Euler Angles

To understand this section a basic knowledge of linear algebra is required.

Definition

In this context we have the following definition:

Euler angles are a set of three angles that describe the orientation of a rigid body in a reference frame in the form of a sequence of three elemental rotations in form: \(Q = R_z(\alpha)R_y(\beta)R_x(\gamma)\)

the angles \(\alpha\), \(\beta\) and \(\gamma\) are called the Euler angles of rotation.

There are other Euler angle conventions (e.g. \(Q = R_x(\alpha)R_y(\beta)R_z(\gamma)\)), but this is the most common.

Example 1

In UAIBot, Euler angles can be calculated using the Utils.euler_angles function in Python.

import uaibot as ub
import numpy as np

Q = np.array([[0,0,1],[1,0,0],[0,1,0]])
alpha, beta, gamma = ub.Utils.euler_angles(Q)
print([alpha, beta, gamma])

Demonstration 1

The following interactive demonstration shows the rotation of a box by the Euler angles \(\alpha\), \(\beta\) and \(\gamma\). This demonstration is made using the UAIbotJS library. You can play around with the code by clicking the “Edit in JSFiddle” button.