# Yaw-Pitch-Roll (YPR) Description

The Yaw-Pitch-Roll (YPR) method of specifying orientation is one of the most difficult methods to understand. This difficulty is due to two factors: the similarity to the Euler angle method and use of a local frame on which the rotations are applied. All rotations in the YPR sequence are applied relative to the original local axes. This is different than the easier to visualize Euler sequence where each rotation occurs about the axes as defined by the prior rotation.

ODTK uses two local frames of reference in the computation of the maneuver direction when YPR is selected as the method of specifying the thrust acceleration direction for a maneuver. To facilitate understanding of these frames, the following directions are defined:

Symbol Name Description
R
Radial Along line from center of Earth through satellite position
I
In-track Normal to radial direction, towards inertial velocity
C
Cross-track Along the orbit angular momentum vector (R x I)
T
Tangential Along the inertial velocity direction
N
Normal Normal to tangential direction, towards radial direction

Sets of axes may now be defined by selecting right handed combinations of orthogonal directions. The convention used below is to specify the axes as (X)(Y)(Z) directions. Simple examples of axes definitions are the Gaussian frame which is defined as (R)(I)(C) and the Frenet frame which is defined as (N)(T)(C).

When the Gaussian frame is selected for input, the local frame of reference for the YPR rotations is defined as (I)(-C)(-R). An analogous frame, (T)(-C)(-N) is used as the local frame of reference for the YPR rotations when the Frenet frames is selected as input. The local frame of frame of reference is the inertial frame if the inertial frame is selected as input.

The YPR rotations (positive in a right handed sense) are defined as follows:

Rotation Description
Yaw Rotation about local Z axis
Pitch Rotation about local Y axis
Roll Rotation about local X axis

Rotations are performed in the sequence specified with the nominal thrust direction, specified via a (0,0,0) rotation, being along the X axis of the local frame. The nominal thrust direction is, therefore, in the (I) direction in the Gaussian frame and in the (T) direction in the Frenet frame. The sense of the rotations in the local frame is shown in the figure below.

Note according the these conventions, a roll rotation by itself does not change the thrust direction, a pitch of 90 degrees will place the thrust acceleration in the radial direction if the Gaussian frame is specified as input and a Yaw of -90 degrees will place the thrust acceleration in the cross-track direction if either the Gaussian or Frenet frames are specified as input.

The following diagram illustrates a Yaw of 30 degrees followed by a Pitch of 30 degrees. Notice that the pitch rotation occurs about the original Y axis of the local frame. There is an interesting relationship between YPR rotation sequences, which perform rotations about the original set of axes, and Euler rotation sequences, which perform rotations about the moving set of axes: a three rotation angle sequence (YPR or 321) about the original axes is equivalent to the three rotation angle sequence (RPY or 123) about the moving axes.

ODTK 6.5