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