=robots =mechanical engineering =biomechanics
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 of
human legs is slightly above the knee, so let's consider a 0.4m pendulum,
swinging at ~0.79/s. That's still slightly
less than 1/2 the typical walking cadence, which makes sense because energy
is added, but ostriches have light legs and walk slower, so
human biodynamics must be leading to a higher natural walking speed than
ostriches have, what with the interaction of arm swinging and hip movement.
By the way, in case someone really wants to suggest that a kangaroo-like
robot would be "better", while kangaroos are fast and reasonably efficient,
contrary to some things I've read, their hopping isn't exactly more
efficient than eg a horse, it just has a different efficient speed relative
to leg length.
How about scaling laws for efficiency? That's...complicated, but generally,
bigger animals have slightly higher locomotive efficiency when walking. The
locomotive efficiency of a 12m bipedal robot running at 23 mph should be
worse than a car but better than a M1 Abrams tank. On roads and trails, bike
riding is more efficient than running, but wheels aren't better than legs on
soft and uneven ground - for animals, that is; current walking robots are
less efficient.
specific torque
Generating 10+ W/kg isn't a problem; some gas turbines and electric motors
do >10 kW/kg. That amount of power needs to be available in several places,
not just one place; the total instantaneous power that all the skeletal
muscles of a human can produce is much greater than 4 W/kg, perhaps
something like 200 W/kg. Multiplying that by a scaling factor, that would be
500 W/kg. If we have an electric motor + hydraulic pump + hydraulic cylinder
for all that, the average specific power for those elements needs to be 1.5
kW/kg. The robot needs other components too, so now maybe you need 3 kW/kg
from the drive system. Still, that's achievable, and that's an overestimate:
a robot would probably have more limited movement, and motors are (unlike
muscles) bidirectional, so the total/average power ratio would be
substantially lower.
The real
problem is the amount of torque required.
Relatively good planetary
gears and cycloidal drives have specific torque of ~200 Nm/kg. The
power/mass of a gear is then specific_torque * rotational_speed in
radians/second.
If we consider the planetary gear mass required to
support the full (12m tall) robot weight with the lever arm length of the
legs, that calculation is:
22 tons * 6m legs * earth gravity / (200
Nm/kg) = ~6.5 tons of gears
That's 30% of the robot's mass, just for
1-axis gears at the hips. Human hip joints have 2-axis movement, which would
be 2x that much gear. Clearly, that approach is problematic.
linear actuators
One
way to get higher specific torque is to use linear actuators. When you see a
big excavator, it has hydraulic cylinders moving the arms.
If you
imagine 2 excavators welded together, upside-down and walking on the
buckets, that probably doesn't seem as effective as the giant robots in
anime. This Gundam statue in
Yokohama uses
hydraulics, and as you can see, its movement is limited and it
moves...very...slowly. The cost was estimated at a few million $.
Excavators don't move smoothly, because the cylinders are connected to
reservoirs through valves, but it's possible to connect them directly to
pumps with electric motors, which can give smooth movement. Other options
for linear actuators include ballscrews and roller screws.
a cheap robot dog
Tesla is aiming for a low price of $20k for
its humanoid
bot, so of course it's using a
bunch of roller screws, the most expensive option for linear actuators. And
for gears, it's using harmonic drives, the most expensive option for that.
How would you go about making a cheap robot? Well, let's look at the
Unitree
Go1,
which is sold for $3700 shipped.
For a dog-sized robot, torque isn't
that big a problem. Here are
some parts of the Unitree
Go1. As you can see, it's just using a gear! A single-stage gear with a
high-torque but normal electric motor! A gear with big teeth, trading some
efficiency and precision for max force.
Industrial robotic arms don't
use normal gears, because they don't give enough positioning accuracy for
factories, but apparently gears are precise enough for a walking robot.
They're probably not good enough for aiming a gun, which
makes the US Marines using a cheap Chinese robot dog to
carry a rocket
launcher extra-bemusing.
Commercial robotic arms typically use cycloidal drives and/or harmonic
drives instead. If the precision of even higher-end planetary gears is good
enough, you can reduce costs quite a bit.
current feasibility
That Yokohama Gundam statue exists, and it can sort of move. What if we just
increase the power level, add a few more actuators, and increase the
structural strength a bit?
Maybe the structure would need to be able
to handle 2x the acceleration and be lighter, but if you consider the
relative strength of modern composites and bone, a 12m humanoid robot
skeleton shouldn't be a major problem.
How about power? Can the power
level be increased that much without making things too heavy? As I said
above, it can, but note that electric motors with a specific power of 10+
kW/kg are a recent development. Power electronics have also improved
substantially.
Back in 2008 some Japanese scientists
estimated a Gundam
would cost $725M to build. They figured: electric motors don't have good
enough specific power, so let's use superconducting motors. But then, the
specific power of available electric motors simply improved, and
superconductors weren't necessary or even helpful. They specified honeycomb
aluminum alloy, which is completely inappropriate. And they specified 7 gas
turbines, which is silly, because it's better to use fewer bigger turbines.
So, if you just pack a 12m robot full of the highest-performance
electric motors and hydraulic pumps you can get, that should be good enough.
But this brings me to why I'm writing this post now. Some people I know
designed a bunch of electromechanical actuators, meant for things like
industrial automation, aircraft, and mining. The extent they were able to
improve on such basic mechanical things was somewhat absurd, and then, they
thought: "these have good enough performance for something as silly as a
giant mecha, lol". Anyway, if I was allowed to use those designs, I would go
electromechanical. They make hydraulics mostly obsolete, and I don't say
that lightly. But I'm already ruining my credibility enough here, so I'll
leave things at that.
cost
How would we
estimate the cost of something like a giant walking robot?
Do we
compare to cars, aircraft, or what? Do we base estimates on mass, power,
force, certain components, or what? Do we make some adjustments? My answer
is: yes. To clarify, my approach here is estimating cost mainly based on the
output power of components that are well-understood from their use in cars
and aircraft, adjusted according to cost/performance tradeoffs for those
component types.
The answer is, of course: it depends. Cost depends
on the desired performance and payload. So, let's suppose the target is a
12m tall 22 ton bipedal robot capable of running at 10+ m/s while carrying a
3+ ton payload. How much would that cost? It depends. Production
cost of such a robot could be as low as $350/kg, but considering a low production
scale I think $600/kg is more reasonable. That's slightly less than the cost per empty mass of a 737 MAX. It's
also ~50% more than the (inflation adjusted) cost/mass of the Stryker - which is overpriced, but it's always
possible to make things expensive.
To clarify,
that number supposes that design is done, and that tooling, facilities, and trained workers already
exist. How much would those things cost? It depends, of course. What's the
location? How good is the management? Will funders insist on expensive
details? Development and tooling costs can get rather expensive. Consider
Formula 1 racing. A F1 car is perhaps $14M and weighs 800 kg. A F1 car team
costs $135M a year. And bigger things require bigger facilities.
I'm
guessing you'd need $30M to $150M worth of facilities and tooling. Some of
that could be rented: big warehouses and gantry cranes for assembly of
ordered components are pretty standard. That cost range depends largely on
the location - eg, USA > Japan > China.
If $100M was really all it
took, this would've already happened. After all, a F1 team or megayacht or a
basketball team costs more than that. But again, electric motors and power
electronics have improved recently. Also, this is all assuming a lead
designer with a decent understanding of mechanical engineering, electrical
engineering, biomechanics and kinematics, material science, and metal &
composite manufacturing techniques. That doesn't seem like a big problem to
me, but apparently such people are hard to find?
Control was another
big issue; adequate software for controlling walking and running of humanoid
robots and robot dogs is fairly recent, but now it's easy enough that lots
of groups have managed it. The same methods are applicable at a larger
scale, but humanoid robots fall over a lot during testing, and again, a 12m
robot falling over is unacceptable. You can do training in a simulation, of
course, but simulations are never quite perfect.