Assumptions

· Rotation Matrix Notation: row major with one row for each Frame B axis unit vector. Rotation matrix is post multiplied by Frame A column vector to get the rotated Frame B column vector.

Other conventions in common use are left handed coordinate systems, clockwise positive rotation, column major rotation matrix format, and quaternion numbering with the scalar component last. The user must exercise care in applying the results from QuatView to ensure that the above conventions are consistent with the user's analysis conventions.

The relationships between the Euler angles, quaternion, and rotation matrix generally depend on the choice of transformation direction and the coordinate system type. Transformation direction really refers to the direction of the angle measurements which can be referenced to the initial frame (Frame A) or the final frame (Frame B). That is, the angles can be measured from Frame A to Frame B or from Frame B to Frame A. The rotation matrix for one choice is simply the transpose of the other choice. The quaternion for one choice is simply the complex conjugate of the other. Choice of reference frame is described further in the quaternion overview section here.

The two different rotation matrix formats (row major or column major) are the transpose of each other. The row major format is most commonly used in dynamic simulation, while the column major format is most commonly used in computer graphics. The row major format is meant to be post-multiplied by a column vector, while the column major format is meant to be pre-multiplied by a row vector as shown below where R_R is the row major form and R_Cis the column major form.

V_I has the vector components referenced to the initial unrotated frame (Frame A), and V_F has the vector components referenced to the final rotated frame (Frame B).