# legged robot scaling laws

Fiction [has lots of giant walking robots](https://tvtropes.org/pmwiki/pmwiki.php/Main/HumongousMecha). Those designs are generally considered impractical or impossible, but they've been discussed for [thousands of years](https://en.wikipedia.org/wiki/Talos), 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](https://www.youtube.com/watch?v=QlizSta3Rw0).) 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.

![](https://39669.cdn.cke-cs.com/rQvD3VnunXZu34m86e5f/images/d9674f0f7874c3570f44223652cd7dace89172d2ddf534d0.jpg)

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](https://www.bhauth.com/blog/flawed%20ideas/mecha.html)

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](https://en.wikipedia.org/wiki/Persistence_hunting).
*   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 of body weight and added energy, 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](https://www.youtube.com/watch?v=qMEMg0QobQ4&t=297s) 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](https://www.youtube.com/watch?v=cpraXaw7dyc), 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](https://shop.unitree.com/products/unitreeyushutechnologydog-artificial-intelligence-companion-bionic-companion-intelligent-robot-go1-quadruped-robot-dog), which is sold for $3700 shipped.

For a dog-sized robot, torque isn't that big a problem. [Here are](https://www.youtube.com/watch?v=KVS0Xwg15Tc) some [parts](https://www.youtube.com/watch?v=4arNA1iPeTM) 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](https://www.youtube.com/watch?v=TYs2URklSGY) 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](http://pinktentacle.com/2008/01/725000000-gundam/) 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.