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direction cosine matrix (Definition)

A direction cosine matrix (DCM) is a transformation matrix that transforms one coordinate reference frame to another. If we extend the concept of how the three dimensional direction cosines locate a vector, then the DCM locates three 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.

$\displaystyle DCM = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \ A_{21} & A_{22} & A_{23} \ A_{31} & A_{32} & A_{33} \end{array} \right]$ (1)

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 $


\includegraphics[scale=0.78]{DCM.eps}


Similarily, the other two unit vectors can be described by

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

$\displaystyle \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 multiplication.

$\displaystyle \left[ \begin{array}{c} \hat{y}_1 \ \hat{y}_2 \ \hat{y}_3 \en... ...eft[ \begin{array}{c} \hat{x}_1 \ \hat{x}_2 \ \hat{x}_3 \end{array} \right]$ (2)

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

$\displaystyle R_{1-2}^{-1} = R_{1-2}^T = R_{2-1} $

$\displaystyle 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

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

$\displaystyle 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

$\displaystyle \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

$\displaystyle v = v_1 \hat{x}_1 + v_2 \hat{x}_2 + v_3 \hat{x}_3$ (3)
$\displaystyle v = v_1\prime \hat{y}_1 + v_2\prime \hat{y}_2 + v_3\prime \hat{y}_3$ (4)

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

$\displaystyle 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

$\displaystyle 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] $

$\displaystyle v = \left[ \begin{array}{ccc} v_1\prime & v_2\prime & v_3\prime \... ...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

$\displaystyle v = \left[ \begin{array}{ccc} v_1 & v_2 & v_3 \end{array} \right]... ...ft[ \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)

$\displaystyle \left[ \begin{array}{ccc} v_1 & v_2 & v_3 \end{array} \right] = \... ... \ 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

$\displaystyle \left[ \begin{array}{ccc} v_1 & v_2 & v_3 \end{array} \right] = (... ...{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

$\displaystyle (AB)^T = B^TA^T $

leads to the relationship

$\displaystyle \left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array} \right] = R... ...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.

$\displaystyle \vec{v} = R_{2-1} \vec{v \prime} $

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



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Cross-references: matrix multiplication, works, unit vectors, vector, direction cosines, concept, reference frame, matrix
There are 14 references to this object.

This is version 15 of direction cosine matrix, born on 2005-08-25, modified 2005-08-28.
Object id is 85, canonical name is DirectionCosineMatrix.
Accessed 23202 times total.

Classification:
Physics Classification: 45.40.-f (Dynamics and kinematics of rigid bodies)
 45.05.+x (General theory of classical mechanics of discrete systems)
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