An inertia matrix (also called an inertia tensor or inertia operator) will be derived by considering as a collection
of particles that are rigidly attached together (all contact forces
between them cancel due to Newton's third law).
The expression
in (13.77) represents the
mass of an infinitesimal particle of . The moment of
momentum of the infinitesimal particle is
. This means that the total moment of momentum of is
|
(13.86) |
By using the fact that
, the expression becomes
|
(13.87) |
Observe that now appears twice in the integrand. By doing some
algebraic manipulations, can be removed from the integrand,
and a function that is quadratic in the variables is obtained
(since is a vector, the function is technically a quadratic form).
The first step is to apply the identity
to obtain
|
(13.88) |
The angular velocity can be moved to the right to obtain
|
(13.89) |
in which the integral now occurs over a
matrix and
is the
identity matrix.
Let be called the inertia matrix and be defined as
|
(13.90) |
Using the definition,
|
(13.91) |
This simplification enables a concise expression of
(13.82) as
|
(13.92) |
which makes use of the chain rule.
Steven M LaValle
2020-08-14