Transformation of angular and linear velocities

[Roughmar]

First and foremost, I decided with putting this in the math sub-forum since I believe my problem has to do with the general maths of this.

I'm trying to study the dynamics of an object. For a whole lot of reasons, it is to my advantage to study those dynamics on a object fixed axis.

For that, I use an Earth fixed axis and subsequently apply a rotation.

I'll use a specific example I found in a known paper:

I'm using a Z-X-Y order for such a rotation, giving me this matrix, expressed in roll, pitch and yaw components:

R =
$\left[cos\psi cos\theta - sin\phi sin\psi sin\theta , -cos\phi sin \psi , cos\psi sin\theta + cos\theta sin\phi sin\psi\right]$

$\left[cos\theta sin\psi + cos\psi sin\phi sin\theta , cos\phi cos\psi , sin\psi sin\theta - cos\psi cos\theta sin\phi\right]$

$\left[-cos\phi sin\theta , sin\phi , cos\phi cos\theta\right]$

(am not really proficient with coding here, so bear with me with the previous and following "matrix", if someone can provide a code example for a matrix I'd be more than willing to make this more elegant)

Now, the components of angular velocity in the body frame are p,q and r.
These values are related to the derivatives of the roll, pitch and yaw angles according to

$\left[p,q,r\right]^{T} =\left[cos\theta , 0 , -cos\phi sin\theta ; 0 , 1 , sin\phi;sin\theta , 0 , cos\phi\cos\theta\right]\left[\dot{\phi}, \dot{\theta},\dot{\psi}\right]^{T}$

...

My question is why.
How is this last matrix calculated? I've been looking at this for the last 4 hours and reached no conclusion whatsoever.

 

[jvicens]

Thank you for sharing your problem with us. I understand the importance of studying the dynamics of an object and the need for a fixed axis for accurate analysis.

The matrix you have mentioned is known as the rotation matrix, which is used to describe the orientation of a body in three-dimensional space. It is commonly used in various fields of science, including mechanics and robotics.

In your specific example, the rotation matrix is expressed in terms of roll, pitch, and yaw components. These components represent the body's orientation in space and are related to the angular velocity of the object.

The last matrix you have mentioned is known as the transformation matrix, which is used to convert the angular velocity components from the body frame to the Earth-fixed frame. This matrix is calculated using the relationship between the roll, pitch, and yaw angles and their derivatives, as shown in the equation you have provided.

To understand this better, let's break down the equation. The first row of the transformation matrix represents the relationship between the angular velocity component p and the derivatives of the roll, pitch, and yaw angles. Similarly, the second and third rows represent the relationships for q and r, respectively.

To calculate the transformation matrix, you can use the trigonometric identities for the roll, pitch, and yaw angles. For example, the first element in the first row can be calculated as cos(theta)*cos(psi). Similarly, you can calculate the other elements using the identities mentioned in the paper you have referred to.

I hope this explanation helps you understand the calculation of the transformation matrix. If you have any further questions, please do not hesitate to ask.