The Chaos Emerald of Invincibility transforms its holder into a buff version of themselves and is granted invulnerability and increased physical strength. However in order to activate the Emerald's power the holder must become King of the Hound Table.
Dr. Robotnik - After threatening Merlynx, Robotnik obtained the Emerald and then stole King Arfur's crown to be king. He was thwarted by Sonic and Tails. Later Robotnik returned to King Arfur's castle and stole back the Emerald. Since Robotnik was not king he was not invincible, but since he had the Emerald it with the other three Emeralds transformed him into Supreme High Robotnik.
Tails - Tails used the Emerald to beat up Scratch and Grounder. Strangely, Tails was able to use it without being King.
King Arfur - After Robotnik was defeated the first time, King Arfur was restored to the throne and kept the Chaos Emerald to make himself invincible. Robotnik later returns and King Arfur gives him the Emerald to ensure the safety of his daughter Princess Gwendolyn.
The Chaos Emerald of Invincibility appears identical to the green Chaos Emerald from the games.