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| Requirements | Software operation | Defining Calibration Tasks | Definitions and Theory |
|---|---|---|---|
Translations
Let \(\mathbf{X}=\left[\begin{array}{ccc}X&Y&Z\end{array}\right]^{\top}\) be a point in \(\mathbb{R}^3\) and let \(\mathbf{X}’=\left[\begin{array}{ccc}X’&Y’&Z’\end{array}\right]^{\top}\) be \(\mathbf{X}\) after a translation by \(\left[\begin{array}{ccc}\Delta X&\Delta Y&\Delta Z\end{array}\right]^{\top}\). Translations may be represented by a single \(4\times4\) matrix acting on a \(4\times1\) homogeneous coordinate.
\(\begin{bmatrix}X’\\Y’\\Z’\\1\end{bmatrix}=\begin{bmatrix}1&0&0&\Delta X\\0&1&0&\Delta Y\\0&0&1&\Delta Z\\0&0&0&1\end{bmatrix}\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}=\begin{bmatrix}X+\Delta X\\Y+\Delta Y\\Z+\Delta Z\\1\end{bmatrix}\)
A translation can be inverted by applying the negative of the translation terms
\(\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}=\begin{bmatrix}1&0&0&\Delta X\\0&1&0&\Delta Y\\0&0&1&\Delta Z\\0&0&0&1\end{bmatrix}^{-1}\begin{bmatrix}X’\\Y’\\Z’\\1\end{bmatrix}=\begin{bmatrix}1&0&0&-\Delta X\\0&1&0&-\Delta Y\\0&0&1&-\Delta Z\\0&0&0&1\end{bmatrix}\begin{bmatrix}X’\\Y’\\Z’\\1\end{bmatrix}\)
Rotations
In \(\mathbb{R}^3\), the rotation of points about the origin are described by a \(3\times3\) matrix \(\mathbf{R}\). Valid rotation matrices obey the following properties:
\(\mathrm{det}\left(\mathbf{R}\right) = +1\)
\(\mathbf{R}^{-1}=\mathbf{R}^{\top}\)
From these properties, both the columns and rows of \(\mathbf{R}\) are orthonormal.
The rotation is applied by left-multipling the points by the rotation matrix.
\(\begin{bmatrix}X’\\Y’\\Z’\end{bmatrix}=\begin{bmatrix}R_{11}&R_{12}&R_{13}\\R_{21}&R_{22}&R_{23}\\R_{31}&R_{32}&R_{33}\end{bmatrix}\begin{bmatrix}X\\Y\\Z\end{bmatrix}=\begin{bmatrix}R_{11}X+R_{12}Y+R_{13}Z\\R_{21}X+R_{22}Y+R_{23}Z\\R_{31}X+R_{32}Y+R_{33}Z\end{bmatrix}\)
Rotations of 3D homogeneous may be defined by a \(4\times4\) matrix
\(\begin{bmatrix}X’\\Y’\\Z’\\1\end{bmatrix}=\begin{bmatrix}R_{11}&R_{12}&R_{13}&0\\R_{21}&R_{22}&R_{23}&0\\R_{31}&R_{32}&R_{33}&0\\0&0&0&1\end{bmatrix}\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}\)
Rotation of axes are defined by the inverse (transpose) of the rotation matrix transforming points by the same amount. A rotation of axes is also referred to as a pose. Unless specified, the rest of this page uses implies rotation to be a rotation of points about the origin.
Basic Rotations
A non-rotation is described by an identity matrix
\(\mathbf{R}_{0}(\theta)=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\)
The right-handed rotation of points about the the \(X\), \(Y\), and \(Z\) axes are given by:
\(\mathbf{R}_{X}(\theta)=\begin{bmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{bmatrix}\)
\(\mathbf{R}_{Y}(\theta)=\begin{bmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{bmatrix}\)
\(\mathbf{R}_{Z}(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}\)
The inverse of these rotations are given by:
\(\mathbf{R}^{-1}_{X}(\theta)=\mathbf{R}_{X}(-\theta)=\begin{bmatrix}1&0&0\\0&\cos\theta&\sin\theta\\0&-\sin\theta&\cos\theta\end{bmatrix}\)
\(\mathbf{R}^{-1}_{Y}(\theta)=\mathbf{R}_{Y}(-\theta)=\begin{bmatrix}\cos\theta&0&-\sin\theta\\0&1&0\\\sin\theta&0&\cos\theta\end{bmatrix}\)
\(\mathbf{R}^{-1}_{Z}(\theta)=\mathbf{R}_{Z}(-\theta)=\begin{bmatrix}\cos\theta&\sin\theta&0\\-\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}\)
The rotation of axes by \(\theta\) radians is equivalent to a rotation of points by \(-\theta\) radians. The choices of rotation of bases or rotation of points and handedness of the rotation(s) should be specified to all relevant parties to avoid ambiguities in meaning.
Chaining Rotations
Rotations may be combined in sequence by matrix-multiplying their rotation matrices. When performing sequences of rotations, later rotations are left-multiplied. For example a transform is defined by first rotating by \(\mathbf{R}_1\), then by \(\mathbf{R}_2\), and finally by \(\mathbf{R}_3\), the single rotation \(\mathbf{R}\) that describes the sequence of rotations is
\(\mathbf{R}=\mathbf{R}_3\mathbf{R}_2\mathbf{R}_1\)
Rotation-Translation Combinations
Rotation-Translation Matrices
A rotation about the origin followed by a translation may be described by a single \(4\times4\) matrix
\(\begin{bmatrix}\mathbf{R}&\mathbf{t}\\\mathbf{0}^{\top}&1\end{bmatrix}\)
where \(\mathbf{R}\) is the \(3\times3\) rotation matrix, \(\mathbf{t}\) is the \(3\times1\) translation, and \(\mathbf{0}\) is the \(3\times1\) vector of zeros.
Since the last row of the \(4\times4\) rotation-translation matrix is always \(\begin{bmatrix}0&0&0&1\end{bmatrix}\), they are sometimes shorthanded to a \(3\times4\) augmented matrix
\(\left[\begin{array}{c|c}\mathbf{R}&\mathbf{t}\end{array}\right]=\left[\begin{array}{ccc|c}R_{11}&R_{12}&R_{13}&t_{1}\\R_{21}&R_{22}&R_{23}&t_{2}\\R_{31}&R_{32}&R_{33}&t_{3}\end{array}\right]\)
Note that when using this shorthand, matrix math is technically being broken as you cannot matrix multiply a \(3\times4\) matrix with a \(3\times4\) matrix. It is the implicit last row that is always the same that allows us to get away with this shorthand.
Rotation-Translation Inverse
The inverse of a rotation-translation matrix is given by
\(\left[\begin{array}{c|c}\mathbf{R}&\mathbf{t}\end{array}\right]^{-1}=\left[\begin{array}{c|c}\mathbf{R}^{-1}&-\mathbf{R}^{-1}\mathbf{t}\end{array}\right]=\left[\begin{array}{c|c}\mathbf{R}^{\top}&-\mathbf{R}^{\top}\mathbf{t}\end{array}\right]\)
Chaining Rotation-Translations
Just like pure rotation matrices, later rotation-translation transforms are left multiplied. Given transform \(\left[\begin{array}{c|c}\mathbf{R}_1&\mathbf{t}_1\end{array}\right]\) followed by \(\left[\begin{array}{c|c}\mathbf{R}_2&\mathbf{t}_2\end{array}\right]\), the resultant transform is
\(\left[\begin{array}{c|c}\mathbf{R}_2&\mathbf{t}_2\end{array}\right]\left[\begin{array}{c|c}\mathbf{R}_1&\mathbf{t}_1\end{array}\right]=\left[\begin{array}{c|c}\mathbf{R}_2\mathbf{R}_1&\mathbf{R}_2\mathbf{t}_1+\mathbf{t}_2\end{array}\right]\)
Rotations About an Arbitrary Point
A rotation about an arbitrary point may broken up into 3 transforms: a translation so that the arbitrary point becomes the origin, followed by the rotation about the origin, and finally a translation back to the arbitrary point.
\(\left[\begin{array}{c|c}\mathbf{I}&\mathbf{t}\end{array}\right]\left[\begin{array}{c|c}\mathbf{R}&\mathbf{0}\end{array}\right]\left[\begin{array}{c|c}\mathbf{I}&-\mathbf{t}\end{array}\right]=\left[\begin{array}{c|c}\mathbf{R}&\mathbf{t}-\mathbf{R}\mathbf{t}\end{array}\right]\)
Key Considerations
- Specify if the meaning is a rotation/translation of points or a rotation/translation of axes.
- When working with rotation matrix decompositions, specify the handedness of the rotation.
- When writing a rotation matrix or a rotation-translation matrix parameters out as a list, specify the order (rows-first or columns-first).
- Rotations matrices are defined about about the origin.
- When chaining rotations or rotations and translations, the later operations are applied on the left.
Further Reading
https://en.wikipedia.org/wiki/Rotation_matrix
http://www.euclideanspace.com/maths/geometry/rotations
http://mathworld.wolfram.com/RotationMatrix.html
S. Altmann. “Rotations, Quaternions, and Double Groups” 2005 Dover Publications.