# Why are two frames of reference needed for attitude determination?

I am trying to understand the global workings of attitude determination, specifically in space, but I don't understand why two reference frames are needed to establish that?

Let's say you have a CubeSat and you put a light sensor on every side. By knowing which side gets the most light you already know, more or less, how it is positioned by its observational frame of reference. Why is an additional inertial reference frame required then?

• I'm not sure I understand your question correctly, but knowing which axis is pointing toward a light source doesn't tell you how you're rolled around that axis. May 24 '15 at 1:28
• Answer this question, and you will have answered your own question: In what frame of reference are you trying to find the attitude of the CubeSat?
– user8406
May 24 '15 at 5:52

Starting with the most basic concept...

Depending on which type of rotation sequence you use, you need three angles to describe orientation (attitude) in space. I'll use the 3-2-1 body sequence (yaw-pitch-roll) since it's the easiest to visualize. (You can also select a 1-2-1 sequence to describe precession, nutation, and spin.)

Each angle implies a change in reference frame. The first angle, yaw, describes the body's heading. The second angle, pitch, then describes its elevation, or how high the "nose" is pointing (using aircraft terminology). Finally, roll describes the rotation about that body about the "nose".

Continuing with the aircraft as an example, the first frame is the inertial reference frame. It's not really inertial because it's fixed to the surface of the Earth, but the forces acting on an aircraft vary far more due to aerodynamics than to changes in gravity, so we can consider it inertial. In the case of spacecraft, the selection of the inertial frame requires greater precision.

*Important: In your question, you ask why an additional inertial reference frame is required. 1) It's simpler to consider a single reference frame "inertial". 2) The only way to have multiple inertial reference frames is if they are motionless with respect to each other. That means a second, third, etc. inertial reference frame is NOT required. 3) I'm assuming you mean to ask why an additional reference frame is required - one that is not stationary with respect to the inertial frame.

Now that we have defined what is and is not "inertial", and determined that three angles are required to describe orientation, let's count the reference frames. Start by naming them:

n - inertial

e - intermediate

f - intermediate

b - body

Yaw angle/heading is the angle between n and e about the aircraft's z axis (pointing down relative to pilot) Pitch is the angle between e and f about the aircraft's y axis (pointing right relative to pilot) Roll is angle between f and b about the aircraft's x axis (pointing forward relative to pilot)

*Note: Rotation about z-y-x is where "3-2-1" comes from.

In total, there are four reference frames, only one of which is inertial, and just to be pedantic, you have to keep in mind that it's not really inertial.

Translating this to a CubeSat...

If you're going to use Euler angles, satellite orientations are typically defined in 1-2-1 or 1-3-1 precession-nutation-spin sequences since they're designed for single missions and need to maintain communication and observe a specific target. I won't get into that.

Some practical considerations...

Suppose your airplane is pointing straight up and rolls to the right. Using the definitions above, is that a roll to the right or a yaw to the left? When using Euler angles, there are some orientations where one rotation sequence will not properly define the orientation of the aircraft/spacecraft and the orientation after that will be unknown as well. This is known as gimbal lock; mathematically, it's a singularity.

Computationally, quaternions are the standard method of calculating orientation and rates. They're based on calculating an axis of rotation and the angle about that axis of rotation. They aren't at all easy to visualize, nor are they intuitive, but you can always translate them to Euler angles, even after gimbal lock (but not during), so if you need to analyze data, you still can.

The way you're thinking of measuring orientation is also very imprecise. Basically useless. The amount of light hitting the spacecraft varies so much depending on the time of day and the orbit that you would never actually know where you're facing.

If you want to learn the math: http://www.swarthmore.edu/NatSci/mzucker1/e27/diebel2006attitude.pdf