For a free-floating 3D rigid body, recall from Section
4.2.2 that its C-space has six dimensions.
Suppose that actions are applied to the body as external forces.
These directly cause accelerations that result in second-order
differential equations. By defining a state to be
,
first-order differential equations can be obtained in a
twelve-dimensional phase space
.
Let
denote a free-floating rigid body. Let
denote the body density at
. Let
denote the total mass of
, which is defined using the density as
Suppose that a collection of external forces acts on (it is
assumed that all internal forces in
cancel each other out). Each
force
acts at a point on the boundary, as shown in Figure
13.10 (note that any point along the line of force may
alternatively be used). The set of forces can be combined into a
single force and moment that both act about the center of mass
.
Let
denote the total external force acting on
. Let
denote the total external moment about the center of mass of
. These are given by
![]() |
(13.79) |
![]() |
(13.80) |
Actions of the form can be expressed as external forces
and/or moments that act on the rigid body. For example, a thruster
may exert a force on the body when activated. For a given
, the
total force and moment can be resolved to obtain
and
.