SVD pseudo inverse
The pseudoinverse gives a “least squared error, minimum-norm” solution: Out of all \(\dot{q}\) vectors at your current \(q\), the vector
satisfies two conditions:
Least squared error: There is no \(\dot{q}\) vector which will get closer to \(\dot{p}\) when passed through \(J(q)\). (I.e. for \(\dot{p}_{s} = J(q)\dot{q}_{s}\), the normed difference \(\lvert \dot{p}_{\text{in}}-\dot{p}_{s} \rvert\) is smaller than any other \(\lvert \dot{p}_{\text{in}} - \dot{p} \rvert\).
Minimum-norm: Out of all \(\dot{q}\) that satisfy the first condition, \(\dot{q}_{s}\) is the smallest of those vectors. (I.e. \(\lvert \dot{q_{s}}\rvert\) is smaller than \(\lvert \dot{q} \rvert\) for all \(\dot{q}\) that minimize error in \(\dot{p}\).
At the singularity, the set of achievable velocities loses one dimension (so that there are \(\dot{p}\) that cannot be exactly produced by a \(\dot{q}\)), but for any \(\dot{p}_{\text{in}}\) there is a set of \(\dot{q}\) that all come equally close to producing \(\dot{p}_{\text{in}}\), and the pseudoinverse picks the smallest (in \(\dot{q}\)-space) of these vectors.