn2R
0
R0R
n1 R0R
n2 R0R
0
Rn n
Y1=m
i=1Re
RnR
nG
Rn1R
nG
R0R
nG
Rn+1
Y2=m
i=1
Re
Rn1R
n
Rn2R
n
R0R
n
R
n
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x(t) = A((t)) x(t) +B((t)) u(t)
y(t) = C((t)) x(t) +D((t)) u(t)
A
B
C
D
(t)
(t)
(t) x(t)
(t) = [Tx (x(t))
Tp(t)]
Tx (x(t))
Tp(t)
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Gi(s) i {0, 1, , v} v
Gi(s)
M
Gi(s)
Gi(s) =bi0s
n +bi1sn1 + +bi(n1)s+bin
sn +ai1sn1 + +ai(n1)s+ain
aij bi R j {1, 2 , n} {0, 1 , n}
Gi(s)
i
yi(t) = yi(t) +yN i
ui(t) = ui(t) +uN i
yN i uN i R i
Gi(s) y(t) u(t)
y(t)
u(t)
Gi(s) Gi(s)
Yi(s) = L[yi(t)]
Yi(s) = L[yi(t) +yN i]
Yi(s) = Yi(s) +yN i
s
Ui(s) =Ui(s) +
uNis
Gi(s) = Yi(s)
Ui(s)=
Yi(s) + yNi
s
Ui(s) + uNis=
s Yi(s) +yN i
s Ui(s) +uN i
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yN i = bi(n+1) uN i = ai(n+1)
Gi(s) =bi0s
n+1 +bi1sn + +bins+bi(n+1)
sn+1 +ai1sn + +ains+ai(n+1)
j {0, 1, , (n+ 1)}
aj = [a1j a2j avj ]T
bj = [b0j b1j bvj ]T
j = [1j 2j vj ]T
ai bi
G(s, ) =b0()s
n+1 +b1()sn + +bn()s+bn+1()
sn+1
+a1()sn
+ +an()s+an+1()
G(s, )
G
G(s, )
A() B()
C() D()
C()[sI A()]1B() +D()
x(t)y(t)
= A() B()C() D()
x(t)u(t)
ai() bi()
f() =0
p + 1p1 + + p p N
p
aj =
p1 p11 1
p2 p12 1
1
pk p1k 1
0
1
p
=Xj
bj
bj =
p1 p11 1
p2 p12 1
1
pk p
1k 1
0
1
p
=Xj
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ai bj
M(q, ) =B(q, )
A(q, )=
b0() +b1() q1 + +bnb() q
nb
1 +a1() q1 + +ana() qna
q1
n= na +nb+ 1
ai()
bj() i [0, 1, , na] j [0, 1, , nb] ai
bj
ai() = a1
i
+a2i
+ +aNi
N1
bj() = b1j +b
2j + +b
Nj
N1
M(q, )
aki b
lj
k, l [1, 2, , N]
a11 a21 a
N1
a12 a22 a
N2
a1na a2na a
Nna
b10 b20 b
N0
b1nb b2nb b
Nnb
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k k k
yk1
ykna
uk
uknb
[1 k 2k
N1k ]
k =(k)
A
B
< A, B >=tra o(A B) =tra
o(B A)
A
A
yk =< , k >
J=J() = 1
T
Tk=0
E{(k, )2}
(k, ) =yk< , k >
k k
= k k1=1
2 g(k1)
g(k1)
g() =dJ()
d
J() =(k, )2
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k+1 = k+1 1
2
d
d
(k, )2
=k
(k, ) =yk< , k >=yk (Tk) d
dX
(XTB) =B
1
2
d
d(2) = (
d
d(yk (
Tk)))
= (k)
=k
k = k k
k = yk< k, k >=yk
k k
k+1 = k+ k k
0
k
k uk yk
ai() bj (
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T
1/T
y(kT)
k
y(k)
yk T
x(t) = limt0
x(t+ t) x(t)t
T 0 x(t)
T
x(k)=x(k+ 1) x(k)
T
A(q) y(k) = B(q)
F(q)u(k) +
C(q)
D(q)v(k)
y(k) = B(q)
F(q)A(q)u(k) +
C(q)
D(q)A(q)v(k)
y(k) = H(q) u(k) +G(q) v(k)
q1
y(k)q1 =y(k 1) v(k)
A(q)
B(q)
C(q)
D(q)
F(q)
A(q) = 1 a1 q1 any q
ny
B(q) = 1 b1 q1 bnu q
nu
C(q) = 1 c1 q1 cnv q
nv
D(q) = 1 d1 q1 dnd q
nd
F(q) = 1 f1 q1 fnf q
nf
H(q)
G(q)
H(q) q = z
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Z
h(k)
H(z)
H(q)
q1
z1
C(q) =D(q) =F(q) = 1
A(q) y(k) = B(q) u(k) +v(k)y(k) =
B(q)
A(q)u(k) +
1
A(q)v(k)
H(q) = B(q)A(q)
G(q) = 1
A(q)
p
q
y(k) = A1y(k 1) +A2y(k 2) + +Anyy(k ny) +
B1u(k 1) +B2u(k 2) + +Bnuu(k nu) + e(k)
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Ai R
p p
Bi Rp r
y(k) = [y1(k) y2(k) yp(k)]T
u(k) = [u1(k) u2(k) ur(k)]T
e(k) = [e1(k) e2(k) ep(k)]T
A(q) = I A1 q1 Any q
ny
B(q) = B1 q1 + +Bnu q
nu
y1(k)
y2(k)
=
0, 4 1, 2
0, 3 0, 7
y1(k 1)
y2(k 1)
+
0, 35 0, 3
0, 4 0, 5
y1(k 2)
y2(k 2)
+
1 0
0 1
u1(k 1)
u2(k 1)
y(k) = T (k 1) + e(k)
= [A1 A2 Any B1 B2 Bnu]T
(k 1) = [yT(k 1) yT(k 2) yT(k ny)
uT(k 1) uT(k 2) uT(k nu)]T
k
xk+1 = Ak xk+Bk uk
yk = Ck xk+Dk uk
a
yk =a1k yk1 a2k yk2 + b0k uk + b1k uk1 + b2k uk2
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Ak1 Bk1
xk = a1k 1
a2(k+1) 0 Ak1
x1(k1)
a2k yk
2+b2k uk
2 xk1
+ b1k a1k b0(k1)
b2(k+1) a2(k+1) b0(k
1) Bk1
uk1
a
Ak =
a1(k+1) 1
a2(k+2) 0
Bk = b1(k+1) a1(k+1) b0(k)
b2(k+2) a2(k+2) b0(k)
Ck = [1 0]
Dk = b0k
n
Ak =
a1(k+1) 1 0 0
a2(k+2) 0 1 0
an(k+n1) 0 0 1
an(k+n) 0 0 0
Bk =
b1(k+1) a1(k+1) b0(k)
b2(k+2) a2(k+2) b0(k)
bn(k+n) a2(k+n) b0(k)
Ck = [1 0 0 0]
Dk = b0k
a
a
yk =a1k yk1 a2k yk2+b0k uk+b1k uk1+b2k uk2
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k
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k= 3
y3 = [1 0]
a13[b12 u1 a12 b01 u1] +b23 u1 a23 b01 u1+b13 u2 a13 b02 u2
a24[b12 u1 a12 b01 u1] +b24 u2 a24 b02 u2
+b03 u3y3 = a13 [b12 u1 a12 b01 u1] +b23 u1 a23 b01 u1+b13 u2 a13 b02 u2+ b03u3
y3 = a13 [a12 b01 u1+b02 u2+b12 u1] y2
a23 b01 u1 y1
+b03u3+b13 u2+b23 u1
y3 = a13 y2 a23 y1+b03u3+b13 u2+b23 u1
Ak Bk Ck Dk k+1 k+2 k+n
n
y= f(x)
N
x
y1 = f(x1)
y2 = f(x2)
=
yN = f(xN)
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y=X
X Rn n i i X i
n i R
n
xi R
y
x
1 x
2
xn
X
X
=X1y
n= N N >> n
X RN n
N > n
X RN n y RN 1 Rn 1 X
XT
XTy=XTX
XTy Rn 1
XTX Rn n
XTX
= [XTX]1XTy
[XTX]1XT
y
y=
T +
N > n
y=X +
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J=N
i=1 (i)2 =T= || ||2
J
y
J
J = (y X)T(y X)
= yTy yTX T
XTy +T
XTX
J
J
= 0
J
= (yTX)T XTy + (XTX+XTX)
= (XTy) XTy + 2XTX
= 2(XTy) + 2XTX
2(XTy) + 2XTX = 0
= [XTX]1 XTy
J 2J
2 = 2XTX >0
2XTX
u(t)
y(t)
e(t)
yk =a1 yk1 a2 yk2 ana ykna+ b0 uk+ b1 uk 1 + + bnb uknb
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ai bj i {1, 2, , na} j {0, 1, 2, , nb}
uk yk
Tk = [yk1 yk2 ykna uk uk1 uknb]
T = [a1 a2 ana b0 b1 bnb]
k R
(na+nb+1) 1
R(na+nb+1) 1
yk =Tk +ek
N
ai bj
y0
y1
yN1
=
T0
T1
TN1
+
e0
e1
eN1
Y = + e
J=N
k=1
(yk Tk )
2
X
(XTX)
X
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X Rn n
X= B
B Rn 1
I Rn n
B
B
B
A= X|B Rn (n+1)
i
n
i
k
aki
i
aii
k= 1
n
k=i akj
akj aij
A = I|
X
B
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G
u
y
uk yk k {1, 2, , m} m
yk
yk =a1 yk1 a2 yk2 an ykn + b0 uk + b1 uk1 + + bn ukn
ai bj i {1, , n} j {0, 1, , n}
G
G
ai
bj
k
yk =a1kyk1 a2k yk2 ank ykn + b0k uk + b1kuk1 + + bnk ukn
k
k
k+ 1
G
[min, max]
ai bj
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aik bjk
aik =i0+i1 k1+ +iN Nk1
bjk =j0+j1 k1+ +jN Nk1
i j [0, , N] N
G
i
j
uk
k
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y
G
y G
y
y
J= 1
m
m
k=1
(yk yk)2
aik bjk
yk =
ni=1
yki aik+
nj=0
ukj bjk
yk = n
i=1
yki (i0+i1 k1+ +iN Nk1) +
nj=0
ukj (j0+j1 k1+ +jN Nk1)
i i
yk =
ni=1
yki [ 1 k1 Nk1 ] [i0 i1 iN]T +
nj=0
ukj [ 1 k1 Nk1 ] [j0 j1 jN]
T
i [1, 2, , n]
Tik yki [ 1 k1 Nk1 ]
i [i0 i1 iN]T
j [0, 1, , n]
T(j+n+1)k ukj [ 1 k1 Nk1 ]
(j+n+1) [j0 j1 jN]T
Tk = [ T1k
T2k
T(2n+1)k ]
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=
1
2
2n+1
yk = Tk
k k yk1 yk2
ykn uk ukn k1
i j G
J= 1
m
mk=1
(yk Tk )
2
m
Y
=
T1
T2
Tm
Y =
y1
y2
ym
Y =
Y m 1
m (2n+1)N
T
T Y P
= T Q
Q (2n+1)N 1 P (2n+1)N (2n+1)N
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G
p
yk = a1k yk1 a2k yk2 ank ykn
+b10k u1k+b
11k u
1k1+ +b
1nk u
1kn
+b21k u2k1+b
21k u
2k1+ +b
2nk u
2kn
+ +
+bp0k upk+b
p1k u
pk1+ +b
pnk u
pkn
yk = n
i=1
yki aik+
pr=1
nj=0
urkj brjk
aik i [1, , n] y
k
brjk j [0, , n] r [1, , p]
y
ur
ai brj
aik = i0+i1 k1+ +iN Nk1
brjk = r
j0+r
j1 k1+ +r
jN Nk1
aik b
rjk
yk =
ni=1
yki (i0+i1 k1+ +iN Nk1) +
pr=1
nj=0
urkj (r
j0+r
j1 k1+ +r
jN Nk1)
i
ri
yk = n
i=1
yki [ 1 k1 Nk1 ] [i0 i1 iN]
T +
p
r=1n
j=0 urkj [ 1 k1 Nk1 ] [rj0 rj1 rjN]T
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i [1, 2, , n]
Tik yki [ 1 k1 Nk1 ]
i [i0 i1 iN]T
j [0, 1, , n] r [1, 2, , p]
rTjk urkj [ 1 k1
Nk1 ]
rj [r
j0 r
j1 r
jN]T
T1k [T1k
T2k
Tnk ]
T2k [1T0k
1T1k
1Tnk ]
T3k [2
T0k 2
T1k 2
Tnk ]
T(p+1)k [pT0k
pT1k
pTnk ]
1 [T1
T2
Tn ]
T
2 [1T
0 1T
1 1T
n ]T
3 [2T0
2T1
2Tn ]
T
p+1 [pT0
pT1
pTn ]
T
Tk [T1k
T2k
T(p+1)k]
1
2
p+1
yk = Tk
Tk
1 N[n(p+1)+p]
N[n(p+1)+p] 1
p= 1
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u
y
G i j
i
rj
vk
wk
(i)k+1=
(i)k +w
(i)k
yk+a1k yk1+ +ank ykn=b1k uk1+ +bmk ukm+vk
{w
(i)k } vk
{w
(i)k } {w
(j)k }
i=j
k+1 = IN(2n+1) N(2n+1)
F
k+wk
yk = Tk
HTk
k+vk
F =I
HTk
xk = Fk1 xk1
Pp(k) = Fk1 Pe(k1) FT
k1+Qk1
Kf(k) = Pp(k) HTk [Hk Pp(k) H
Tk +Rk]
1
xk = xk+Kf(k)[yk Hk xk]
Pe(k) = [I Kf(k) Hk] Pp(k)
x
x
Pp
Pe Q
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R
Kf
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yk
n
n
na
nb
na nb
yk
uk
D= 0
JId JV al
a
na nb1
nb2
b0
a
JId
JV al
na nb1 nb2 N b0 JId
m1 13
13
m2 16
16
T
T
uk =cu yk =cy k =c k T
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u
y
n
n
t1 t2 t3 tn y y1 y2
yn u u1 u2 un 1 2 n
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t1 y1 u1 1
G1
G1 y1
G2 t2
y1
T y2T
T
y1 y2
Gn
y1 y2 yn1 yn
na
nb
N
k
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y=
1 0
0 1
x1
x2
+
0
0
F
k= cos(x)
=x(t)
m = 1Kg
b = 1Ns/m
k = sen(x)
3
10
m
k= cos(x)
JId
JV al
JId JV al
a
na nb N b0 JId JV al
k = cos(x)
8
5
8 6
k= sen(x)
10
3
9 6
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JId JV al
m
k = cos(x)
na nb N b0 JId JV al
5
7
3
8 5
m
k=cos(x)
m
k=cos(x)
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a
k= cos(x)
m
k=cos(x)
m k= cos(x)
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m
k= sen(x)
m
k= sen(x)
a
k = sen(x)
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m
k= sen(x)
m
k= sen(x)
F1 F2
m1
m1 xm1=F1+Fk2+Fb2 Fk1 Fb1
Fk1 = k1 xm1
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a
Fk2 = k2 (xm2 xm1)
Fb1 = b1 xm1
Fb2 = b2 (xm2 x1)
m1 xm1 = F1+k2(xm2 xm1) +b2(xm2 xm1) k1 xm1 b1 xm1
xm1 = 1
m1[F1+k2(xm2 xm1) +b2(xm2 xm1) k1 xm1 b1 xm1]
m2
m2 xm2 = F2 Fk2 Fb2
m2 xm2 = F2 k2(xm2 xm1) b2(xm2 xm1)
xm2 = 1
m2[F2 k2(xm2 xm1) b2(xm2 xm1)]
x1 = xm1 x1 = xm1=x2
x2 = xm1 x2 = xm1
x3 = xm2 x3 = xm2=x4
x4 = xm2 x4 = xm2
x1
x2
x3
x4
=
0 1 0 0
k1+k2m1
b1+b2m1
k2m1
b2m1
0 0 0 1
k2m2
b2m2
k2m2
b2m2
x1
x2
x3
x4
+
0 0
1m1
0
0 0
0 1m2
F1
F2
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m1 m2
y=
1 0 0 0
0 1 0 0
0 0 1 00 0 0 1
x1
x2
x3
x4
+
0 0
0 0
0 00 0
F1
F2
k1 m1
k1 = cos(xm1) = cos(x1)
m1 =
m2= 3Kg
k1 = 1000 cos(x1)N/m
k2 = 1000N/m
b1=b2= 1Ns/m
na nb1 F1 nb2
F2 N
xm1 xm2 m1 m2
vm1 vm2
9
16
JId JV al
a
na nb1 nb2 N b0 JId JV alm1
13
5
9 2
m2 16
6
12
5
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m1 x1
m2 x2
m1
v1
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m2 v2
c(t) (t)
q(t)
(t)
q
=
Z 1 Z 0
M 0 M 0
0 0 0 1
0 0
2
a 2a
q
+
0
0
0
2
a
c
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q
=
N 0 N 0
0 1 0 0
q
Z Z M M N N
Z = K M cos() [an 2 +bn || +cn(2 M/3)]
Z = K M dn cos()
M = Kq M2 [am
2 +bm || +cm (7 + 8M/3)]
M = Kq M2 dm
N = Kz M2 [an
2 +bn || +cn (2 M/3)]
Nz = Kz M2 dn
(t)
= 0
=||
K = 0, 7 P0180Smvs
m = 13, 98slugs
Kz = 0, 7 P0 Smg vs = 1036, 4 p s/s
Kq = 0, 7 P0180SdIy
Ax = 0, 7 P0SCam
an = 0, 000103 3
d = 0, 75 p s bn = 0, 00945 2
Iy = 182, 5 slug.p s2
cn = 0, 1696 1
Ca = 0, 3 dn = 0, 034 1
= 0, 7
am = 0, 000215 3
a = 150rad/s bm = 0, 0195 2
cm = 0, 051 1
g = 32, 2 p
s/s2
dm = 0, 206
1
P0 = 973, 3 lbs/p s2
S = 0, 44 p s2
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Z Z M M N N
c
G(s, )|=0 = b2(0) s
2 +b3(0) s +b4(0)
s4 +a1(0) s3 +a2(0) s2 +a3(0) s +a4(0)
a1() = 2 a Z a2() = a M 2 a Z
a3() = Z a M 2 a a4() = M a
b2() = 0
b3() = Z 2a
b4() = M 2a
b2() = N 2a
b3() = N Z 2a
b4() = N M 2a
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BZ
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G(s, t)|t=t0 = b1(t0) s
2 +b2(t0) s
s3 +a1(t0) s2 +a2(t0) s+a3(t0)
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JId
na nb N b0 JId
6
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a
6
m
k= cos(x)
m
k= sen(x)
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JId JV al
a
na nb N b0 JId JV al
k= cos(x) 5 5
k= sen(x)
4
3
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JId JV al
na nb N b0 JId JV al
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m
u
y
yk
na nb N
na nb N
a
yk uk
Gn
n 1
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Y
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2
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M
K=SEN(X)
m
k= sen(x)
10 10 30
11 11 31
12 12 32
13 13 33
14 14 34
15 15 35
16 16 36
20 20
21 21
22 22
23 23
24 24
25 25
26 26
M
K=SEN(X)
m
k = sen(x)
10 40 70
11 41 71
12 42 72
13 43 73
14 44 74
15 45 75
20 50 10
21 51 11
22 52 12
23 53 13
24 54 14
25 55 15
30 60 20
31 61 21
32 62 22
33 63 23
34 64 24
35 65 25
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M
K=C OS(X)
m
k= cos(x)
10 10 40
11 11 41
12 12 42
13 13 43
14 14 44
15 15 45
16 16 46
20 20 50
21 21 51
22 22 52
23 23 53
24 24 54
25 25 55
26 26 56
30 30 60
31 31 61
32 32 62
33 33 63
34 34 64
35 35 65
36 36 66
M
K=C OS(X)
m
k= cos(x)
10 50 90 20
11 51 91 21
12 52 92 22
13 53 93 23
14
54
94
24
15 55 95 25
16 56 96 26
17 57 97 27
18 58 98 28
19 59 99 29
110 510 910 210
20 60 100 30
21 61 101 31
22 62 102 32
23 63 103 33
24 64 104 34
25 65 105 35
26 66 106 36
27 67 107 37
28 68 108 38
29 69 109 39
210 610 1010 310
30 70 00 40
31 71 01 41
32 72 02 42
33 73 03 43
34 74 04 44
35 75 05 45
36 76 06 46
37 77 07 47
38 78 08 48
39 79 09 49
310 710 010 410
40 80 10
41 81 11
42 82 12
43 83 13
44 84 14
45 85 15
46 86 16
47 87 17
48 88 18
49 89 19
410 810 110
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M
K=C OS(X)
m k=cos(x)
10 50 90 20
11 51 91 21
12 52 92 22
13 53 93 23
14 54 94 24
15 55 95 25
16 56 96 26
20 60 100 30
21 61 101 31
22 62 102 32
23 63 103 33
24 64 104 34
25 65 105 35
26 66 106 36
30 70 00 40
31 71 01 41
32 72 02 42
33 73 03 43
34 74 04 44
35 75 05 45
36 76 06 46
40 80 10 50
41 81 11 51
42 82 12 52
43 83 13 53
44 84 14 54
45 85 15 55
46 86 16 56
M
K=SEN(X)
m
k= sen(x)
10 40 70 100
11 41 71 101
12 42 72 102
13 43 73 103
14 44 74 104
15 45 75 105
20 50 80 10
21 51 81 11
22 52 82 12
23
53
83
13
24 54 84 14
25 55 85 15
30 60 90 20
31 61 91 21
32 62 92 22
33 63 93 23
34 64 94 24
35 65 95 25
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M1
6
m1 x1
10 40 2
20
11 41 2
21
12 42 2
22
13 43 2
23
14 44 2
24
15 45 2
25
20 1
10 2
30
21
1
11
2
31
22 1
12 2
32
23 1
13 2
33
24 1
14 2
34
25 1
15 2
35
30 2
10 1
40
31 2
11 1
41
32 2
12 1
42
33 2
13 1
43
34 2
14 1
44
35 2
15 1
45
M1 4
m1 v1
10 1
00 1
50 2
40
11 1
01 1
51 2
41
12 1
02 1
52 2
42
13 1
03 1
53 2
43
20 1
10 2
00 2
50
21 1
11 2
01 2
51
22 1
12 2
02 2
52
23 1
13 2
03 2
53
30 1
20 2
10
31 1
21 2
11
32 122 212
33 1
23 2
13
40 1
30 2
20
41 1
31 2
21
42 1
32 2
22
43 1
33 2
23
50 1
40 2
30
51 1
41 2
31
52 1
42 2
32
53 1
43 2
33
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M2
m2 x2
10 1
00 1
40 2
20
11 1
01
1
41
2
21
12 1
02 1
42 2
22
20 1
10 1
50 2
30
21 1
11 1
51 2
31
22 1
12 1
52 2
32
30 1
20 2
00 2
40
31 1
21 2
01 2
41
32 1
22 2
02 2
42
40 1
30 2
10 2
50
41 1
31 2
11 2
51
42 1
32 2
12 2
52
M2
10
m2 v2 10 40
2
10 2
40
11 41 2
11 2
41
12 42 2
12 2
42
13 43 2
13 2
43
14 44 2
14 2
44
15 45 2
15 2
45
16 46 2
16 2
46
17 47 2
17 2
47
18 48 2
18 2
48
20 1
10 2
20
21 111 221
22 1
12 2
22
23 1
13 2
23
24 1
14 2
24
25 1
15 2
25
26 1
16 2
26
27 1
17 2
27
28 1
18 2
28
30 1
20 2
30
31 1
21 2
31
32 1
22 2
32
33 1
23 2
33
34 1
24 2
34
35 1
25 2
35
36 1
26 2
36
37 1
27 2
37
38 1
28 2
38
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10 60 30 80
11 61 31 81
12 62 32 82
13 63 33 83
14 64 34 84
15 65 35 85
16 66 36 86
17 67 37 87
18 68 38 88
20 70 40 90
21 71 41 91
22 72 42 92
23 73 43 93
24 74 44 94
25 75 45 95
26 76 46 96
27 77 47 97
28 78 48 98
30 80 50
31 81 51
32 82 52
33 83 53
34 84 54
35 85 55
36 86 56
37 87 57
38 88 58
40 10 60
41 11 61
42 12 62
43 13 63
44 14 64
45 15 65
46 16 66
47 17 67
48 18 68
50 20 70
51 21 71
52 22 72
53 23 73
54 24 74
55 25 75
56 26 76
57 27 77
58 28 78
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10 40 70
11 41 71
12 42 72
20 50 10
21 51 11
22 52 12
30 60
31 61
32 62
10 30 00
11 31 01
20 40 10
21 41 11
10 40 00
11 41 01
12 42 02
13 43 03
14 44 04
20 50 10
21 51 11
22 52 12
23 53 13
24 54 14
30 60 20
31 61 21
32 62 22
33 63 23
34 64 24
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10
30
00
11 31 01
20 40 10
21 41 11
10 40 00
11 41 01
12 42 02
13 43 03
14 44 04
20 50 10
21 51 11
22 52 12
23 53 13
24 54 14
30 60 20
31 61 21
32 62 22
33 63 23
34 64 24