I wonder how self-driving cars determine the path to follow. Yes, there's GPS, but GPS can have hiccups and a precision larger than expected. Suppose the car is supposed to turn right at an intersection on the inner lane, how is the exact path determined? How does it determine the trajectory of the inner lane?
As you say, GPS is not precise enough for the purpose (until recently it was only accurate within 5m or so, since 2018 there are receivers that have an accuracy of about 30cm). Instead, autonomous vehicles have a multitude of sensors, mostly cameras and radar, which record the surrounding area and monitor the road ahead. Due to them being flat, mostly one colour, and often with lines or other markers on them, roads are usually fairly easy to spot, which is why most success has been made driving on roads as opposed to off-road. Once you know exactly where you are and where you want to go, computing the correct trajectory is then just a matter of maths and physics.
For an academic paper on the subject of trajectory planning see Local Trajectory Planning and Tracking of Autonomous Vehicles, Using Clothoid Tentacles Method.
It quickly becomes more complex when other road users and obstacles are taken into account; here machine learning is used to identify stationary and movable objects at high speed from the sensor input. Reacting to the input is a further problem, and one reason why there aren't any self-driving cars on the roads today.
This is all on driving automation level 2 and above; on the lower levels things are somewhat easier. For example, the latest model Nissan LEAF has an automatic parking mode, where the car self-steers, guided by camera images and sonar, but still requires the driver to indicate the final position of the vehicle. Apart from that, it is fully automatic.