| Element | Name | Meaning |
|---|---|---|
| $w$ | Scalar part | Related to the cosine of half the rotation angle |
| $x,y,z$ | Vector part | Defines the rotation axis and the sine of half the rotation angle |
q=cos(θ/2)+sin(θ/2)(uxi+uyj+uzk)
where:
- θ is the rotation angle
- u=[ux,uy,uz] is a unit vector along the axis of rotation
| Value | Interpretation |
|---|---|
| $w \approx 1$ | Rotation angle $\theta \approx 0$ |
| $[x,y,z] \approx 0$ | Rotation axis contribution is negligible |
| Quaternion $\approx [1,0,0,0]$ | Identity quaternion (no rotation) |
“z is close to 1” means “A 180° rotation around the z-axis”。
rotation
rotation matrix
intrinsic frame rotation
- Rotate the frame around its current Y
- Then rotate around the updated X
For passive rotations, intrinsic frame rotations compose in the same order: \(R_A = R_x(-\alpha)\,R_y(-\beta)\) \(\mathbf{p}_{new}^{(A)} = R_x(-\alpha)\,R_y(-\beta)\,\mathbf{p}_{old}\)
extrinsic frame rotation
- Rotate the frame around world Y
- Then rotate around world X
Extrinsic frame rotations apply in reverse order: \(R_B = R_y(-\beta)\,R_x(-\alpha)\) \(\mathbf{p}_{new}^{(B)} = R_y(-\beta)\,R_x(-\alpha)\,\mathbf{p}_{old}\)