Utility that transforms 3x3 rotation matrix into rotation axis-angle 4-vector

First Generation/Utilities

A rotation with respect to an initial orientation has many equivalent
representations. A common and important one is the 3-by-3 orthogonal
rotation matrix *R*, where *R*^{-1}* =
R*^{T} and *R*^{T}*R
= RR*^{T}* = I*,
the 3-by-3 identity matrix. Another important representation is the
combination of rotation axis (a unit vector ** n**)
and angle of rotation θ about that axis. The sign of rotation follows the right-hand-rule.

The RotationMatrix2VR block converts the 3-by-3 rotation matrix
representation of orientation to its equivalent representation as
a rotation axis and angle about that axis, the form used in Virtual
Reality Modeling Language (VRML) for orienting bodies. The input and
output signals are bundled Simulink^{®} signals.

The most common use of rotations is to represent the orientation of a body with respect to some coordinate system (CS) axes.

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The rotation matrix *R* has the form:

$$\left(\begin{array}{ccc}{R}_{11}& {R}_{12}& {R}_{13}\\ {R}_{21}& {R}_{22}& {R}_{23}\\ {R}_{31}& {R}_{32}& {R}_{33}\end{array}\right)$$

The input signal to the RotationMatrix2VR block is the *R* matrix
components passed column-wise and bundled into a single 9-component Simulink signal: ```
[R11
R21 R31 R12 ...]
```

.

The output signal is the equivalent rotation represented as
the axis of rotation, a unit vector ** n** =
(

** n·n** =

and the angle of rotation θ about that axis. The sign of the rotation follows the right-hand rule. The output signal is bundled into a single 4-component Simulink signal:

`[n`

_{x}` n`

_{y}` n`

_{z}` `

θ`]`

.

See Representations of Body Motion and Representations of Body Orientation for more details on representing body rotations.

See entries on axis-angle rotation, Euler angles, quaternion, and rotation matrix in the Glossary for summaries of body orientation representations.

For more on virtual reality and VRML, see the 3D Animation™ documentation.

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