Here is the full list of all 40 Project AETHER: First Contact achievements.
Clear Mission 1
Clear Mission 2
Clear Mission 3
Clear Mission 4
Clear Mission 5
Clear Mission 6
Clear Mission 7
Clear Mission 8
Finish the campaign
Clear all missions on Hard
Clear all missions on Intense
Complete a Trial
Complete 10 Trials
Complete all Trials
Clear Boss Rush on any difficulty
Clear 20 waves in any Survival mode
Achieve a 3.0x combo
Get a kill with 8 different weapons
Get 100 kills with the Assault Rifle
Get 100 kills with the Shotgun
Get 100 kills with the Railgun
Get 100 kills with the Energy Blade
Get 100 kills with the Particle Beam
Get 100 kills with the Concussive Shot
Get 100 kills with the Scatter Rockets
Get 10 kills with any EMP
Get 200 kills with any EMP
Hit 5 enemies or missiles with a single EMP detonation
Kill a Piercer with the Energy Blade while it is dashing
Using the Ram Dash, hit an enemy after charging through 8 bullets
Hit 4 enemies with a single Railgun shot
Kill 3 enemies with a single Scatter Rocket salvo
Defeat any boss after destroying all of its components
While powered up, kill 4 enemies before returning to normal
Using the Concussive Shot, hit a Ninja during its dash
Destroy 6 missiles or bombs using one continuous shot of the Particle Beam
Clear any Mission (except Mission 1) on Normal or higher without healing or using a continue
Evade 100 enemy shots by using the dash or Phase EMP
Get an S Rank on every Mission
Complete all other achievements