A control moment gyroscope, CMG, or гиродин is a device used, typically in pairs, to control the attitudes of large spacecraft, as an alternative to reaction wheels. Someone has built a cute little cube-shaped robot called Cubli that can get up and walk by rolling from one corner to another using reaction wheels, so the robot has no externally protruding components or pivoting joints, other than the bearings for the reaction wheels themselves. It would be interesting to do something similar using CMGs.
The idea is that you’d have something like an opaque, matte tetrahedron with rounded corners, or perhaps some more irregular shape, of a few hundred mm in diameter, mostly made of some very light material; however, inside of it would be hidden two or more gimbaled gyroscopes, whose rims would contain most of the mass of the whole device, as well as several motors, a battery, and control electronics. If the walker decides to start walking, it spins up its CMGs and starts torquing them in order to stand up on, for example, one corner, and move around.
As a concrete example, perhaps the height of the tetrahedron would be 720 mm, but the last 100 mm of the point are rounded off. The inscribed sphere has diameter 360 mm, so perhaps the largest gyroscope has diameter 350 mm and can rotate to any angle within; the rim of its perfectly toroidal rotor is, say, 100 mm in minor diameter, with a circular cross section centered at 170 mm from the center, thus 340 mm major diameter. The torus has a volume of about 8.4 liters, so if it is mostly or entirely made of lead, 11.34 g/cc, then it will weigh about 95 kg, too heavy for most humans to lift. I think it should be possible to build the rest of the machine — frame, bearings, smaller gyros, gimbals, motors, cables, etc. — under 5 kg. So 95% of the machine’s mass will be in its primary gyro, which can safely be spun at some 30 m/s.
At this speed its kinetic energy would be some 43 kJ, enough to drain a 2400-milliamp-hour USB power pack just to spin it up, on the order of 50 g of Li-ion battery. (The 10050-mAh USB power pack next to me weighs 205 g.) So probably 1 kg or more would need to be battery.
So clearly this beast could have an angular momentum to be reckoned with, and with the appropriate gearing, motors, and secondary CMGs, would have no trouble at all slowly lifting itself off the floor to stand on one point, or walking across the floor on two of its points. It could perhaps walk up and down stairs, light up, vibrate, make sounds, and, by balancing on one point, serve as a cocktail-party coffee table, though keeping it from being a very noisy and vibration-heavy table would take substantial engineering of the bearings.
In addition to walking, it could tilt a bit to one side and rotate on its rounded point, which would cause it to roll across the floor rather than merely walking.
Equipped with a sense of touch to feel things placed on top of it, it could balance a ball on its center, constantly tilting slightly to nudge the ball back toward its center. It might even be able to simultaneously engage in such a motion while balancing an object on its top.
If you wanted it to carry things around, though, a more useful polyhedral shape would have an edge between two vertices, usable for walking, opposite a flat face, so that it could walk while objects remained on its upper surface mostly by friction, minimally tilting back and forth to shift its weight between these two feet. An equilateral triangular prism, for example, would work; so, too, would a square pyramid, though there is only one angle for such a pyramid at which one of its triangular faces will be horizontal when its center of gravity is over the opposite edge.
More irregular shapes would offer more versatility.
A prototype of 1% the mass could probably be constructed. Instead of weighing 100 kg, it would weigh about 1 kg. You’d scale it down by a linear factor of, say, 0.22. So the tetrahedron would be 158 mm tall, the incribed sphere 79 mm diameter, the rotor rim 22 mm thick, centered 37 mm from the center (74 mm major diameter). This rim has a volume of 89 ml, 1.01 kg. If we also scale down the rotor linear speed and leave its angular speed alone, it’s only going 6.6 m/s, which is still 1700 rpm. (I guess I should work out what the scaling laws for CMGs are; I think that small CMGs are worse than reaction wheels.) The kinetic energy has dropped even more: now it’s only 22 J. I feel like this would still probably work but you might need to spin up the motors.
The total mass left over, if it scaled the same way, would be about 50 g.
A collection of such contrivances possessed of concave surfaces, or even surfaces that could be horizontal near floor level, could climb atop one another; with adequate friction, they could then function as if parts of a single body, with rolling contact between them rather than flexible joints or bearings.