3D Rotation Matrices

The orientation of a rigid body in 3D can be described by a $3\times 3$ rotation matrix which has these properties:

$$\begin{array}{r}{R}^{T}R=I\\ \det R=1\end{array}$$

$R\in {\mathbb{R}}^{3\times 3}$ is said to be in the special orthogonal group $SO(3)$.

The parameterization of $SO(3)$ is tricky.

• A general $3\times 3$ matrix requires 9 parameters (with constraints) for 3 DOF motion
• Euler Angles need only 3 angles but suffers from singularities (gimbal lock)