Electric motor scaling laws and inertia in robot actuators

3/16/2026 at 5:35:50 AM

Aaed Musa blew my mind about 18 months ago with his capstan drive video:

Eight months ago he built a quadrupedal robot that could step sideways using three of them per leg. I’m not going to link that, you’ll have to find it from his YouTube page because you should look around.

by hinkley

3/16/2026 at 7:23:15 AM

Sorry for going off topic. "Electric Motor Scaling Laws and Inertia in Robot Actuators" by Ben Katz who designed the MIT Mini Cheetah in 2018 is very well known in the legged robotics community. His master’s thesis on actuator design is also widely referenced.

During the COVID period, some Chinese companies even sold variants of actuators inspired by the Mini Cheetah design.

Aaed Musa has also mentioned in some of his videos that his actuator designs were inspired by the Mini Cheetah actuator. Yes, His capstan drive video is especially impressive.

For example, in Aaed Musa’s video "I Built a Rubik's Cube Solving Robot" (https://www.youtube.com/watch?v=m0bMMALYMYk), he states in the description that the design was inspired by Ben Katz’s work.

Ben Katz master thesis, is worth reading: "A low cost modular actuator for dynamic robots" 2018 https://dspace.mit.edu/handle/1721.1/118671 and also has a good post https://robot-daycare.com/posts/2019-12-16-the-mini-cheetah-...

And also, The Rubik's Contraption (2018), 297 points, the work done by same author https://news.ycombinator.com/item?id=16561049

by num42

3/16/2026 at 11:01:16 AM

I did a bit of a deep dive into motor control and actuators in building a two axis gimbal for tracking satellites with an RF antenna (with the eventual goal of building a mount for optical tracking).

Ben's vids were kind of mind-blowing for me at that time. I couldn't believe some of the control that was possible with relatively pedestrian electronics. Aaed's vids do a wonderful job of making it accessible in an applied way.

It's something I think a lot of the folks on HN would find interesting to tinker with. Nice mix of software and hardware that actually does work in physical reality. It also gives me a level of appreciation for the advances in humanoid robots that I don't think I would have had otherwise. (If you *do* get into it, I'd highly recommend getting into field oriented control with brushless motors and encoders. Small hobby servos are fun but they encapsulate a lot of the interesting parts and tend to have limited options available for things like the capstan vid linked above)

by jcims

3/16/2026 at 6:45:14 AM

Thanks for this. I almost never have the patience for any videos, but this one hooked me and kept me engaged throughout. Worth it for that random "yo mama" joke alone.

by keeda

3/16/2026 at 7:01:26 AM

This channel about retrofitting industrial robotic arm control systems is quite practical. Could always run the playback at 1.25 speed for slow talkers/edits =3

https://www.youtube.com/@ExcessiveOverkill/videos

by Joel_Mckay

3/16/2026 at 2:02:55 PM

I hate the phrase "zero backlash" that gets used right in the intro. Nothing has zero backlash. If it did, that would mean the material is incompressible & unstretchable: it would have an infinite speed of sound. You could implement FTL communication using zero-backlash actuators. Capstan drives have backlash because the rope stretches. It's very low if using a low-stretch rope, but nonzero.

by SAI_Peregrinus

3/16/2026 at 2:24:30 PM

What you are referring to is stiffness, which is different than backlash. Backlash, or play, is due to gaps.

by fusionadvocate

3/16/2026 at 7:15:31 PM

Backlash is specifically the slack between directional change. The forces you describe are proportional to force. Backlash is independent of force

by vablings

3/16/2026 at 6:00:22 AM

Same issue covered on HN a few weeks ago.[1] This one has more motor theory but less machine learning theory.

Too much gear reduction, and you can't back-drive or sense forces from the motor end. Too little gear reduction, and your motors are too bulky or too weak. Reflected inertia goes up as the square of the gear ratio, as the article points out, because the gear ratio gets you both coming and going. So high gear ratios really hurt.

Robots, like drones, need custom motors sized for the specific requirements of the joint. For a long time, the robotics industry was too tiny to get such custom motors engineered, and had to use motors designed for other purposes. This will become a non-problem as volume increases. Especially since 3-phase servomotor controllers, which drones need, are now small and cheap. They used to be the size of a paperback book or larger.

(I've been out of this for years. I've used hydraulic robots and R/C servo powered robots. The newer machinery sucks a lot less.)

[1] https://news.ycombinator.com/item?id=47184744

by Animats

3/16/2026 at 6:20:10 AM

Reflected inertia does scale as the square of the gear ratio but it's a bit misleading unless you also consider the change in rotor inertia, which scales as a cube of the rotor radius (as the article points out).

The other side of the scaling laws say that motor torque scales as a square of air gap radius (roughly rotor radius), and output torque scales as linearly with gearing ratio.

When you balance these out, the reflected inertia depends on the inverse of power dissipated for a fixed output torque.

In an ideal world, your total reflected inertia is independent of the gearbox and largely depends on the motor fill factor and how hot you can run it.

by blueblisters

3/16/2026 at 4:14:48 PM

I think electric motors are just barely suitable for mobile / humanoid robots. Compared to human muscles, they're heavy and they overheat quickly. Current humanoid robots are spending most of their energy in carrying around just themselves, which is a huge amount of dead weight, while being unsafe for anyone to be around.

If someone invents a new type of 'artificial muscle' which has low inertia, high force/torque density, and can work without overheating, that would instantly kill all other robotics companies.

by Geee

3/16/2026 at 3:29:50 PM

I'm used to approaching these problems from a slightly different angle. There are two simple cases that can be used to establish guide points.

If you have a load that is purely inertial, the optimum gear ratio (to minimize I-squared-R motor loss) is found by picking a gear ratio which matches the reflected inertia of load and motor. At this point, on every acceleration you put just as much energy into the rotor inertia as into the load inertia.

In contrast, for a steady-state load which is all friction (e.g. a mixer such as for paint or food), a gear ratio which balances friction loss in the motor with the load friction will minimize the armature loss.

Most applications have live between these points, and these optimizations ignore gearing losses and expense and noise, but they can serve as guide posts.

There's also the issue of separating winding choice from gearing choice. For each candidate motor there exists an optimum gear ratio which will minimize the heat produced when driving a given load (friction and inertia) over a given velocity profile. That gearing can be found by trial and error in a simulation. These aren't crazy difficult simulations (can be done in a spreadsheet) but do need to take temperature dissipation and change of motor performance with temperature into account. Once that gearing is found, the V-I requirements of the motor at that gearing will be known and then winding adjusted to fit requirements of driver circuitry (i.e. trade current for voltage).

by klaff

3/16/2026 at 2:57:49 PM

What is the "ELI5" summary of the practical limits & scaling laws that govern robotics?

The current "futurist" vision is one of humanoid robots taking over many/most jobs done by humans today, but - as someone that routinely hires human welders & assemblers - the dexterity required for most ad-hoc tasks seems many many decades (if not more?) away from what I see robots do--yes, even the fancy chinese jumping ones.

This has led me to think one of two things:

1. The robotics revolution will not come. It's predicated on the idea that advances in robotics will follow a curve of the same shape as advances in compute/ai, which will not happen. OR...

2. There has been some paradigm-shift or some breakthrough that has put robotics improvement on a new curve.

To an outsider, what I see in robots is not categorically different than like, the sony AIBO dog in 1999. It's significantly better of course, but is it really that different? (Whereas what we can do in compute-land today is categorically diffrent because of the transformer model breakthrough).

So:

1. Have there been any breakthroughs that would lead us to believe that a robot will be able to like, look under a table to adjust a screw?

2. What are the scaling laws & practical limits to present-day robotic dexterity? Is it materials? Energy density? What?

3. What is the real rate of improvement along these key dimensions? Are robots improving linearly? Geometrically? Exponentially?

4.Or should I keep discounting robotics until we get our first robots that are made of meat? That I'd believe would result in exponential change!

by elbasti

3/16/2026 at 6:20:54 PM

From your early point -- both 1) and 2) are true. True human level dexterity is ver far (few decades surely), it would require further advancements in hardware, learning approaches etc. Recent approaches provide a glimmer of hope and maybe we can have some intermediate robots -- to be honest even waymo's and tesla's are robots and we will see much more of such robots with vision, working with humans etc. in narrow settings - chinese dancing robots are examples of this.

by robots0only

3/16/2026 at 3:15:11 PM

I'm fairly ignorant on this but robots that are teleoperated seem completely capable of doing basically any household task and using tools like screw drivers. It may be slow, sure, but autonomy and speed seems like a solvable software problem.

There's also endless welding and assembly robots and have been for a long time. Sure they're huge and weight 3 tons or whatever, but it's not like we're building humanoid robots to do work like that anyway.

by brynnbee

3/16/2026 at 4:04:52 PM

Consider 2 welding systems, a hungover human on a 3 legged ladder with a scratched up welding helmet doing an overhead TIG weld holding the filler rod a foot away from the weld pool, and a 6 DOF Kuka bot doing a weld in the same position on a completely rigid work piece clamped down to a precision machined fixture table which is clamped down to a precision machined floor that the robot is also mounted to.

The human system weighs 250lbs and can be placed anywhere. Let's ask what it takes to walk the factory robot in that direction. First let's have the work piece be moving, let's say on a conveyor belt. The old robotics way of thinking would be to introduce this variable into the programming of the bot/station, create simple sensors for either the work piece or conveyor itself to indicate to the programming loop where the part is with as little error as possible, and continue to keep accuracy while maintaining as much precision as possible using rigidity (which equals mass and space). Now the whole system is functionally 7 DOF, and you add in the error and failure modes of the 7th DOF (the conveyor system) and accumulate some error. Now just imagine instead of a conveyor the part is on a rolling table with random Z height, and so it the robot arm, and you can see this will fall apart, you can't fight this battle with deterministic programming, machining precision, and rigidity. Obviously if you extended this system to be a humanoid robot on a 3 legged ladder which would be 30+ DOF between the weld and the ground, it couldn't possibly work.

But back to the hungover human, why does this system work so well? The human has very good eyes and a very good internal IMU. They are looking at the end of the filler rod and the weld pool, and even though the information isn't that good coming through the scratched welding helmet, they can compensate for all that error and run an internal function that holds the torch and filler rod in the optimum position to do a good TIG weld while ignoring or automatically adjusting for tons of other variables. Now to address your original question, in our system 1. Are current cameras good enough to get an equivalent amount of information about the weld that the hungover welder has? Yes, in fact can get more information than a human can 2. Are IMUs as good as a hungover human has? Hard to really know, but seems like it, though if you need many IMUs attached to different limbs on a robot its probably not as good as humans yet 3. Is the power density of actuators and power storage good enough to approximate this 250lb system of a human on a ladder with some combination of DOF that reaches a sufficient range of motion to emulate the humans hands (whether the robot looks like a human or not?) - yeah, plus in this case the welder is plugged into the ground for the human anyway so that system is already attached to mains power

So given all this, seems like the limiter is just software, which is the bull case for this prospected robotics revolution

by beau_g