Creating a small scale bumper car ground power grid
What you're describing is already a available product for charging mobile phones:






Basically, the metalic strips are alternatively power or ground.

The arrangement of contacts on the mobile device is somewhat arbitrary, but the idea is to ensure that no matter what the physical orientation, there will always be at least one pin contacting one of the V+ strips, and one contacting a ground strip.

Then, you just feed all the contacts into a multi-phase rectifier (like the schematic in Nick Alexeev'a answer), and you get power out.