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