Angular velocities using Euler angles
In this blog post, I explore the concept of angular velocity for a rigid body undergoing rotations, specifically when described by Euler angles. I detail how to calculate the total angular velocity by summing individual contributions of rotation, and I then derive expressions for this total angular velocity in two frames: the body-fixed and the space-fixed frames. This process involves expressing intermediate rotation axes in terms of the respective frame’s basis vectors, leading to explicit formulas for angular velocity using Euler angle rates. This provides a clear understanding of how rotations accumulate to create a body’s overall angular motion.
When a rigid body rotates, its angular velocity, which I denote as \mathbf{\omega}, describes how fast it is rotating and around which axis. If I consider rotations described by Euler angles (\psi, \theta, \phi), the total angular velocity is the sum of the individual angular velocities due to each rotation. Using the angular velocity addition theorem, I can write the total angular velocity of a body-fixed frame (frame 3) with respect to a space-fixed frame (frame 0) as:
\mathbf{\omega}_{3/0} = \mathbf{\omega}_{3/2} + \mathbf{\omega}_{2/1} + \mathbf{\omega}_{1/0}
This represents the sum of the angular velocity of frame 3 with respect to frame 2, the angular velocity of frame 2 with respect to frame 1 and the angular velocity of frame 1 with respect to the fix frame 0. In terms of Euler angle rates, this is given by:
\mathbf{\omega}_{3/0} = \dot{\phi}\mathbf{k}_1 + \dot{\theta}\mathbf{n}_{12} + \dot{\psi}\mathbf{k}_2
To express this in the body-fixed frame (frame 3), I need to write \mathbf{k}_1 and \mathbf{n}_{12} in terms of the body-fixed basis vectors \mathbf{i}_2, \mathbf{j}_2, \mathbf{k}_2. After some geometrical considerations, I arrive at:
\begin{aligned} \mathbf{\omega}_{3/0} &= (\sin(\psi)\dot{\theta} - \sin(\theta)\cos(\psi)\dot{\phi})\mathbf{i}_2 \\ &+ (\cos(\psi)\dot{\theta} + \sin(\theta)\sin(\psi)\dot{\phi})\mathbf{j}_2 \\ &+ (\dot{\psi} + \cos(\theta)\dot{\phi})\mathbf{k}_2 \end{aligned}
This shows how the angular velocity is expressed in the rotating frame.
Alternatively, I can express the same angular velocity in the space-fixed frame (frame 0). This time, I need to express \mathbf{n}_{12} and \mathbf{k}_2 in terms of the space-fixed basis vectors \mathbf{i}_1, \mathbf{j}_1, \mathbf{k}_1. Using the same expression for the total angular velocity in terms of Euler angle rates, and after geometrical considerations, I arrive at:
\begin{aligned} \mathbf{\omega}_{3/0} &= (-\dot{\theta}\sin(\phi) + \dot{\psi}\sin(\theta) \cos(\phi)) \mathbf i_1 \\ &+ (\dot{\theta}\cos(\phi) + \dot{\psi}\sin(\theta)\sin(\phi)) \mathbf j_1 \\ &+ (\dot{\phi} + \dot{\psi}\cos(\theta)) \mathbf k_1 \end{aligned}
This gives me the expression for angular velocity as observed from the non-rotating frame. These expressions provide insight on how rotations translate to overall angular motion.
For more insights into this topic, you can find the details here.