Every platform disappears quickly so the key is to move swiftly through the platforms while avoiding incoming attacks.
- Activate the pedestal and walk into the first platform.
- Leave the platform and go back to the solid platform.
- Kill all the Mephistos while in there.
- Once, you have killed the Mephistos, wait for the 3rd set of platforms then quickly Streak towards it, Air Hike, then Calibur.
- Quickly Streak towards the direction of the Blue Orb Fragment then Jump and Calibur.
- Activate the Pedestal.
- Wait until the platforms extend out to where the Fragment is hiding.
- Parallel to the wall, Stinger then Sky Star and then Double Jump and you should land on where the Fragment is located.
Navigate through the platforms while blocking the Mephistos' attacks. Blocking them doesn't cause you to fall off the platforms.
Have a full Disaster Gauge before initiating the mission, then simply use Argument to float through the chasm towards the Blue Orb Fragment.