# Documentation

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# Direction Cosine Matrix to Rotation Angles

Convert direction cosine matrix to rotation angles

## Library

Utilities/Axes Transformations

## Description

The Direction Cosine Matrix to Rotation Angles block converts a 3-by-3 direction cosine matrix (DCM) into three rotation angles R1, R2, and R3, respectively the first, second, and third rotation angles. The DCM matrix performs the coordinate transformation of a vector in inertial axes into a vector in body axes. The block Rotation Order parameter specifies the order of the block output rotations. For example, if Rotation Order has a value of `ZYX`, the block outputs are in the rotation order `z-y-x` (psi theta phi).

## Parameters

Rotation Order

Specifies the output rotation order for three rotation angles. From the list, select `ZYX`, `ZYZ`, `ZXY`, `ZXZ`, `YXZ`, `YXY`, `YZX`, `YZY`, `XYZ`, `XYX`, `XZY`, or `XZX`. The default is `ZYX`.

Limitation

The `'Default'` limitations for the `'ZYX'`, `'ZXY'`, `'YXZ'`, `'YZX'`, `'XYZ'`, and `'XZY'` implementations generate an `R2` angle that lies between ±90 degrees, and `R1` and `R3` angles that lie between ±180 degrees.

The `'Default'` limitations for the `'ZYZ'`, `'ZXZ'`, `'YXY'`, `'YZY'`, `'XYX'`, and `'XZX'` implementations generate an `R2` angle that lies between 0 and 180 degrees, and `R1` and `R3` angles that lie between ±180 degrees.

The `'ZeroR3'` limitations for the `'ZYX'`, `'ZXY'`, `'YXZ'`, `'YZX'`, `'XYZ'`, and `'XZY'` implementations generate an `R2` angle that lies between ±90 degrees, and `R1` and `R3` angles that lie between ±180 degrees. However, when `R2` is ±90 degrees, `R3` is set to 0 degrees.

The `'ZeroR3'` limitations for the `'ZYZ'`, `'ZXZ'`, `'YXY'`, `'YZY'`, `'XYX'`, and `'XZX'` implementations generate an `R2` angle that lies between 0 and 180 degrees, and `R1` and `R3` angles that lie between ±180 degrees. However, when `R2` is 0 or ±180 degrees, `R3` is set to 0 degrees.

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-3 matrixContains the direction cosine matrix.

OutputDimension TypeDescription

First

3-by-1 vectorContains the rotation angles, in radians.