How to connect the GND with the metal case of the robot
Having charge build-up on the case isn't itself a problem, even if the board is grounded to the case. They're at the same potential.

The problem arises when the robot comes into contact with something at a different charge potential: the charge path may make its way though the sensitive stuff. So you need some countermeasures to route the charge away from where it can do harm.

Floating the electronics isn't such a great idea, because you're adding impedance to the discharge path. This will make your system more vulnerable to ESD, not less.

Instead, you should have two ground systems: shield (enclosure), and system (logic board). The two grounds should star-tie at the I/O connectors with a reliable, low-impedance connection. In this way no potential difference develops between the two, and there is a low-impedance path for ESD protection.

Speaking of which, the connector signals should have ESD protection that makes its own short tie to the shield and system GND star-tie. This isolates the discharge path away from the board and back to the enclosure.

You can also consider having the robot drag a ground strap to prevent build-up.