Robust robot docking with lidar only

edit

Generally, the solutions I saw in the wild on the Internet used additional infrared sensors coupled to a USB camera, with possible QR/AR tags on the charging station.

If additional sensors aren't possible, perhaps you could try adding an easily identifiable object next to the dock so that you could control the angle as you approach the station? Perhaps this is not even needed if you can calculate the difference between the robot's current position and the position needed at the end of the path - I can see use of Rotation Shim Controller here (if you use Nav2, of course), combined with special behavior when the robot is close to the target, e.g., the robot approaches the station when it is within 1 m of the station, plans a midpoint just before the station and an end point with a tight angle error + a possible angle correction with the other waypoint.

How do you detect this triangle-shaped dock now? Did you transform the LiDAR scan to point clouds and use some known packages to detect objects? You can add a description of your setup as well :)