Escape the palace is a mission in Daxter. After discovering the tanker was meant for the Baron's palace at the end of the "Investigate tanker" mission, Daxter decided to investigate, likely out of suspicion and an inclination to find Jak. Soon after gaining entrance he triggered part of the security system and was forced to escape, though not before stealing a map of the prison and clothes from a locker room.
You start in the large entrance hall with computer terminals, the same one from Jak II. A handful of metal klaws can be seen here and in fact they make up the majority of bugs encountered in the palace. Take the elevator to a different floor, only to trigger the security system and officially begin the mission. In this first hallway the pots will turn into automated turrets, firing at you if you get too close. The flower beds are changed with deadly spikes, and machines spring forth from behind the paintings to reveal hall-crossing lasers.
Dodge these as you head through the fairly linear palace. Eventually you enter a library and must dodge a very large number of lasers as you progress through this area. Be prepared to rely on the pressure boost attachment, both to fly over lasers and spiked floors, but also to quickly close in on electricity-inducing metal klaws.
Eventually you will run into the treasury. Use large crates moving above an impenetrable blue barrier to enter a second vault area. Here you must use several more stacked crates to climb to the highest level in the room and eventually trigger a crane to malfunction and drop a crate down a large vent. Use Precursor orbs to guide your way to the highest crates if needed.
Drop down through the vent, witness a cutscene, and you will be nearly done. All that stands between you and the exit are several large boxes and metal backs.