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