Gimbal lock easier to control with quaternions?

Question

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come? They use less memory, and are commutable, and provide an smooth rotation along nonlinear rotation but gimbal lock wise i don't see the benefit. Could someone elaborate?

Answer 1

Using quaternions doesn't resolve the issue of gimbal lock, but make is more controllable... how come?

Where are you getting this information?--it is false. There is absolutely no gimbal lock with the quaternions and the proof of this fact is straightforward. Gimbal lock occurs because there is no local homeomorphism of three copies of the circle onto three dimensional real projective space, which is why gimbal lock occurs for the Euler angles or any 3-parameter set which is topologically the former, but which seeks to parameterize $SO_3$ which is topologically the latter. The quaternions are a natural parameterization of $SU_2$ which is obvious when you send the quaternion units $i,j,k$ to the traceless basis of $SU_2$ and $1$ to the identity matrix, thus $SO_3$ inherits a parameterization by the unit quaternions by applying the $SU_2\rightarrow SO_3$ universal covering map. Since the covering map induces a projection $S^3\rightarrow R\mathbb{P}^3$ it must be a local homeomorphism thus the quaternions, homeomorphic to $S^3$, must nondegenerately parameterize $SO_3$.

Alternatively you can write down the quaternion parameterization of $SO_3$, its inverse (you can find it in any attitude determination book) and see manually if there are any places where either map is undefined (I assure you there are not).