Planar rigid body kinematics: Euler angles
In this blog post, I explore the concept of Euler angles, a method I use to describe the orientation of a rigid body. I will show how Euler angles use three successive rotations to transform a space-fixed coordinate system to a body-fixed one.
When we analyze the motion of a rigid body, it is necessary to consider both its translation and its rotation. While the translation is defined by how a single point moves, rotation requires a convention due to its non-commutative nature. In short, the order in which I perform rotations affects the final orientation of the object.
Euler angles offer a way for me to represent any orientation using three angles. I achieve this through a sequence of three rotations:
- precession: we rotate this frame around the \mathbf k_1 axis by an angle \phi. This transforms the frame to a new orientation with axes \mathbf n_{11}, \mathbf n_{12}, \mathbf n_{13}. The angular velocity of this rotation is \frac{\mathrm d \phi}{\mathrm d t},
- nutation: we then rotate around the \mathbf n_{12} axis by an angle \theta, leading to another intermediate frame with axes \mathbf n_{21}, \mathbf n_{22}, \mathbf n_{23}. The angular velocity of this rotation is \frac{\mathrm d \theta}{\mathrm d t},
- spin: finally, we rotate around the \mathbf n_{23} axis (which aligns with the z axis of the second frame) by an angle \psi. This rotation brings me to the body-fixed coordinate system, with axes \mathbf i_2, \mathbf j_2, \mathbf k_2. The angular velocity is \frac{\mathrm d \psi}{\mathrm d t}.
We can use a gyroscope to help visualize these rotations. The precession corresponds to the rotation of the gyroscope’s axis around the vertical. Nutation is the tilting of the axis, and spin is the rotation of the gyroscope about its own axis.
For more insights into this topic, you can find the details here.