Assume we have a rigid body with mass and a coordinate frame located at the body’s center of mass. If and give the translational and rotational velocity of the coordinate frame, then the body’s linear and angular momentum and are given by
where is the rotational inertia with respect to the center of mass. These relationships can be combined into a single equation
where and are the spatial momentum and spatial inertia:
The spatial momentum satisfies Newton’s second law, so that
which can be used to find the acceleration of a body in response to a spatial force.
When the body coordinate frame is not located at the center of mass, then the spatial inertia assumes the more complicated form
where is the center of mass and is defined by (A.22).
Like the rotational inertia, the spatial inertia is always symmetric positive definite if .