It depends on the target tradeoffs.

4 wheels can be inherently statically stable but requires more space and requires more energy when running (counter-intuitive, but if your balance algorithm is efficient, you use less energy wheeling around on 2 than on 4 because you can basically play games with gravity to get some acceleration tradeoffs... And you have half the number of powered wheels, assuming you're 4-wheel drive).

If they solve the hard math, the end result may very well be a more performant machine (though as someone else in thread has noted, depending on the overall goals, a zero-wheels statically-mounted arm could be better still).

BD's machines actually have terrible energetics. They use very heavy and inefficient hydraulic actuation. I can't find a source for it, but IIRC Atlas has a specific resistance of around ~40, compared to a human which is around 0.2. I don't know an RC Car is at, but direct drive motors are quite efficient.

Whatever BD's angle is, it's not efficiency.

BD is a technology company. I don't think they set out to just build robots that can move boxes. I bet this is a stepping stone towards building an advance machine that can do more than that.

I assume it's a combination:

* If the robots need to "reach" for items, you have to figure out balance/stability issues regardless of how many wheels you have.

* The fewer the wheels used, the smaller the footprint of the robot.

It needs to swing that counterweight around maybe?

The thing is: that counterweight looks very heavy, but the boxes being carried look fairly light (something that a human could lift). It doesn't really seem like there's any reason they needed to design this robot to have two wheels and a counterweight. As others have said, the counterweight makes the robot more unpredictable and dangerous. What happens when someone steps behind the robot, and the robot needs to swing the counterweight back to keep its balance? Does it hit the person with that massive weight or fall flat?

I suspect that this is an exercise in learning how build a two-wheel balancer that doesn't "hunt" around the vertical position. A normal two-wheel balancing robot is continuously controlling to keep the robot at an unstable equilibrium point (the "inverted pendulum" problem). The only actuators it has to do that are the wheels, so a normal two-wheel balancer is continuously rocking back and forth as it controls for vertical stability. The counter-weight gives a second actuator to control for equilibrium, so that it doesn't have to use the wheels for small corrections.

Also, recall from freshman physics that the pendulum problem is non-linear. Most balancing robots linearize the problem around the stable point. Which means if the robot gets far away from the equilibrium point, you either have to switch to a different controller or live with the consequences. One of the consequences is the robot racing off to the horizon as it tries to get the wheels under the CG again. (Which may happen anyway whether or not you switch controllers.)

Thus, my standard advice to all newbie robot builders: Your first robot should have a low enough mass that a missing semicolon will not punch a hole in your living room wall.

If this kind of thing is interesting to you, MIT 6.832 Underactuated Robotics by Russ Tedrake is posted on YouTube going a few years back, and the Spring 2019 version is being live-streamed every Tuesday and Thursday, picking up again next Tuesday after spring break.

I imagine the same thing that happens when a person steps behind a forklift, and the forklift operator backs up without looking.

Static doesn't necessarily mean safer. But the article also notes these aren't intended to share space with humans yet.

If I had to guess, the counterweight area also contains a big bulky lithium ion or lipo battery pack. Perhaps if they've determined they need it to carry around a 35 kilogram battery pack, they're accomplishing two purposes by also putting the battery in the counterweight location.