Euler angles and angular velocity

In summary: A}}{dt}=\frac{dA_x}{dt}\hat{i}+\frac{dA_y}{dt}\hat{j}+\frac{dA_z}{dt}\hat{z}+\left( A_x \frac{d\hat{i}}{dt}+A_y\frac{d\hat{j}}{dt}+A_z \frac{d \hat{z}}{dt}\right)If you know the velocity of a vector in a frame you can solve for the time derivative by using the dot product.
  • #1
Curl
758
0
How do you prove that angular velocity is just the time derivative of each Euler angle times the basis vector of its respective frame?
I remember it used to be perfectly clear to me a while back, but now I don't remember how the result was derived, and I couldn't find it in any of my books I looked so far.
Does anyone remember how the result was derived?

Thanks
 
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  • #2
Hi Curl! :smile:

Each euler angle is measured in a plane through the origin.
In other words, one specific Euler angle behaves exactly as if it was a 2 dimensional polar coordinate.
If an angle changes by an amount ##d\phi## in an time interval ##dt##, the position changes by ##r d\phi##.
In other words:
$$\frac{ds}{dt}=\frac{r d\phi}{dt}=r \frac{d\phi}{dt} = r \omega$$
 
  • #3
the problem I have is that in order to add the 3 angular velocity vectors (one in each frame) implies that the rotation can be represented as the sum of the three individual rotations. But rotation order matters, which makes things confusing.
 
  • #4
Exactly. The time derivatives of a set of Euler angles (better said: Tait-Bryan angles, Bryan angles, or Cardan angles; Euler angles are a z-x-z rotation) are not angular velocity.
 
  • #5
How do you prove that angular velocity is just the time derivative of each Euler angle times the basis vector of its respective frame?

I think you mean the 'cross product', not times.
 
  • #6
Well that sentence is flawed actually.
Say you have a vector [itex] A=A_x\hat{i}+A_y\hat{j}+A_z\hat{z} [/itex] in a Cartesian frame. Knowledge of its velocity in this frame is obtained through its time derivative. This yields
[tex]
\frac{d\hat{A}}{dt}=\frac{dA_x}{dt}\hat{i}+\frac{dA_y}{dt}\hat{j}+\frac{dA_z}{dt}\hat{z}
[/tex]
which can then be deduced to
[tex]
\frac{d\hat{A}}{dt}=\dot{A} \vec{A}+\vec{\omega} \times \vec{A}
[/tex]
In the above, [itex] \vec{\omega} [/itex] is what you call angular velocity.

Cheers,
 
  • #7
Oh jeeze. I came to check this just now and I realized I forgot one line. :)

Here it is:
[tex]
\frac{d\hat{A}}{dt}=\frac{dA_x}{dt}\hat{i}+\frac{d A_y}{dt}\hat{j}+\frac{dA_z}{dt}\hat{z}+\left( A_x \frac{d\hat{i}}{dt}+A_y\frac{d\hat{j}}{dt}+A_z \frac{d \hat{z}}{dt}\right)
[/tex]

Change this line in my above post. This can then deduce the last equation after realizing that the unit vectors have a fixed length.
 

1. What are Euler angles and how are they used in science?

Euler angles are a set of three angles that describe the orientation of an object in 3D space. They are used in science, particularly in physics and engineering, to represent the rotational motion of an object.

2. How do Euler angles differ from other methods of representing orientation?

Unlike other methods, such as quaternions, Euler angles are intuitive and easy to visualize. They also allow for a unique representation of orientation, unlike other methods which may have multiple representations for the same orientation.

3. How are Euler angles related to angular velocity?

Euler angles are directly related to angular velocity, as they describe the rotational motion of an object. The angular velocity vector can be derived from the time derivatives of the Euler angles.

4. Can Euler angles be used to describe any type of rotation?

No, Euler angles have limitations and cannot be used to represent all types of rotations. They are most commonly used for rotations around a fixed axis, and can cause issues with singularities when trying to represent certain types of rotations.

5. How are Euler angles used in real-world applications?

Euler angles have various applications in fields such as aerospace engineering, robotics, and computer graphics. They are used to describe the orientation of objects, such as aircraft and spacecraft, and are also used in animations and simulations to create realistic movements.

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