Conversion between quaternions and Euler angles: Difference between revisions

Revision as of 05:10, 3 July 2005

Revision as of 22:34, 6 July 2005

Line 1:

Spatial rotations in three dimensions can be parametrized using both [[Euler angles]] and [[Quaternions and spatial rotation|unit quaternions]]. This article explains how to convert between the two representations. Actually this simple use of "Quaternions" was first presented by Euler some thirty years earlier than Hamilton to solve the problem of "Magic Squares." For this reason the dynamics community commonly refers to the quaternions as "Euler Parameters" in this application.

A unit quaternion can be described as:

:<math>\mathbf{q} = \begin{bmatrix} q_0 & q_1 & q_2 & q_3 \end{bmatrix}^T</math>

where

:<math>|\mathbf{q}|^2 = q_0^2 + q_1^2 + q_2^2 + q_3^2 = 1</math>

:<math>q_0 = cos(\alpha/2)</math>

:<math>q_1 = sin(\alpha/2)cos(\beta_x)</math>

:<math>q_2 = sin(\alpha/2)cos(\beta_y)</math>

:<math>q_3 = sin(\alpha/2)cos(\beta_z)</math>

where <math>\alpha</math> is a simple rotation angle (radians) and <math>\beta_x</math>, <math>\beta_y</math>, <math>\beta_z, </math> are the "direction cosines" of the axis of simple rotation (Euler's Theorem).

Similarly for Euler angles, we use:

Line 10 ⟶ 14:

* Pitch - <math>\theta</math>: rotation about the Y-axis

* Yaw - <math>\psi</math>: rotation about the Z-axis

where the X-axis points forward, Y-axis to the right and Z-axis downward and in the example to follow the rotation occurs in the order Yaw, Pitch, Roll (about body fixed axes).

== Rotation matrices ==