Proper Euler angles use the same axis for both the first and third elemental rotations (e.g., z-x-z, or z-x’-z″).
There are six possibilities of choosing the rotation axes for the Yaw, Pitch and Roll angles if we consider that they are Tait–Bryan angles representing rotations about three distinct axes (e.g. The rotation matrix to rotate an offset vector in the object reference frame by a Heading angle around the Z-axis is given by: The HPR rotation sequence about the original set of non-moving reference axes can be explained in terms of the Euler angle rotation sequence about the moving set of body axes via the following relationship: three angle rotations in the sequence ijk taken about the reference axes results in the same net orientation as performing the same three rotations but in the sequence kji about the moving axes. Of the six possible combinations of Yaw, Pitch and Roll, this combination is the only one in which the Heading (direction of the Roll axis) is equal to one of the rotations (the Yaw) and the Elevation (angle of the Roll axis with the horizontal plane) is equal to one other of the rotations (to the Pitch). The matrix of the composed rotations is M = Yaw In that case, heading is the direction that the aircraft's nose is pointing. In literature, the HPR sequence is often called the YPR method when describing the principal axes of an aircraft. the rotation matrix using the negative angle.
In order to rotate an offset vector x' in the local level reference frame to an offset vector x in the object reference frame, the transpose of the above equations would be required as:įor rotation matrices holds that the transpose matrix is equal to the inverse matrix, i.e. When seen from outside the object, first the Heading is applied (around the Z-axis), then the Pitch (around the rotated X-axis), and then the Roll (around the rotated Y-axis). Heading, pitch and roll angles are measured about the local level reference frame axes. Internally, Qinsy uses the so-called HPR sequence of rotation matrices to rotate an offset vector x in the object reference frame (body coordinates) to an offset vector x' in the local level reference frame (world coordinates), where H denotes the Heading matrix, P is the Pitch matrix, and R is the Roll matrix: