Break the tethers to the zeppelin is a set of missions in The Precursor Legacy that Jak and Daxter must complete to receive four power cells (for each tether) in Boggy Swamp. Lurkers were trying to use a dirigible to remove a Precursor robot's arm from the swamp. Jak destroyed the tethers keeping the aircraft in place after a request from Samos Hagai.
The PlayStation 3 and Vita versions of this mission are related to a trophy named "The Lead Zeppelin". Completing this mission will reward you with a silver trophy.
Due to the fact that this is not one mission, but four with the same name and objective, there are four different walkthrough sections; each for one of the tethers. Due to the linearity of the level you will more or less be going to encounter each of them in this order.
After the first swamp rat nest, where you make a U-turn and fight a kermit you will encounter a cave entrance to the left. Ignore this one for now and continue on ahead to a more open space behind some spikes. Here is another two swamp rat nests and some yellow eco (as well as a blocked vent), behind the trees on a raised platform above the mud is the first tether.
You can use the lone cloud of yellow eco to unblock the vent, which helps in taking down the rats and their nests. From there you can take down the first tether by aiming while channeling yellow eco. After that you can jump on top of either of the two sinking pillars (the other will rise instead) and jump up to the platform to get the power cell.
Moving on, you will go through a short section where the camera moves to the side, like a 2D sidescroller. At the end of this section you use one of the swinging poles to jump up to a high platform. In front of you is a lower platform with the tether as well as a kermit hopping around. Break the treasure chests on the first platform to get some yellow eco to destroy the tether.
Straight after the second one you go through a small cave section where a wall with a row of dark eco boxes blocks your path (there is also a blocked launcher to the right). Use yellow eco to destroy the boxes and jump over the wall (grab the blue eco), on the other side is another launcher. Use this one to reach a lone section with the third tether, as well as a yellow eco vent and a swamp rat nest.
The last tether is near the end of the level, past Boggy Billy and the flut flut. You will eventually reach a lone yellow eco vent in front of another mud lake and on the other side a cave entrance with more swamp rats. Ignore those for now and instead use the yellow eco to aim up and to the left and destroy the last tether, causing the dirigible to float away.
Then, just jump onto the mushroom shaped pillars (where one sinks and the other rises up) and make your way to the high platform where the tether was located.