Avoiding gimbal lock I am not really sure if I understand the phenomenon of gimbal lock correctly. Say I have a vector $\begin{pmatrix} x\\\ y\\\ z \end{pmatrix}$. And I want to keep the vector's length fixed but move it in a given direction with respect to the $x, y$ or $z$ axis - i.e. rotate it in that direction. So, for instance, if I want to rotate it $30$ degrees about the $z$-axis, I would multiply by the matrix $$\begin{pmatrix} \cos(30°) & -\sin(30°) & 0\\\ \sin(30°) & \cos(30°) & 0\\\ 0 & 0 & 1\end{pmatrix}_.$$ And likewise for the other two axes. Will some sequence of these rotations eventually cause "gimbal lock?" Or will no problem arise using this method?

Gimbal lock occurs when one of the rotation matrices reduces to the identity. Then you effectively reduce one degree of freedom.

Let $R_x(\alpha)$ denote a rotation matrix around $x$ by $\alpha$.

Then, a general rotation can be written as $R = R_x(\alpha) R_y (\beta) R_z(\gamma)$. Suppose that $R_x(\alpha)$ becomes the identity map. Then $R = R_y(\beta) R_z(\gamma)$ in the new coordinate frame, and hence there is no longer any notion about "rotation around $x$."