There are plenty of places showing how to fix wandering analog PSP joysticks. Most of these repairs involve buying a replacement joystick. I found using some isopropyl alcohol (rubbing alcohol 90%+) and a lot of squinting, you maybe able to repair your joystick without purchasing a replacment. (And who really wants to wait for the shipping anyway!)
You will need:
Isopropyl alcohol 90%+ (rubbing alcohol)
Petroleum jelly (optional)
Disassembled PSP (can be found elsewhere)
Nerves of steel.
Step 1: Analog Joystick Removal
Prestep 1: Remove the 5 screws (4 black screws - 1 silver screw on the side) that hold the PSP faceplate on. You can find videos of this on youtube, or other instructables.
Prestep 2: Flip the faceplate over and remove the two silver screws that retain the analog joystick. Remove the joystick from the faceplate. Be careful not to lose the cap.
Step 1: Flip the joystick upside down, and gently pry the silver metal plate tabs on the side. You will remove it to find the PCB as in the picture.
Step 2: PCB Removal
Step 2: Seperate the PCB from the joystick. It will lift off with out the retaining clip.
Step 3: Slide Removal
Step 3: Remove the slides from the joystick. Be careful not to bend or break the whiskers on the slides.
Step 4: Slide Removal and Cleaning.
Step 4: Pull the metal slide out of the joystick. At this point, you can clean all the parts with isopropyl alcohol using Q-tips. You can skip to step 9 for simplicity, or if you really want to clean it all, you can continue to step 5. You may want to continue dissemble if the shaft tends to not center properly.
Step 5: Spring Removal.
Step 5. Remove the spring from the cap and plate by gently prying the clam shelled pieces apart.
Step 6: Total Disassembly and Cleaning.
Step 6. Clean all the pieces with 90%+ isopropyl alcohol using Q-tips. Remove all grime and oil. You should have 9 pieces. From top left to bottom: the base plate, shaft metal slide, the spring, the LR slide, the metal retaining clip, the cap, the UD slide, the shaft, and the PCB (looks like Wheatstone bridge). We are ready for reassembly.
Step 7: Intall Spring and Shaft
Step 7: Take the base plate, the shaft and the spring in that order. Prepare to sandwich the cap to retain the spring. If you like a very smooth moving joystick, take a tooth pick and smear a very small amount of petroleum jelly on the shaft. (Less is more here!) You can skip using the lubricant.
Step 8: Seat the Spring and Cap.
Step 8: This is a bit tricky, align the 4 posts of the base plate with the cap. Sandwich the spring between them. You will need to gently rotate the shaft around to properly seat the spring. It works best if you center the spring, then start to press the pieces together. Rotate the shaft in a circle with your thumb, and it should seat. This may take patience. Don't force it or you will break the posts off your base plate.
Step 9: Insert the Slides
Step 9: Insert the metal slide onto the shaft and seat it.
Step 10: Align the Slide Whiskers
Step 10: Prepare the UD and LR slides to be installed in the joystick. Note the whiskers, they have to align perfectly with the traces on the PCB. You may need to gently arch them to make contact. I believe this is what causes the joysticks to eventually wander. This alignment is critical to the functionality. You also want to make sure they are clean.
Step 11: Install the Slides
Step 11: Install the slides in the assembly.
Step 12: Check Your Alignment
Step 12: Now install the circuit board on top. Note the traces on the PCB. The whiskers on the slides must hit in the center of traces. It may take several attempts to get them aligned up perfectly. If you reassemble the PSP and you do not have up-down or left-right movement, you probably do not have the whiskers aligned properly.
Step 13: Reassebly and Reinstallation
Step 13: Install the PCB, and lastly clip the retaining clip back on. (I didn't picture it here.) It snaps on the assembly lengthwise so that it doesn't cover the contacts.
Lastly, reinstall the joystick in the PSP. Before replacing the cover, clean the contacts on the PCB with alcohol. Also clean the silicon rubber contact it meets with on the PSP.
If everything went smoothly, you should have a functional analog joystick. Good luck.