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
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 .