Coupled Equations Motion/Rotation

[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 won't 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!

 

[eljose79]

Thank you for your inquiry regarding the equations that describe an untethered rigid body about a body fixed axis in relation to a reference axis. This is a complex problem and the equations you are looking for are known as the Euler equations of motion. These equations describe the rotational motion of a rigid body about a fixed axis and are commonly used in spacecraft dynamics and control.

The Euler equations of motion are based on the principles of angular momentum and torque. They can be written in terms of the body's angular velocity and the moments of inertia, and they take into account the noncommutative property you mentioned in your post. The equations are coupled and take into account the motion of all three axes.

In your specific case, where you are trying to drive two axes with sinusoidal inputs, the equations would involve additional terms for the applied torque and the resulting motion on the third axis. These equations can be solved numerically using simulation software or through analytical methods.

I would recommend consulting with a specialist in spacecraft dynamics or a professor in a related field for further assistance with these equations. They can also provide guidance on how to incorporate the effects of continuous inputs on the rigid body's motion.

I hope this information helps and wish you the best of luck with your research. Please let me know if you have any further questions.