Scaling Robots for the Moon
The Moon is a terrible place for human habitability. 0.1654 gees gravity and 1738 km ( 0.2725 Earth) equatorial radius, zero radiation protection, no beneficiated resources, and practically no water (< 100ppm compared to 25% humidity in the Sahara) where it has been detected. Compared to the Moon, deserts, polar icecaps, and deep ocean trenches are human paradises. Even Low Earth Orbit is relatively benign - closer to resources, closer to home, protected by the Earth's magnetic field, and suitable for one gee rotating cylinder habitats.
But what about robots? Robots already have a much longer track record on the Moon than humans, are one-way expendible, and are vastly less expensive. Humanoid robots with interchangeable parts and very high bandwidth to Earth probably make more sense. And robots can be scaled. Scaling robots for human response times and capabilities is the subject of this web page.
Assumptions
Predictive-Adaptive Telepresence - The Moon is dead, making it much more predictable than the Earth. The Moon is 400,000 km away; with satellite and fiber relay, the speed-of-light round trip is under three seconds. A direct, synchronous remote controlled robot or waldo manipulator would be slow and impractical, but the human can be operating a local simulation, replicated 1.5 seconds later on the Moon, with anomalies transmitted back to Earth 1.5 seconds later to update (and if necessary, retrain) the simulation.
External computation - A robot does not need a large on-board computer "brain"; instead, a tight radio link to a fixed-location computer less than 5 kilometers away. The speed-of-light round trip would be 30 microseconds, vastly less than the neural path delay from finger-to-brain and back in a human. The fixed-location computer (and there will someday be millions) operates the terabit laser data link to MEO relays, and then to telepresence operators on Earth.
Scaling - On the Moon, nobody knows you are small. Miniature humanoid robots should be scaled to human reflex response times. What is the scaling factor, and how does resource consumption scale to that?
Scaling Factors
Sizes, speeds and accelerations should all be scaled to scaling factor s = g_m / g_e = 0.1654, so that perceived velocities and "the time it takes to fall" are the same as Earth:
t_m = t_e ~~~~~ time is the same
\rho_m = \rho_e ~~~~~ mass density is the same
l_m = s l_e ~~~~~ length scales by s
v_m = s v_e ~~~~~ velocity scales by s
a_m = s a_e ~~~~~ acceleration scales by s
Implying that:
Area_m = s^2 Area_e ~~~ Area scales by s^2
Vol_m = s^3 Vol_e ~~~~~ volume scales by s^3
m_m = s^3 m_e ~~~~~ mass scales by s^3
KE_m = s^5 KE_e ~~~~~ kinetic energy ( mv^2) scales by s^5 (!!!)
Interestingly, lunar mean radius is 1737 km compared to Earth's 6371 km. In scaled units, the lunar radius is a factor of 1731 / ( 0.1654 * 6371 ) = 1.65 times larger, and the surface area is 2.717 times larger. Since the Earth is only 30% land (and the rest 70% water), the scaled lunar "land" area is nine times larger than Earth, perceived from the vantage point of a scaled robot.
The scaled power level is VASTLY larger. Surface illumination on Earth is reduced by clouds; the Moon receives full illumination, perhaps 50% more. Since scaled lunar object kinetic energy is 1/s^5 more effective, the scaled energy density is 12,000 times higher perceived by one robot, or 330 times higher perceived by a scaled community of robots.
Of course, robots will fail and break. Unlike humans, identical robots can be disassembled and the remaining functional parts used to fix other robots. Since their "brains and personality" are the local computers and the humans back on Earth (perhaps elsewhere, in space colonies), they cannot "die" because nothing is lost - just updated, repaired, recycled.
Not all is grins and giggles - our plucky little robots will be battery powered, which means they must carry the equivalent of both fuel and oxidizer, and they must also carry the "waste equivalent" of human excreted CO2 and water. However, the batteries can be swapped at charging stations in seconds. The equivalent of a "crop season" will be battery recharging time, not the once-or-twice per year maximum crop frequency on Earth.
In reality, there won't be enough human operators on Earth for all these robots. Most will be autonomous, engaged in repetitive tasks that do not need human creativity.