Rigid Body Motion

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Summary

The following are the key concepts covered in this chapter:

  1. Rotational motion of a rigid body is represented by an element of the special orthogonal group
<amsmath> SO(3) = \{ R \in {\mathbb R}^{3 \times 3} \mid R^T R = I, \det R = 1 \} </amsmath>

which is often parameterized by the exponential map

<amsmath> \exp: so(3) \longrightarrow SO(3): \skew{\omega}\theta \mapsto e^{\skew{\omega} \theta}. </amsmath>

Other parameterizations of SO(3) include fixed and Euler angle sets, and unit quaternions.

  1. The configuration of a rigid body is represented as an element <amsmath>\textstyle g \in \SE(3)</amsmath>. An element <amsmath>\textstyle g \in \SE(3)</amsmath> may also be viewed as a mapping <amsmath>\textstyle g:{\mathbb R}^3 \to {\mathbb R}^3</amsmath> which preserves distances and angles between points. In homogeneous coordinates, we write \begin{displaymath} g = \bmatrix R & p \ 0 & 1 \endbmatrix \qquad \aligned R &\in SO(3) \ p &\in {\mathbb R}^3. \endaligned \end{displaymath} The same representation can also be used for a rigid body transformation between two coordinate frames.