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).
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
By using the fact that
, the expression becomes
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
The angular velocity can be moved to the right to obtain
in which the integral now occurs over a
Let be called the inertia matrix and be defined as
Using the definition,
This simplification enables a concise expression of
which makes use of the chain rule.
Steven M LaValle