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Utility that transforms 3x3 rotation matrix into rotation axis-angle 4-vector
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_{x},n_{y},n_{z}), with
n·n = n_{x}^{2} + n_{y}^{2} + n_{z}^{2} = 1,
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™ User's Guide.