What is the PID output controlling, torque or velocity?
From the video it looks like you are controlling torque, in which case
you either need cascaded PID loops (inner velocity --> torque and
outer position --> velocity loops) or a single loop with a large amount of D term
I would first try adding D term to your existing setup. the more D term you add,
the more P term will be stable.
A torque loop --> position loop with no D term, that is, the restoring force
is only proportional to displacement is basically a pendulum...
