How can I compute the attitude of a satellite given its yaw, pitch, roll, and velocity?

Question

How can I compute the attitude of a satellite given its yaw, pitch, roll, and velocity?

Yaw, Pitch, and Roll are given with respect to the velocity vector (A vector whose origin is the satellite's center towards to the direction of motion).

The velocity is given in ECEF.

By attitude I mean the direction of the satellite's body after the rotations along different axes caused by yaw/pitch/roll steering.

Answer 1

You can combine the 3 successive rotations to obtain the rotation matrix from the initial to the final attitude.

Let's name $\vartheta$, $\varphi$ and $\psi$ the angles respectively about roll, pitch and yaw. And suppose we want to combine the rotations in exactly this order. The rotation matrices related to each rotation are: $$ A_{\rm r} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c_\vartheta & s_\vartheta \\ 0 & -s_\vartheta & c_\vartheta \end{bmatrix}, \; A_{\rm p} = \begin{bmatrix} c_\varphi & 0 & -s_\varphi \\ 0 & 1 & 0 \\ s_\varphi & 0 & c_\varphi \end{bmatrix}, \; A_{\rm y} = \begin{bmatrix} c_\psi & s_\psi & 0 \\ -s_\psi & c_\psi & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Then just perform the product of the 3 matrices in the reverse order (successive rotations rule): $$ A(\vartheta, \varphi, \phi) = A_{\rm y}A_{\rm p}A_{\rm r} $$

A couple of notes

• If you change the order, the final attitude will be different
• I suppose that RPY reference frame rotates while the satellite moves along the orbit (i.e. the roll axis is always aligned with the velocity vector). This means that the attitude matrix you compute describes actually the orientation of a body fixed reference frame (initially coincident with RPY) wrt RPY related to a particular position of the satellite.

Hope this helps