Direction Cosine Matrix

The difference, in three dimensions, between any given orthogonal frame and a base coordinate frame is the orientation or attitude.

Rotations are linear operations preserving vector lenght and relative vector orientation, and a rotation operator acting on a vector \(\mathbf{v}\in\mathbb{R}^3\) can be defined in the Special Orthogonal group \(SO(3)\), also known as the rotation group.

The rotation operator is a linear transformation represented by a \(3\times 3\) matrix:

\[\begin{split}\mathbf{R} = \begin{bmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{bmatrix} \in \mathbb{R}^{3\times 3}\end{split}\]

where

\[\begin{split}\begin{array}{lcr} \mathbf{r}_1 = \begin{bmatrix}r_{11}\\ r_{12} \\ r_{13} \end{bmatrix} \; , & \mathbf{r}_2 = \begin{bmatrix}r_{21}\\ r_{22} \\ r_{32} \end{bmatrix} \; , & \mathbf{r}_3 = \begin{bmatrix}r_{31}\\ r_{23} \\ r_{33} \end{bmatrix} \end{array}\end{split}\]

are unit vectors orthogonal to each other. All matrices satisfying this orthogonality are called orthogonal matrices.

The transformation matrix \(\mathbf{R}\) rotates any vector \(\mathbf{v}\in\mathbb{R}^3\) through the matrix product,

\[\mathbf{v}' = \mathbf{Rv}\]

We observe that \(\mathbf{RR}^{-1}=\mathbf{RR}^T=\mathbf{R}^T\mathbf{R}=\mathbf{I}\), indicating that the inverse of \(\mathbf{R}\) is its transpose. So,

\[\mathbf{v} = \mathbf{R}^T\mathbf{v}'\]

The determinant of a rotation matrix is always equal to \(+1\). This means, its product with any vector will leave the vector’s lenght unchanged.

Matrices conforming to both properties belong to the special orthogonal group \(SO(3)\). Even better, the product of two or more rotation matrices yields another rotation matrix in \(SO(3)\).

Direction cosines are cosines of angles between a vector and a base coordinate frame [WikipediaDCM]. In this case, the difference between orthogonal vectors \(\mathbf{r}_i\) and the base frame are describing the Direction Cosines. This orientation matrix is commonly named the Direction Cosine Matrix.

DCMs are, therefore, the most common representation of rotations [WikipediaRotMat], especially in real applications of spacecraft tracking and location.

References

[WikipediaDCM]

Wikipedia: Direction Cosine. (https://en.wikipedia.org/wiki/Direction_cosine)

[WikipediaRotMat]

Wikipedia: Rotation Matrix (https://mathworld.wolfram.com/RotationMatrix.html)

[Ma]	Yi Ma, Stefano Soatto, Jana Kosecka, and S. Shankar Sastry. An Invitation to 3-D Vision: From Images to Geometric Models. Springer Verlag. 2003. (https://www.eecis.udel.edu/~cer/arv/readings/old_mkss.pdf)

[Huyhn]

Huynh, D.Q. Metrics for 3D Rotations: Comparison and Analysis. J Math Imaging Vis 35, 155–164 (2009).

[Curtis]

Howard D Curtis. Orbital Mechanics for Engineering Students (Third Edition) Butterworth-Heinemann. 2014.

[Kuipers]

Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality. Princeton; Oxford: Princeton University Press, 1999.

[Diebel]

Diebel, James. Representing Attitude; Euler Angles, Unit Quaternions, and Rotation. Stanford University. 20 October 2006.