Should Null space be computed using SR-Inverse?

[garaemon]

The current implementation of ImpedanceController utilizes null space based on SR-Inverse.

I'm wondering whether null space should be spanned using an SR-inverse ( $J^{\sharp}$ ) or a simple pseudo-inverse ( $J^{+}$ ).

$$ dq = J^{\sharp}dx + (I-J^{\sharp}J)z $$

$$ Jdq = JJ^{\sharp}dx + J(I-J^{\sharp}J)z $$

The second term should be 0 according to the definition of null space.
However, If $J^{\sharp}$ is an SR-inverse, $J(I-J^{\sharp}J)z$ cannot be 0 because of the damping factor.
Instead of $J^{\sharp}$, if we modify the second term with $J^{+}$ like the following, the second term can be 0.

$$ dq = J^{\sharp}dx + (I-J^{+}J)z $$

$$ Jdq = JJ^{\sharp}dx + J(I-J^{+}J)z $$

I think null space should be computed using a simple pseudo-inverse. Is my understanding correct?
If we can use simple a pseudo inverse matrix for null space, the performance of convergence would be better.

CC: @k-okada