Quaternion

Intro

Quaternions are a mathematical representation used to describe orientations and rotations in three-dimensional space. Unlike Euler angles (roll, pitch, yaw), quaternions avoid issues such as gimbal lock and provide smoother interpolation of rotations.

• See KiboRPC- Quaternion

A quaternion is composed of four components: $q=(w,x,y,z)$, where:

• $w$ is the scalar part.
• $x,y,z$ are the vector parts.

A unit vector $\lambda =({\lambda }_x,{\lambda }_y,{\lambda }_z)$ represents the rotation axis, and a quaternion $q$ represents the rotation by an angle $\theta$ radians as follows:

$$q=({\lambda }_x\sin (\theta /2),{\lambda }_y\sin (\theta /2),{\lambda }_z\sin (\theta /2),\cos (\theta /2))$$

Quaternion

Concept	Representation
Quaternion (4D)	Rotation axis (x, y, z) and scalar (w) q = (x, y, z, w)
Axis-Angle → Quaternion	Given unit axis λ = (λₓ, λᵧ, λ_z) and angle θ: q = (λₓ·sin(θ⁄2), λᵧ·sin(θ⁄2), λ_z·sin(θ⁄2), cos(θ⁄2))

For example, a quaternion representing a 90-degree rotation around the Z-axis would be $q=(\cos (45^{\circ }),0,0,\sin (45^{\circ }))$.

Euler Angles vs Quaternions

https://youtube.com/shorts/rh63Ykbw9Bw?feature=shared

Visualizing quaternions

Quaternion tool