Euler Angle from Body Frame to Inertial Frame

[Billwaa]

Hi,

This is not really a homework problem, but a project I'm working on.
So, I am trying to build a Simulink model for my quadcopter.

I derived the equations of motion using the Newtown-Euler method in the body frame to get transnational and angular acceleration.

For the transnational part, I can simply use a rotation matrix to convert the accelerations back into the inertial frame.

But what about the rotational part? Can I use the rotation matrix to get my angular acceleration in inertial frame? If so, how do I obtain the angles inside the rotation matrix since they are in inertia frame to start with I assume...

Can this problem be solved by starting the simulation aligning the Body Frame and Inertial Frame, using all 0 degrees as the initial inertial angle. Then in the next iteration, just use the inertial frame angles solved in the previous iteration?

Thanks,

 

[FactChecker]

But what about the rotational part? Can I use the rotation matrix to get my angular acceleration in inertial frame?

Rotational rate is a vector and can be transformed like any other vector, But as you stated, the problem is that the Euler angles have a rate of change.

Can this problem be solved by starting the simulation aligning the Body Frame and Inertial Frame, using all 0 degrees as the initial inertial angle. Then in the next iteration, just use the inertial frame angles solved in the previous iteration?

Yes. The rotational forces and rates can be analysed in the inertial coordinate system that is instantaneously aligned with the body axis.

 

[Billwaa]

Thanks for the response! So just to make sure I understand this correctly since this stuffs is kinda confusing.

Assuming that subscript n is the inertial frame and subscript c is the body frame.
And then in the rotation matrix, c is cos and s is sin.

The equations on the top right are derived in body frame. I want to convert to inertial frame by transposing the rotation matrix in lower right. The rotation matrix is derived using Euler ZYX multiplication. The angles in the rotation matrix are inertial frame?

To run this process in a simulation loop, I will use the inertial angles from the previous loop for the rotation matrix to get current inertia angle?

Thanks again :)

 

[FactChecker]

If you are trying to calculate the motion of an airplane for analysis or simulations, you can not avoid the problems with a rotating coordinate system, even for the linear accelerations. You should look at some references for the equations of motion and follow those equations. In atmospheric flight, some standard references are "Aircraft Dynamics and Automatic Control" by McGruer et al, or "Dynamics of Atmospheric Flight" by Etkin.

If you are worrying about the control system and the gyro/accelerometer inputs, then you can probably simplify things greatly. The measured accelerations and rotations are often what you want to use directly.