Microcontrollers. Separate power to board and servos. Why common ground?
 

I am working on an Arduino based robot. Being converted from a rather ancient OOPic controller.

You don't want to control the logic end of things using same battery as the servos, given that they draw rather huge amounts of juice. Forgive my ignorance, but why do you have to have a common ground for the logic vs. the locomotion sides of the beast? I've never quite understood this.

I'm thinking noise issues with two floating grounds ?

The battery's chemical ability to absorb might also fall into play...

I you have two grounds, one ground may be "better" than the other ground and the current runs across the circut boards to reach the other ground, burning out electrical thingies along the way.

The pure logic does not necessarily need a common ground with the locomotion components. But often the rather high current carrying driver circuits are part of the logic circuit board. Those current carrying driving circuits need a common ground with the other high current electro-mechanical locomotion parts. That could be a justification for common ground everywhere.

Hard to get a good impedance match in high speed switching circuits unless the grounds are tied together. Spurious noise can introduce signal faults and incorrect data.

So if I understand this correctly, if you have a separate grounds, and one of the circuits is operating at a much higher power level, you can have power 'leaking' across the board and causing problems?

No. You use a common ground so they are at the same zero. If the grounds are not connected, the input circuit for the servo has no reference for the signal from the arduino.

You can also run servo at higher voltage for more power and speed. Most are rated for 6volts, some more.

A (typical hobbyist/RC) servo contains two (electrical) elements within its enclosure. The motor, and the positioning circuitry. BOTH utilize the same power source and BOTH utilize the same ground. Hence the common 3-terminal interconnect from the servo is made up of: power, ground, and PWM control signal input (referenced to the mentioned ground).

In order for your MCU board to provide a control signal to the servo, which the servo can "interpret", 2 key criteria must be satisfied.
First the (voltage based) digital logic level of the control output from the MCU board must be compatible with the (voltage based) digital logic level of the control input into the servo. Second, the ground reference for both must be the same (presumption: no isolation circuitry on the servo outputs at the MCU board). The digital logic levels are always in reference to the associated ground. Hence the grounds being shared permit the logic level from the output side (the MCU board) to be correctly "interpreted" by the input side (the servo).