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About repeatability.

The basic pseudoinvertion metods of invers kinematics suffer the problem of non-repeatability of the trajectory or so-called non-cyclic behaviour. The conditions of repeatability were putted in mathematical form by Shamir & Yomdin (Shamir, Y. & Yomdin, Y. (1988) Repeatability of redundant manipulators: Mathematical solution of the problem. IEEE Transactions on Automatic Control. 33(11), 1004-1009).
We have dq=K*dx, where K is our inverted J. Collumns of K span a linear N-dimentional subspace, Range(K), above each non-singular point in joint space. It would be tangent to our integrable surface, where our joints' trajectories should be embedded not crossing each other. It is possible if there is such an integrable surface. Let's derive conditions for existence of integrable surface.
Suppose that it exists. Let's take a shift along any direction to the nearest point, being on integrable surface. We must be sure for our path to be repeatable backward, that each of this two tangent spaces at two neighbouring points have in common this shifting vector. So the tangent space at each point of our cyclic trajectory should contain two neighbouring points of this trajectory, for this trajectory to be repeatable in each direction. And now let's consize our cyclic trajectory to very small square of 4 points in joint space, that includes our "base point".
Let's take two orthogonal shifts, it will be vectors k indexed i and k indexed j, that start at base point. So we have a condition for tangent space of each of this 4 points contains it's two neighbours:

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It is generalized for all pairs of collumns of K due to linearity. This Lie bracket:

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of each pairs of collumns of K should be a linear combination of the other collumns of K.
But that isn't all of it. We must be sure that our cyclic trajectory lays in simply connected region of the workspace (that could be transformed to ever smaller region without crossing the boundary of the workspace), so it cannot contain singularity points inside. Breaking this limit, we aquire such a strange cases of repeatability but with a multiplicated periodicity.