Rotation d`un objet rigide - CaMUS

~ω(t)
t
~ω(t)

~a
~a kak
~a
kakˆa
~a(t)t
~a
~a = [a1, a2, a3] [ax, ay, az]~a(t)
d
dt~a(t) = ~
˙a(t)
∆t∆~a ~a ~ b
~a ·~ b
~a~ b s r
~a ×~ b=
a2·b3−b2·a3
a3·b1−b3·a1
a1·b2−b1·a2
.
~ω(t) = [ωx(t), ωy(t), ωz(t)]
kω(t)k
ˆω(t)

~ω(t) = [a1+b1·t, a2+b2·t, a3+b3·t] = ~a +~ bt
~
θ(t) = kθ(t)k · ˆn(t) ˆn(t)
kθ(t)k
t
~
˙
θ(t)6=~ω(t)
~
θ(t)
~ω(t)~ θ(t)
t
R
0
~ω(τ)dτ =~ θ(t)

ˆu= [ux, uy, uz]
θ
R=
u2
x+ (1 −u2 x) cos θ uxuy(1 −cos θ)−uzsin θ uxuz(1 −cos θ) + uysin θ
uxuy(1 −cos θ) + uzsin θ u2
y+ (1 −u2 y) cos θ uyuz(1 −cos θ)−uxsin θ
uxuz(1 −cos θ)−uysin θ uyuz(1 −cos θ) + uxsin θ u2
z+ (1 −u2 z) cos θ
= ˆuˆuT+ (I−ˆuˆuT) cos θ+
0−uzuy
uz0−ux
−uyux0
sin θ
RaRbRc~v
Rc·Rb·Ra=RrRr
~
v0=Rr·~v
I R
R·Rt=I
q
q=q0+qx·~ i+qy·~ j+qz·~ k
q0qxqyqz
~
i2=~ j2=~ k2=−1
~
ij =~ k

~
jk =~ i
~
ki =~ j
q= [q0, ~v]
~v =qx·~ i+qy·~ j+qz·~ k
~v qq0
q+p=(q0+p0)+(qx+px)·~ i+
(qy+py)·~ j+ (qz+pz)·~ k
q−p=(q0−p0)+(qx−px)·~ i+
(qy−py)·~ j+ (qz−pz)·~ k
q·s= [q0·s, ~v ·s]
q·~ h= [−~v ·~ h, ~v ·q0+~v ×~ h]
q×p=[q0·p0−~vq·~vp,
q0·~vp+p0·~vq+~vq×~vp]
q∗= [q0,−~v]
=q0−qx·~ i−qy·~ j−qz·~ k
kqk=qq2
0+q2
x+q2 y+q2 z
q−1=q∗
kqk2
kθ(t)k
ˆu(t)
q= [q0= cos kθ(t)k
2, v = sin kθ(t)k 2·ˆu(t)]