b - NTNU

3.1.2 Rotational Motion about CG
The derivation starts with the Euler’s 2nd axiom:
m g i d
dt
I g b/i
i d
dt
I g b/n
b d
dt
I g b/n b/n I g b/n
I g b/n I g b/n b/n #
Rotational Motion about CG Expressed in {b}
b
I g b/n
SI g b b
b/n b/n
m g
b
where I g is the inertia matrix
where I x , I y , and I z are the moments of inertia about {b} and I xy =I yx , I xz =I zx and I yz =I zy are the
products of inertia defined as:
I g :
I x I xy I xz
I yx I y I yz , I g I g 0 #
I zx I zy I z
I x V
y 2 z 2 m dV;
I y V
x 2 z 2 m dV;
I z V
x 2 y 2 m dV;
I xy V
xy m dV V
yx m dV I yx
I xz V
xz m dV V
zx m dV I zx
I yz V
yz m dV V
zy m dV I zy
6
Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)