Transforming angular velocity between different coordinate frames

In summary, the conversation discusses two coordinate frames, A and B, and their relationship through a fixed rigid body transformation given by the rotation matrix R_BA. The goal is to calculate the 3D angular velocity of frame B, w_B, given the 3D angular velocity of frame A, w_A. Two solutions are presented, with the second one being the correct approach. The first solution involves converting w_A into a delta rotation, applying the transformation, and converting back to w_B, while the second solution directly applies the rotation. The second solution is the correct one because it follows the correct method of transforming a vector between frames using the rotation matrix.
  • #1
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Homework Statement


There are two coordinate frames i.e. frame A and frame B. The relationship between them is that frame A is rotated w.r.t to frame B . This relationship remains fixed i.e. rigid body. This rigid body relationship is given by rotation matrix R_BA which transforms the vector in frame A to a vector in frame B. Now i have a 3D angular velocity describing rotation of frame Ai.e. w_A. I want to calculate 3D angular velocity of frame B i.e w_B. How can i do this . I have two solutions(given below) which don't agree with each other.

Homework Equations



R_BA rotation from frame A to frame B
R_AB rotation from frame B to frame A. This is the transpose of R_BA
Delta R_B delta rotation in frame B
Delta R_A delta rotation in frame A
w_A 3D angular velocity in frame A
w_B 3D angular velocity in frame B

The Attempt at a Solution


Solution One
I can convert this angular velocity w_A into rotation matrix i.e. Delta R_A, then do the following
Delta R_A= R_BA*Delta R_A *R_AB

Now i can convert Delta R_A back to angular velocity w_B.

Solution two

w_B= R_BA * w_A

Now these two solution don't agree. In solution one i converted the angular velocity to delta rotation, applied the transformation and converted the rotation back to angular velocity.

In the second i directly applied the the rotation.
 
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  • #2
Can someone please tell me which solution is correct. And why is the other wrong. A:The second is correct, because if you want to transform a vector from one frame to another you just multiply it by the rotation matrix and vice versa.The first is wrong, because the expression $R_{BA} \Delta R_A R_{AB}$ means that you rotate the vector $\Delta R_A$ in frame A with rotation matrix $R_{AB}$, then you rotate the resulting vector with the rotation matrix $R_{BA}$. This is not what you want.
 

1. What is angular velocity and why is it important?

Angular velocity is a measure of how fast an object is rotating around a fixed point. It is important because it helps us understand the motion of objects in circular or rotational motion and is a key concept in fields such as physics, engineering, and astronomy.

2. How do you calculate angular velocity?

Angular velocity is calculated by dividing the change in angular displacement (in radians) by the change in time. This can be represented by the equation ω = Δθ/Δt, where ω is the angular velocity, Δθ is the change in angular displacement, and Δt is the change in time.

3. What is the difference between angular velocity and linear velocity?

Angular velocity is a measure of how fast an object is rotating, while linear velocity is a measure of how fast an object is moving in a straight line. Angular velocity is expressed in radians per second, while linear velocity is expressed in meters per second.

4. How do you convert angular velocity between different coordinate frames?

To convert angular velocity between different coordinate frames, you can use the formula ω₂ = R ω₁, where ω₁ is the angular velocity in the initial frame, ω₂ is the angular velocity in the new frame, and R is the rotation matrix that describes the transformation between the two frames.

5. Can angular velocity be negative?

Yes, angular velocity can be negative. A negative angular velocity indicates that the object is rotating in the opposite direction compared to a positive angular velocity. The direction of rotation is determined by the right-hand rule, where the direction of the fingers curling in the direction of rotation corresponds to a positive angular velocity.

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