Gimbal Lock - Gimbal Lock in Applied Mathematics

Gimbal Lock in Applied Mathematics

The problem of the gimbal lock appears when one uses the Euler angles in an application of mathematics, for example in a computer program (3D modeling, embedded navigation systems, 3D video games, metaverses, ...).

In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T3 to the real projective space RP3) is not a covering map – it is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any rotation in three dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles). This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space; the only (non-trivial) covering map is from the 3-sphere, as in the use of quaternions.

To make a comparison, all the translations can be described using three numbers, and, as the succession of three consecutive linear movements along three perpendicular axes, and axes. That's the same for rotations, all the rotations can be described using three numbers, and, as the succession of three rotational movements around three axes that are perpendicular one to the next. This similarity between linear coordinates and angular coordinates makes Euler angles very intuitive, but unfortunately they suffer from the gimbal lock problem.