First of all, you have to have Ky and Sol's Ex versions unlocked. Play Robo Ky II's Boost Mode on Normal difficulty, and beat Stage 5-E (Last Dance) with 1,500,000 points or more, and a giant Robo Ky II will appear. (Robo Ky II: Substance.) Defeat him to unlock Kakusei Ky and Kakusei Sol. Select them by highlighting Ky or Sol on the character select screen, and press Start twice.
Defeat Leopaldon at level 200 in Arcade mode.
Beat boost mode in AK II factory
You have to clear Boost Mode Once.
Accumulate 100 points in Arcade mode; now EX mode characters will challenge you randomly (their names will be in red). Defeat them to unlock them.
Defeat Leopaldan in arcade mode, at level 100.