xiyiz (xi, yi) i
x̂0 x̂i
θi
r ρ
ρ = Qr
Q
r ρ Q xiyiz
x̂i · r
r i=
ŷi · r
⎡
x̂i · Qx̂i
⎢
Q i = ⎣ ŷi · Qx̂i
⎤
x̂i · Qŷi
x̂i
ŷi · Qŷi
⎥
⎦
ŷi
Qx̂i = cos θi x̂i + sin θi ŷi
Qŷi = − sin θi x̂i + cos θi ŷ
∴
cos θi − sin θi
Q i=
sin θi cos θi
(3.6)
ρ = Qr
⇒
ρ i= Q i r i
(3.7)
! x̂j = Qj/ix̂i
ŷj = Qj/iŷi
ρ = Qj/ir
ρ j= r i
ρ = Qj/i r
=
⇒
a
b
(3.7)
⇒
ρ i = Qj/i i r i = Qj/i i ρ j
ρ i = Qj/i i ρ j ,
Qj/i i =
ρ
cos θj/i − sin θj/i
sin θj/i
cos θj/i
(3.9)
"
ρ 0 = Qj/0 0 ρ j
# $ Pi
rPi
rPi = ri + sPi
(3.11)
ṡPi s̈Pi %
&
s i
∴
⇒
ṡ = ω i × s = θ̇i ẑ × s
'()(*
'()+*
ṡ = θ̇is̃
q
s̃ = Q s
s̈ = θ̈i s̃ + θ̇is̃˙
s̃˙ =
s̃
i
s̃˙ = −θ̇is
= θ̇i ẑ × s̃ = −θ̇is
(3.17)
i
ṙ r̈ %
Pi
Pi
∴
ṙPi = ṙi + ṡPi = ṙi + θ̇i s̃Pi
r̈Pi = r̈i + s̈Pi = r̈i + θ̈i s̃Pi − θ̇i2sPi
, '())* ⇒
rPi
0
= ri 0 + sPi 0
⎡ ⎤
x
⎣ i⎦
yi
&
s i ⇒
(3.9) s 0
= Qi/0 0
s i
sx0
s
Qi
s=
s = xi
sy0
syi
-
↑
t.
↑
/.
s = Qi s
(3.13) ṡ 0 = θ̇i s̃ 0 ,
(3.14)
s̃ 0 =
q
−sy0
s 0
=
Q 0
sx0
s
0 −1
s
s = x0
I =
sy0
1 0
0
&
s i
ṡ = θ̇is̃
s̃˙ = −θ̇is
s = Qi s
−sy0
s̃ =
sx0
cos θi − sin θi
Qi =
sin θi cos θi
⎡ ⎤
x1
⎢y ⎥
⎢ 1⎥
⎢θ ⎥
⎢ 1⎥
0
q = ⎢⎢⎢x2⎥⎥⎥ ∈ Rnc ,
⎢ y2 ⎥
⎢ ⎥
⎣ θ2 ⎦
nc 1 '*
H(q, t) = 0
2 (H(q) = 0)
3 ⎡
k
H (q) = ⎣
H1k (q)
Hnkk (q)
⎤
⎦=0
nk
2 nf = nc − nk