PlanetPhysics/Direction Cosine Matrix

A direction cosine matrix (DCM) is a transformation [[../Matrix/|matrix]] that transforms one coordinate [[../CosmologicalConstant2/|reference frame]] to another. If we extend the [[../PreciseIdea/|concept]] of how the three dimensional [[../DirectionCosines/|direction cosines]] locate a [[../Vectors/|vector]], then the DCM locates three [[../PureState/|unit vectors]] that describe a coordinate reference frame. Using the notation in equation 1, we need to find the matrix elements that correspond to the correct transformation matrix.

$DCM=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]$

The first unit vector of the second coordinate frame can be located in the first frame by normal vector notation. See figure 1 for relationship.

${\hat{y}}_1=A_{11}{\hat{x}}_1+A_{12}{\hat{x}}_2+A_{13}{\hat{x}}_3$

\medskip \begin{figure} \includegraphics[scale=0.78]{DCM.eps} \end{figure} \medskip

Similarily, the other two unit vectors can be described by

${\hat{y}}_2=A_{21}{\hat{x}}_1+A_{22}{\hat{x}}_2+A_{23}{\hat{x}}_3$ ${\hat{y}}_3=A_{31}{\hat{x}}_1+A_{32}{\hat{x}}_2+A_{33}{\hat{x}}_3$

It is easy to see how equation 1 works as a transformation matrix through simple [[../Matrix/|matrix multiplication]].

$\left[\begin{array}{c}{\hat{y}}_1\\ {\hat{y}}_2\\ {\hat{y}}_3\end{array}\right]=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\\ {\hat{x}}_3\end{array}\right]$

Once this transformation matrix is found, it can be used to transform vectors from the second frame to the first frame and vice versa. Equation 2 transforms the x frame to the y frame and can be denoted as $R_{1-2}$. In order to get $R_{2-1}$, which transforms the y frame to the x frame, we use a property of transformation matrices of orthonormal reference frames (a frame that is described by unit vectors and are perpindicular to each other). See the entry on a transformation matrix for more info on its properties. We use the properties that

$R_{1-2}^{-1}=R_{1-2}^T=R_{2-1}$ $R_{1-2}{R}_{1-2}^T=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

so using these properties and rearranging equation 2 $\hat{y}=R_{1-2}\hat{x}$ yields

$R_{1-2}^{-1}\hat{y}=R_{1-2}^{-1}{R}_{1-2}\hat{x}$

giving the transformation of the y frame to the x frame

$\hat{x}=R_{2-1}\hat{y}$

So to extend this concept to transform vectors from one frame to another a closer examination of a vector being represented in both frames is needed. If we denote the second frame as the prime ($\prime$) frame, then a vector expressed in each of these is given by

$v=v_1{\hat{x}}_1+v_2{\hat{x}}_2+v_3{\hat{x}}_3$

$v=v_1\prime {\hat{y}}_1+v_2\prime {\hat{y}}_2+v_3\prime {\hat{y}}_3$

Since both equations describe the same vector, let us set them equal to each other so

$v_1{\hat{x}}_1+v_2{\hat{x}}_2+v_3{\hat{x}}_3=v_1\prime {\hat{y}}_1+v_2\prime {\hat{y}}_2+v_3\prime {\hat{y}}_3$

This notation is clumsy so we want to represent it in matrix notation. This is simple enough if you have an understanding of multiplying a column vector by a row vector. This allows us to describe equations 3 and 4 by

$v=\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\\ {\hat{x}}_3\end{array}\right]$ $v=\left[\begin{array}{ccc}{v}_1\prime & v_2\prime & v_3\prime \end{array}\right]\left[\begin{array}{c}{\hat{y}}_1\\ {\hat{y}}_2\\ {\hat{y}}_3\end{array}\right]$

Setting them equal and substituting equation 2 in for the second coordinate frame yields

$v=\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\\ {\hat{x}}_3\end{array}\right]=\left[\begin{array}{ccc}{v}_1\prime & v_2\prime & v_3\prime \end{array}\right]\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]\left[\begin{array}{c}{\hat{x}}_1\\ {\hat{x}}_2\\ {\hat{x}}_3\end{array}\right]$

Then by inspection (or go through the matrix manipulation to cancel the x frame)

$\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]=\left[\begin{array}{ccc}{v}_1\prime & v_2\prime & v_3\prime \end{array}\right]\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]$

Representing the transformation matrix as $R_{1-2}$ as the transformation from the first frame to the second frame and transposing the previous equation gives

$\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]=(\left[\begin{array}{ccc}{v}_1\prime & v_2\prime & v_3\prime \end{array}\right]R_{1-2}{)}^T$

Performing the transposition and using a transposition property for two matrices A and B such that

$(AB{)}^T=B^T{A}^T$

leads to the relationship

$\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]=R_{1-2}^T\left[\begin{array}{c}{v}_1\prime \\ v_2\prime \\ v_3\prime \end{array}\right]$

Finally giving us the ability to transform a vector from the second (prime) frame to the first frame.

$\overrightarrow{v}=R_{2-1}\overrightarrow{v\prime }$

Much much more can be found in the general entry about the Transformation matrix.