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5.1. CONCEPTS 41
Let us follow the derivation from [13]. From the definition of the linear momentum for particles and the rigid
body we get P (t) = N mi · ri(t). ˙ This can be rewritten according to (5.4) as i=1 P (t) = N mi · (v(t) + ω(t) ×
i=1
(ri(t) − x(t))) = N i=1 mi · v(t) + ω(t) × N i=1 mi · (ri(t) − x(t)) = ( N
i=1 mi) · v(t) + ω(t) × 0 = M · v(t) because
N i=1 mi · (ri(t) − x(t)) = N i=1 mi · R(t)rb N i=1
i (t) = R(t) ·
mi·r b i (t)
· M = R(t) · r M
b cm · M = 0.
(t) = mi · ¨ ri(t) = mi · ai(t) for any particle i and therefore
Ftotal(t) = N i=1 F total
i (t) = N N
i=1
mi ·ai(t). Since x(t) =
i=1 mi·r i(t)
, we get M
¨ x(t) = ˙ N i=1 v(t) =
mi·a i(t)
, finally
M
obtaining Ftotal = M · ˙ v(t) = ˙ P (t).
From the Newton’s Second Law we have F total
i
Equations (5.10) and (5.11) describe the effect of exerted external force on the translational
motion of the rigid body and “mimic” the Newton’s Second Law for particles — this effect is the
same as if the rigid body was replaced by a single particle with mass M, which Ftotal(t) acted
on. Note that the external torque τtotal(t) has no effect on the translational motion of the body.
To describe the rotational effects we will define the concept of angular momentum.
Angular Momentum
The angular momentum of particle i is defined as Li(t) = (ri(t)−x(t))×(mi · ˙
ri(t)−mi · ˙ x(t)) and
the angular momentum of the rigid body L(t) as the sum of angular momentums due to individual
particles. Angular momentum describes the rigid body state with respect to the rotational motion
of the body and it can be shown that
L =
N
Li(t) = I(t) · ω(t), (5.12)
i=1
where I(t) is a 3 × 3 positive definite matrix called the world space inertia tensor, which characterizes
the distribution of mass over the body, relative to the body’s center of mass, along the
world space axes. I(t) is independent of the body position x(t), but it depends on the orientation
R(t). Let us define Ibody as
⎛
⎞
Ibody = ⎝
Ixx −Ixy −Ixz
−Ixy Iyy −Iyz
−Ixz −Iyz Izz
⎠ , (5.13)
where Ixx, Iyy, Izz, Ixy, Ixz, Iyz are the moments and products of inertia distilled from the rigid
body’s density function ρ(rb). Ibody is called the body space inertia tensor, it is evaluated in the
body space and equals the world space inertia tensor I(t) if the current body’s orientation R(t)
is an identity matrix (Ibody characterizes the distribution of mass over the body, relative to the
body’s center of mass, along the body space axes). It can be shown that the relation between
the world space and body space tensors is given by
I(t) = R(t) · Ibody · R(t) T
I −1 (t) = R(t) · I −1
body · R(t)T , (5.14)
which provides us with a formula for the evaluation of I(t) and I −1 (t) for arbitrary body orientation
because Ibody is a constant matrix, [10]. Note that Ibody is a positive definite matrix and
so I −1
body , I(t) and I−1 (t) are also positive definite for arbitrary orientation R(t) 3 .
Since the body’s angular momentum describes the body’s kinetic state with respect to its
rotational motion, we are naturally interested in what its time derivative is and what affects its
3 This special property is utilized by the constraint solver to compute constraint forces more efficiently.