Lamp after re-rigging:
Animated lamp after re-rigging
[Note: The video is paused at times when the lamp is in the air.]
This part quickly became complicated. The bones, arms, shade and base moved apart in Edit, Pose and Object mode. So, ended up deleting the entire armature and re-creating it from scratch. Not sure where things went wrong. It seems that when ‘origin’ is moved it moves everything with it, because each meshed part has a different origin!
This is a video of just the solid object and not rendered view.