13.3.3 Motion of a Rigid Body

For a free-floating 3D rigid body, recall from Section 4.2.2 that its C-space ${\cal C}$ 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 $(q,{\dot q})$, first-order differential equations can be obtained in a twelve-dimensional phase space $X$.

Let ${\cal A}\subseteq {\mathbb{R}}^3$ denote a free-floating rigid body. Let ${\sigma}(r)$ denote the body density at $r \in {\cal A}$. Let ${m}$ denote the total mass of ${\cal A}$, which is defined using the density as

$$m = \int_{\cal A}{\sigma}(r) dr ,$$ (13.77)

in which $dr = dr_1dr_2dr_3$ represents a volume element in ${\mathbb{R}}^3$. Let ${p}\in {\mathbb{R}}^3$ denote the center of mass of ${\cal A}$, which is defined for ${p}= ({p}_1,{p}_2,{p}_3)$ as

$${p}_i = \frac{1}{m} \int_{\cal A}r_i {\sigma}(r) dr .$$ (13.78)

Figure 13.10: A force $f$ acting on ${\cal A}$ at $r$ produces a moment about $p$ of $r \times f$.

Suppose that a collection of external forces acts on ${\cal A}$ (it is assumed that all internal forces in ${\cal A}$ cancel each other out). Each force $f$ 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 $p$. Let ${F}$ denote the total external force acting on ${\cal A}$. Let ${N}$ denote the total external moment about the center of mass of ${\cal A}$. These are given by

$${F}= \sum f$$ (13.79)

and

$${N}= \sum r \times f$$ (13.80)

for the collection of external forces. The terms ${F}$ and ${N}$ are often called the resultant force and resultant moment of a collection of forces. It was shown by Poinsot that every system of forces is equivalent to a single force and a moment parallel to the line of action of the force. The result is called a wrench, which is the force-based analog of a screw; see [681] for a nice discussion.

Actions of the form $u \in U$ 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 $u$, the total force and moment can be resolved to obtain ${F}(u)$ and ${N}(u)$.