After his failure against the initial Mugann assault on Earth and his subsequent breakdown over condemning his best friend to death, Rossiu left for Adai Village, where he intended to end his life. With the help of Lordgenome, Simon and Kinon used perceptual teleportation - with Rossiu being the perceptual focus - to jump Gurren Lagann from their location over the wasteland where they were searching for him into the amphitheater deep underneath Adai. Gurren Lagann only partially exited the corridor up to its shoulders before Simon jumped out of Lagann's cockpit to sock Rossiu in the face.
Super Galaxy Dai-Gurren
When Team Dai-Gurren left Earth to search for Nia, Lordgenome once again used Simon's perception of Nia's engagement ring to jump Super Galaxy Dai-Gurren. Using this method, the ship was able to traverse the inter-galactic distance in the blink of an eye. The Anti-Spirals used a projection of Nia to cloud Simon's perception, which is how they were able to lure Super Galaxy Dai-Gurren into an ambush of Anti-Spiral vessels.
Being the size of a small planet, it is impossible to use conventional means to traverse Super Galaxy Dai-Gurren's immense bulk. Crew must use a form of perceptual teleportation to transport themselves to distant parts of the ship. Boota uses this to transport to the power room and supply spiral power to Gurren's cockpit - currently occupied by Viral - to help Simon transform the ship into Super Galaxy Gurren Lagann.
Tengen Toppa Gurren Lagann
While the team were trapped in their own false universes by the the Anti-Spiral, the spirit of Kamina came to each member of the team and showed them an opening in their perception of these fantasy universes: a projection of Gurren Lagann, piloted by Simon. Simon used perceptual teleportation to traverse the space between universes, and each member of the team used perceptual teleportation to jump aboard Gurren Lagann once he had. With the eighteen members of Team Dai-Gurren aboard, they all jumped back into real space, at the same time coalescing all spiral power in the universe to create Tengen Toppa Gurren Lagann, which exited from the space-time corridor with Gurren Lagann, Arc-Gurren Lagann, and Super Galaxy Gurren Lagann already aboard. During the final battle Perceptual Teleportation allowed the Earth to be seen from the Anti-Spiral dimension bigger than it was due to the massive energy interference given off during the fight. It also allowed the Granzeboma to briefly enter the Universe in an attempt to destroy the Earth.