The Nonlinear Library LW - legged robot scaling laws by bhauth
Jan 22, 2024
13:47
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: legged robot scaling laws, published by bhauth on January 22, 2024 on LessWrong.
Fiction has lots of giant walking robots. Those designs are generally considered impractical or impossible, but they've been discussed for thousands of years, there must be something appealing about them. So, let's consider exactly what's impractical about large walking robots and what properties they'd have if they could be made.
practicality
Suppose you have a humanoid robot that operates in a factory. It never needs to leave the factory, so it can just sit in a wheelchair, which means it doesn't need legs, thus reducing costs. (Or you could give it tracks.) Better yet, it could just stay one place on an assembly line, so you don't even need the wheels. And then maybe it only needs one arm, so you could just take the arm.
Now you're down to 1/4 the limbs of the original robot, and the legs would've been heavier because they handle more weight. And then maybe the hand can be replaced with something much simpler, like a vacuum gripper or pincer. So the result of all the cost reduction is cheap, right? Not really; commercial robotic arms are fairly expensive. Industrial equipment does only what's necessary, and it's still expensive.
A lot of people designing stuff don't really understand costs. Large-scale production of goods has been heavily optimized, and the costs are very different from what they are for individuals. I've seen chemists who develop a lab-scale process using something expensive like palladium catalyst and expect it to be a good idea for industrial plants.
Making a giant humanoid robot wouldn't be practical, but that's part of the point. Going to the moon wasn't practical. Giant robots are difficult, so maybe they're good for developing technology and/or showing off how good the stuff you designed is.
scaling laws
Still, it is possible to make walking machines with hydraulics; they're just slow and inefficient. So, that only makes sense where movement speed and efficiency don't matter much, but it turns out that those are usually important.
me
The scaling laws for walking animals and robots are:
mass ~= height^3
sustained_power/mass ~= height^(1/2)
walk_speed ~= height^(1/2)
run_speed ~= height^(1/2)
walk_cadence ~= height^-(1/2)
run_cadence ~= height^-(1/2)
joint_torque/mass ~= height
structural_mass/mass ~= height/material_strength
As height increases, the potential energy of falls also increases. Current humanoid robots fall over a lot during testing, but a giant robot would probably be destroyed if it fell over, and could damage property or kill someone. So, safety and reliability becomes more of an issue.
Now, let's use those scaling laws to go from human numbers to a giant robot.
human baseline:
height = 1.8m
mass = 75 kg
sustained_power/mass = 4 W/kg
walk_speed = 1.45 m/s
run_speed = 4 m/s
walk_cadence = 1.7/s
run_cadence = 2.4/s
giant robot:
height = 12m
mass = 22 tons
sustained_power/mass = 10.33 W/kg
sustained_power = 230 kW
walk_speed = 3.74 m/s
run_speed = 10.3 m/s
walk_cadence = 0.66 Hz
run_cadence = 0.93 Hz
Some animals run faster than humans, of course. If we apply those scaling laws to ostriches, this 12m robot would have a run_speed more like 35 m/s. But humans do have some advantages over ostriches and other faster-running animals:
Humans can run long distances.
Humans can carry heavier backpacks than most animals. (But that's probably bad for you. Abolish textbooks etc etc.)
Lots of humans can reach 9 m/s while sprinting. The above numbers are for a long-distance run.
While ostriches run fast, their efficient walking speed is actually slightly slower than human walking.
Natural walking speed is related to pendulum frequency. Human leg bone length is ~50% of height. If we consider a 0.9m pendulum, its natural frequency is ~0.525/s. The center of gravity...
Fiction has lots of giant walking robots. Those designs are generally considered impractical or impossible, but they've been discussed for thousands of years, there must be something appealing about them. So, let's consider exactly what's impractical about large walking robots and what properties they'd have if they could be made.
practicality
Suppose you have a humanoid robot that operates in a factory. It never needs to leave the factory, so it can just sit in a wheelchair, which means it doesn't need legs, thus reducing costs. (Or you could give it tracks.) Better yet, it could just stay one place on an assembly line, so you don't even need the wheels. And then maybe it only needs one arm, so you could just take the arm.
Now you're down to 1/4 the limbs of the original robot, and the legs would've been heavier because they handle more weight. And then maybe the hand can be replaced with something much simpler, like a vacuum gripper or pincer. So the result of all the cost reduction is cheap, right? Not really; commercial robotic arms are fairly expensive. Industrial equipment does only what's necessary, and it's still expensive.
A lot of people designing stuff don't really understand costs. Large-scale production of goods has been heavily optimized, and the costs are very different from what they are for individuals. I've seen chemists who develop a lab-scale process using something expensive like palladium catalyst and expect it to be a good idea for industrial plants.
Making a giant humanoid robot wouldn't be practical, but that's part of the point. Going to the moon wasn't practical. Giant robots are difficult, so maybe they're good for developing technology and/or showing off how good the stuff you designed is.
scaling laws
Still, it is possible to make walking machines with hydraulics; they're just slow and inefficient. So, that only makes sense where movement speed and efficiency don't matter much, but it turns out that those are usually important.
me
The scaling laws for walking animals and robots are:
mass ~= height^3
sustained_power/mass ~= height^(1/2)
walk_speed ~= height^(1/2)
run_speed ~= height^(1/2)
walk_cadence ~= height^-(1/2)
run_cadence ~= height^-(1/2)
joint_torque/mass ~= height
structural_mass/mass ~= height/material_strength
As height increases, the potential energy of falls also increases. Current humanoid robots fall over a lot during testing, but a giant robot would probably be destroyed if it fell over, and could damage property or kill someone. So, safety and reliability becomes more of an issue.
Now, let's use those scaling laws to go from human numbers to a giant robot.
human baseline:
height = 1.8m
mass = 75 kg
sustained_power/mass = 4 W/kg
walk_speed = 1.45 m/s
run_speed = 4 m/s
walk_cadence = 1.7/s
run_cadence = 2.4/s
giant robot:
height = 12m
mass = 22 tons
sustained_power/mass = 10.33 W/kg
sustained_power = 230 kW
walk_speed = 3.74 m/s
run_speed = 10.3 m/s
walk_cadence = 0.66 Hz
run_cadence = 0.93 Hz
Some animals run faster than humans, of course. If we apply those scaling laws to ostriches, this 12m robot would have a run_speed more like 35 m/s. But humans do have some advantages over ostriches and other faster-running animals:
Humans can run long distances.
Humans can carry heavier backpacks than most animals. (But that's probably bad for you. Abolish textbooks etc etc.)
Lots of humans can reach 9 m/s while sprinting. The above numbers are for a long-distance run.
While ostriches run fast, their efficient walking speed is actually slightly slower than human walking.
Natural walking speed is related to pendulum frequency. Human leg bone length is ~50% of height. If we consider a 0.9m pendulum, its natural frequency is ~0.525/s. The center of gravity...