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)
Cleared 20 Challenges fully (obtaining 2 Bots)
Got all the toys in the Astro Ship
Defeated an enemy by retracting the rope under it
Blasted the President's name in the credits
Defeated the Alien and completed the game
Defeated the Shark Boss
Found all 4 chameleons in world 5
Found all Bots in world 5
Found all Bots in all worlds
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
Cleared 10 Challenges fully (obtaining 2 Bots)
Defeated 8 enemies with a single spin attack
Defeated the Bird Boss
Got half the toys in the Astro Ship
Found a forgotten prize inside the crane machine
Found all 4 chameleons in world 3
Found all Bots in world 3
Defeated the Octopus Boss
Found all 4 chameleons in world 2
Found all Bots in world 2
Get shocked while splashing water
Scored a goal on the beach
Get Astro stuck to your face underwater
Splashed yourself with the water gadget
Got your first toy in the Astro Ship
Defeated the Gorilla Boss
Found all Bots in world 1
Found all 4 chameleons in world 1
Hit Astro with the controller in the VR Menu
Cleared your first Challenge fully (obtaining 2 Bots)
Completed the first stage