Explore palace was a mission in Jak II. After having entered Haven Palace Jak met Ashelin Praxis who threatened him, wondering on whose side he was. Jak in turn told her the truth about Baron Praxis and the Precursor Stone. Ashelin gave Jak a black security pass allowing access into the weapons factory.
You will have to escape through the innards of the Palace, fighting past many red and yellow guards. When you get past the poles and dropping platforms at the start there's three right away on your right. Next is an automated turret that looks similar to the sentry gun except it cannot be destroyed and repeatedly shoots from left to right. After this bit you'll be shot at by several guards up ahead, use the Peace Maker (or Blaster or Vulcan Fury if out of ammo) to get them without much trouble. The Peace Maker is excellent in this mission due to its homing shots and arc lightning that will also kill any nearby enemies.
Go to where the guards were and then shoot the cracked glass window nearby; you can use the JET-Board to grind on the lower rail and then jump off at the other side. Move on and jump into a room covered by another turret, get past it and then take a left before the elevator (it is locked). Use the jet board again to get over these rails and engage some more guards on your right (you might want to use the Peace Maker against the guys on the other side). After this shoot down the big ring in the middle, after which you can use it as an improvised grind rail to cross to the other side.
Another turret and some more yellow guards behind some glass windows past that one. Now comes the last stretch over some dropping platforms and some more guards on the other side. Take these out and you'll have a clear route to the oddly placed exit elevator.