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Most humanoid bodies are hybrid: rotary joints for arms and waists, linear actuators for the heavy joints that push against the ground. The part sitting inside nearly every one of those linear joints is a planetary roller screw. This piece explains how it works, why the inverted version won, what its lead does to force and speed, how the L10 cube law turns catalogue ratings into kilometres of travel, and why the internal thread of the nut is the reason this part used to cost more than the motor around it.

Open almost any humanoid teardown from the last two years and the legs will not look like the arms. The arms carry compact rotary modules — motor, reducer, encoder on the joint axis. The hips, knees and ankles carry cylinders: a motor spinning a threaded shaft that pushes a rod, swinging the limb through a lever the way a hydraulic cylinder swings an excavator arm. At its 2022 AI Day, Tesla showed an Optimus body with 28 structural actuators in six designs — three rotary classes rated 20, 110 and 180 N·m, and three linear classes rated 500, 3,900 and 8,000 N. Supplier reporting since then, including a 36Kr teardown analysis, puts roller screws in all 14 linear joints of that body — inverted types throughout, with the heaviest eight in the hips, knees and ankles and the remainder in the elbows and wrists.

The screw inside those cylinders is not a lead screw and not a ball screw. It is a planetary roller screw, and it is quietly the reason a 2-kilogram motor can deadlift a machine. Here is the engineering picture a joint buyer actually needs.

Two Ways to Bend a Knee

A rotary actuator puts the torque exactly where the knee pivot is: wide range of motion, the same torque curve at every angle, easy to mount. A linear actuator sits beside the limb and changes length. The rod's push converts into joint torque through a lever arm, and that lever arm changes with the joint angle — which sounds like a drawback until you notice it is also what human legs do: near full extension the effective lever is short and the joint needs speed, in deep squat the lever is long and the joint needs force.

The screw does the reduction. One turn of the motor advances the nut by one lead — a few millimetres — so a small, fast motor produces thousands of newtons of push without any gear stage in front of it. That is why locomotion-first robots keep choosing it for the joints that hit the ground: the same property that multiplies force makes the joint naturally tolerant of impact, and the push rod loads the screw in pure compression along its axis rather than bending a gearbox output flange.

The trade-offs are speed and packaging. A screw-driven knee needs stroke space along the limb, and its force envelope is angle-dependent. That is why almost nobody goes all-linear: the industry settled on mixed bodies, as we covered in our comparison of direct-drive and reducer routes for rotary joints. For builders who standardise on rotary modules for the upper body, integrated joint actuators remain the reference architecture — the linear screw complements them rather than replacing them.

Five Parts, One Motion

A planetary roller screw has five part families: a screw, a nut, a ring of threaded rollers between them, two ring gears fixed inside the nut, and two carriers that space the rollers.

The screw and nut carry multi-start triangular threads. The rollers carry a single-start thread with a rounded profile, so each roller touches both the screw and the nut along many small contact zones instead of resting on a few balls. Spur teeth at each end of every roller mesh with the ring gears, keeping the rollers parallel to the screw axis; the carriers hold them at even spacing. Turn the screw and the rollers orbit it like planets, spinning on their own axes as they go.

That geometry is the whole argument. A ball screw concentrates load at a handful of point contacts; a roller screw spreads it across dozens of line contacts on every roller. In the same envelope a roller screw carries several times the dynamic load of a ball screw — commonly cited at around three times, with some sources quoting more — runs efficiency in the same band (roughly 75–85 percent, per manufacturer data), and tolerates shock far better — point contacts dent under impact, line contacts just spread the load. For a leg joint that lands every step, that difference is the difference between a scheduled-life part and a consumable.

Standard, Inverted, Recirculating

Catalogues sort roller screws into three families, and the split explains most humanoid design decisions.

Source: figures consolidated from a major European manufacturer's roller screw catalogue; speed limits are short-term maxima, n in rpm, d in mm.

The inverted family is the humanoid one. Here the roles swap: the motor spins the nut, and the screw becomes the output rod, advancing one lead per nut turn. Only the threaded section of the screw needs rollers, and the nut can sit inside a hollow-shaft motor rotor — so the whole actuator collapses into a short cylinder: motor ring, nut, and a push rod sliding out of the middle. That is the architecture behind the linear joint classes in the Optimus body and in most linear-first designs since — and it pairs naturally with the hollow-axis layout we described in our piece on how humanoid joints route power and data through the axis.

The recirculating type swaps threaded rollers for grooved ones that are returned to their start position by cams after each orbit. It allows leads down to 1 mm — finer resolution, higher stiffness — but gives up speed, which is why it shows up in machine tools and precision stages rather than legs.

Lead Is the New Gear Ratio

In a rotary joint you choose a gear ratio; in a linear joint you choose a lead. Ignoring friction, the thrust from a motor torque τ through a lead L is:

thrust ≈ 2π × efficiency × τ ÷ L

A worked example shows the scale. The largest Optimus linear class is rated 8,000 N. Through a 5 mm lead at 80 percent efficiency, delivering 8,000 N needs roughly 8.0 N·m at the nut. Through a 2 mm lead, the same force needs about 3.2 N·m — small-frameless-motor territory, not geared-joint territory. Halve the lead, double the thrust, halve the speed.

That speed term is the honest cost. At a 2 mm lead, pushing the rod at 100 mm/s means spinning the nut at 3,000 rpm. High-speed running also heats the nut faster than a rotary reducer, because all of the sliding-and-rolling contact is concentrated in a short internal thread. Real catalogue efficiency sits above 80 percent for roller screws, and manufacturers recommend designing with a friction coefficient about 30 percent above the reference value — roughly five points below the theoretical number — so the honest thrust margin is smaller than the formula suggests.

One more property matters for legs: most roller screws back-drive. Push on the rod and it turns the motor. That is good for energy return and for controllers that sense force through motor current, and it is bad for holding a pose with power off — which is why catalogue guidance tells designers to add a brake or a non-back-driving stage wherever the joint must hold position unpowered.

Life Runs on a Cube Law

Every catalogue quotes a dynamic load rating, Cₐ: the constant centred axial load under which the calculated life — per the ISO 3408-5 method — reaches one million revolutions. Life then scales with the cube of Cₐ divided by the actual load. Run the screw at half its rating and it lasts eight times longer; at a quarter, sixty-four times longer.

For a concrete screw — 15 mm diameter, 5 mm lead, Cₐ = 26 kN (a catalogue value from a major European manufacturer's preferred range), a small general-purpose size — the cube law converts directly into distance:

Source: basic L10 calculation per ISO 3408-5 as referenced in roller screw catalogue documentation. L10 is the life 90 percent of identical screws reach; it ignores contamination, misalignment and lubrication faults.

Two practical readings of that table. First, a humanoid leg mostly runs far below its peak: a 500 N-class ankle screw doing walking duty lives effectively forever at average loads, and the rating is there for landings. Second — and this is the part people get wrong — mixed duty cycles do not average linearly. Life follows a cube-weighted mean load, so the two seconds of impact when the foot lands count orders of magnitude more than the thirty seconds of easy swing between them. Sizing a leg screw on its average load is how projects discover the cube law the expensive way. We made the same point about rotary joints and their L10 ratings and failure modes earlier this month; the physics is identical, only the direction of the load has changed. The same cube-weighted caution applies to the energy side of joint sizing — see the real energy budget of a robot joint.

The Grinding Problem

Why did this part, essentially a threaded shaft, cost more than the motor driving it? Because of one feature: the inverted nut.

Its multi-start internal thread is long, narrow and hardened — industry machining sources describe typical robotics nuts around 150 mm long with a 20 mm bore, length-to-diameter ratios of 5:1 to 8:1, hardened to 60 HRC, surface finish Ra 0.4, and single-digit-micrometre tolerances on roundness and concentricity. Grinding an internal thread at the bottom of a deep narrow bore invites tool overhang, vibration and chatter, which forces low speeds and scrapes yield. The process chain — soft turning, heat treatment, hard turning, finish grinding — is where the money goes.

Lead accuracy is graded under ISO 3408-3: the standard G5 grade allows 23 µm of travel variation over 300 mm, G3 allows 12 µm, G1 allows 6 µm. Chinese suppliers quote the equivalent C-classes, and both Chinese and Western commentators agree that the top precision grades are where domestic production still trails.

This is also where the cost story is moving fastest — and the direction matters more than any number we could print today. Domestic Chinese suppliers brought volume production online in 2026, with one widely quoted domestic retail listing reported at roughly one-tenth of the traditional European price for a comparable specification (a single-size comparison reported by China Daily in January 2026, not an across-the-board figure), and one domestic humanoid already using such screws in more than twenty of its joints per the same report. At least one Chinese plant is being tooled for million-unit annual screw capacity, according to 36Kr reporting. The engineering consequence is bigger than the price: when the screw stops being the scarce part, linear joints stop being the premium option and become a default architecture — with consequences for rotary-module vendors that deserve their own analysis. The screw alone is generally estimated to take roughly 60–65 percent of a linear actuator's cost in one teardown breakdown, and most industry estimates agree it is over half, so compressing it compresses the whole joint.

What a Joint Buyer Should Ask

For teams specifying humanoid linear joints, the checklist that falls out of the physics:

• Cₐ and C₀ₐ for the exact size and lead — then compute life against your own cube-weighted load cycle, including landings, not against the average load.
• Precision class and how it is measured — ISO 3408-3 class and travel variation over 300 mm, and whether it is inspected per part or by sampling.
• Preload and backlash — backlash-free versions exist but add friction and cost; say which your force-control loop actually needs.
• Back-driving behaviour at your lead — it decides whether the joint needs a brake to hold a pose unpowered.
• Thread process and capacity — ground or rolled internal threads, which grinding line, and committed annual volume; the answer sets ripple, noise and your lead time.

The planetary roller screw is not exotic anymore. It is a mature mechanism whose three governing numbers — lead, Cₐ, and thread grade — map one-to-one onto thrust, life and price. Teams that treat it like a gear ratio with a lifetime attached will size it right; teams that treat it like a black-box cylinder will rediscover the cube law on someone else's warranty. And for the rotary half of a mixed body — wrists, elbows, waists — a lightweight integrated joint module keeps doing what the screw cannot: delivering torque exactly on the pivot axis.

Source: humanoid.guide planetary roller screw and linear-vs-rotary explainers (2026); Ewellix roller screw catalogue data as cited therein; Tesla AI Day 2022 actuator figures; 36Kr teardown reporting (2026); China Daily, January 2026; ISO 3408-5 / ISO 3408-3 methods; EMAG machining-chain commentary via industry sources.