# Coupled Equations Motion/Rotation

1. Mar 19, 2013

### aerowenn

I'm looking for the set of equations that describe an untethered rigid body about a body fixed axis in relation to a reference axis (ECI or etc.). I'm not sure what they are called. I was trying to use DCM or Quaternions to do this, but the discrete motion nature of these wont work for my application. Let me give a brief description of the problem I'm looking to solve.

It's some research I'm doing for small spacecraft. Basically, there is a noncommutative property to angular rotation sets about the body axis of an untethered rigid body. The best way I know to explain this without diagrams is like this:

If you have a rotation sequence some set angular motion in both +/- direction (say 5 degrees or something): +Roll, +Pitch, -Pitch, -Roll (1)

With rotation sequence (1), you end up back where you started, assuming the reference coordinates and the body coordinates were aligned originally, the DCM representation would be the identity matrix. The net motion about either of those two axes is zero.

Take this rotation sequence: +Roll, +Pitch, -Roll, -Pitch (2)

In sequence (2) the same motions were performed as before, but the last two are in a different order. The net change about either axis is still zero! HOWEVER, the rigid body has now changed attitude on all axes relative to the reference coordinates. Roll and Pitch axes change is extremely small, but the Yaw axis has experienced a significant change (depending on the size of that set angular motion about Yaw and Roll +/-).

This is a well known phenomenon and it's not really why I'm here. I know that it works and how it works (conceptually), but I need a way to determine the change on all three axes due to a continuous input.

I'm wanting to drive any two of these axes, in this example let's keep it to Roll and Pitch, with a sinusoidal input. Meaning that the body is oscillating on each of those two axes with time, the same distance in each direction. Those input sinusoids will differ by phase as to attempt to reproduce the behavior in the step motions described above, perhaps by pi/2 or some other phase, whatever is needed for optimal change.

I can't seem to locate the equation sets that define this. I know they would be coupled somehow, if motion occurs on only one axis at a time, only that axis is changed. But what if two are being changed as I described? In continuous time, oscillating. There should be some torque component I'm sure (though small in this case), an something that links the motion of two of those axes with the third?

Any help with this would be greatly appreciated. It doesn't seem to be a well documented scenario, or maybe I'm not sure what I'm looking for. Thanks!