I suspect that the best answer for automatic plug in is an additional socket, probably under the car, only for automatic plugging, and the other plugs for manual plug-in can remain on the side.

If the plug is on the side, then the car probably can't do all the work though there are some tricks that might be possible there with the car possibly doing a sharp turn to plug in. Or the car could have a socket which pops in and out so it faces backward. In that case it could be spring loaded, so the car can release it to make it pop out, but a human is needed to push it back in -- it would drive from the charging station to the human with the port popped out a few inches, less than the mirrors are today.

Alternately to use today's layouts the cord has to be on a basic robot arm. The arm would need actuators to position the height, and another to thrust the plug in and out. That's the complexity I am trying to avoid.

If you are designing from scratch, it is better to put your complexity in the charging station than the car, because there are far more cars than stations. However, since the car is a robot that can accurately position itself and that's a given, you want to make use of that where you can.

My current thinking is a box on the ground the car drives over. Two large pins or a plug come out (it could be from the box, or could be from the car) and the car moves to put them in the socket(s). The car does need a cover plate it can retract over its end for weather protection. The box if outside needs one too.

There are other options to explore, including the car just pressing against plates (after all plate on overhead wire works for pantographs at high power.) That's one reason to look under the car as you can put exposed live wires down there where nobody (but animals perhaps) can touch them, along with modern GFCI safety.

But the key is to find the cheapest solution that exploits the car's inherent hardware as much as possible. During the transitional phase, cars must be able to plug into both old style manual and automatic. Perhaps on the future they can just do the automatic but that's sadly fairly far away as long as people want to do things like plug in to RV parks etc.

But am critical when I think they make the wrong choices.