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5.2. EQUATIONS OF MOTION 45
If m and I(t) are the mass 6 and world space inertia tensor of the rigid body, then the mass
matrix of the rigid body M(t) is a 6 × 6 block diagonal matrix defined as
M(t) =
⎛
m 0 0 0 0 0
⎞
m · E
0
⎜
⎜ 0
0
⎜ 0
= ⎜
I(t) ⎜ 0
⎜
⎝ 0
m
0
0
0
0
m
0
0
0
0
I11(t)
I21(t)
0
0
I12(t)
I22(t)
0 ⎟
0
⎟ ,
I13(t) ⎟
I23(t) ⎠
0 0 0 I31(t) I32(t) I33(t)
where E is a 3 × 3 identity matrix. Since I(t) is positive definite and m > 0, M(t) and M −1 (t)
are also positive definite. Since M(t) is a block diagonal matrix, M −1 (t) is also a block diagonal
matrix whose blocks are the inverses of the corresponding blocks of M(t),
M −1 (t) =
From (5.20) and (5.21) we see that
m−1 · E 0
0 I−1
.
(t)
m · a(t) = Ftotal(t)
I(t) · α(t) = τtotal(t) + τcoriolis(t),
which can be rewritten using the generalized notation and mass matrix M as
M(t) · ˙ vgen(t) = F total
gen (t) + F coriolis
gen
where ˙ vgen(t) = (a(t), α(t)) is the time derivative of the generalized velocity vgen(t), F total
gen (t) is
the generalized external force and F coriolis
gen (t) = (0, τcoriolis(t)) is the generalized coriolis force, or
equivalently,
(t),
˙v(t) = M −1 (t) · Ftotal(t) + M −1 (t) · Fcoriolis(t).
We have removed the gen subscripts this time to improve readability. Further on, when it is clear
that the generalized notation is used, the subscripts will be removed implicitly and “generalized”
adjective omitted as well (v in the generalized notation will refer to both the linear and angular
velocities).
This equation describes how the acceleration of the rigid body changes if Ftotal(t) is applied
to it. The acceleration M −1 (t) · Fcoriolis(t) due to the coriolis force is independent of other forces
and can be treated as if it was a part of the acceleration due to the total external force, that is,
the coriolis force can be incorporated into Ftotal(t) and can be ignored elsewhere 7 . From now on
we will assume that the coriolis force is implicitly included into Ftotal(t) whenever working with
generalized forces.
This allows to write
M(t) · ˙ v(t) = F (t) (5.23)
for any force F (t) and acceleration ˙ v(t) due to F (t). This equation generalizes (5.20) and (5.21)
and describes the dynamics of the rigid body, [9]. As can be seen, it resembles the Newton’s
6 Until now we used M to denote the rigid body’s mass. This would clash with our generalized notation and so
the mass would be denoted m.
7 This is generally true for any inertial force.