Here is the full list of all 15 Astrodogs achievements.
Pass the tutorial
Get "S" Rank in HUNGER CT
Get "S" Rank in VERY SATELLITE
Get "S" Rank in WOOFAVELA
Get "S" Rank in THE CLOUD
Get "S" Rank in BARKADIA
Get "S" Rank in METROPOLI-A
Complete Toy Collection
Get "S" Rank in BARKSTILLE
Get "S" Rank in METROPOLI-B
Get "S" Rank in DARK SOURCE
Get "S" Rank in PRIME JUNGLE
Complete Garage
Complete mission 1 - 8
Complete Final Mission