Jenny Dykes 12/31/2007

Derivation of angular velocity components (pqr) in body coordinates. This derivation applies to Euler angles relative to the right handed coordinate system defined by a set of axes fixed to the Earth. Yaw is the first rotation about the z axis, pitch is the second rotation about the y axis that formed following yaw, and roll is the third rotation about the x axis that was formed following pitch.

Yaw is the first rotation about the Earth's Axis. This gives,, , and body axes in Earth coordinates:

1

The matrix is the direction cosine matrix and defines the axes components after the yaw rotation. The and matrices will define the axes components after the pitch and roll rotations. The Earth axes in, , and body coordinates are given by transposing the A matrix:

2

This says that:

3

Pitch is defined as the rotation about the new axis that was created when we yawed about the axis. We pitch about the axis to get , , body axes.

4

Body , , axes in, , coordinates are given by:

5

This says that

6

Roll is defined as the rotation about the new axis that was created when we pitched. We roll about the axis to give us the final , , body axes in Earth coordinates.

7

Body , , axes in , , coordinates are given by:

8

This says that:

9

And

10

Each rotation can be considered independently about the , , axes respectively. Therefore, their rates can be summed to derive an angular velocity vector.

11

Where is the yaw rate, is the pitch rate, and is the roll rate.

To create a velocity vector in body coordinates, we need to express the angular velocity components in Equation 11 in body coordinates (, , ) to create the angular velocity (pqr) vector. Consider Equations 3 and 6:

12

Substituteand from Equations 9, you get

13

Now, we have in terms of body coordinates. Now consider the second component of Equation 11. Substitute the using Equations 6 and 9:

14

Now, we have in terms of body coordinates. And finally consider the third component of Equation 11. Substitute the with from Equation 9:

15

Substitute the unit vectors in equation 10 with equations 13, 14, and 15.

16

Now rearrange Equation 16 to combine the, , components so

17

Angular velocity is a vector with components that are functions of pitch, roll, yaw rate, pitch rate, and roll rate. The resulting components agree with the , , and values given in the IEEE std. P1278.1/d8.