Here is the full list of all 37 ASTRO BOT Rescue Mission trophies - there are 14 bronze, 19 silver, 3 gold and 1 platinum.
You collected all ASTRO BOT Rescue Mission trophies. You rock! Now, time to get out and see the real world!
Rescued all crew in levels and challenges
Cleared all Challenges fully (obtaining 2 Bots)
Got all the toys in the Astro Ship
Cleared 20 Challenges fully (obtaining 2 Bots)
Defeated 8 enemies with a single spin attack
Blasted the President's name in the credits
Defeated the Alien and completed the game
Cleared 10 Challenges fully (obtaining 2 Bots)
Defeated the Shark Boss
Found all Bots in world 5
Found all Bots in all worlds
Found all 4 chameleons in world 5
Finished off the Spider Boss while looking away
Defeated the Spider Boss
Found all 4 chameleons in world 4
Found all Bots in world 4
Got half the toys in the Astro Ship
Found a forgotten prize inside the crane machine
Cleared your first Challenge fully (obtaining 2 Bots)
Defeated an enemy by retracting the rope under it
Defeated the Bird Boss
Found all Bots in world 3
Found all 4 chameleons in world 3
Defeated the Octopus Boss
Get shocked while splashing water
Found all 4 chameleons in world 2
Found all Bots in world 2
Splashed yourself with the water gadget
Scored a goal on the beach
Get Astro stuck to your face underwater
Got your first toy in the Astro Ship
Defeated the Gorilla Boss
Hit Astro with the controller in the VR Menu
Found all 4 chameleons in world 1
Found all Bots in world 1
Completed the first stage