Planetary Gearboxes in Humanoid Joints: How a Sun, Three Planets and a Ring Set the Ratio
Inside the planetary gearbox: how a sun, three planets and a ring set a humanoid joint’s ratio
The planetary stage is the reducer behind most quasi-direct-drive legs, from the MIT Mini Cheetah to the Unitree G1 knee. Its ratio comes straight from two tooth counts, its geometry fixes how far one stage can go, and the gap between catalogue efficiency and what labs measure decides how well a joint can sense force through its own motor.
Sources: Katz 2018 (MIT thesis); China-Amass G1 knee teardown; Apex Dynamics AD series data; Crispel et al., IEEE/ASME AIM 2024.
Four parts, one held still
A planetary (epicyclic) stage has four members. A small sun gear sits on the axis. Several planet gears, usually three, mesh with it and run on pins held by a carrier. The planets also mesh with a ring gear that has its teeth on the inside. Fix any one member, drive a second and take power from the third, and you get a fixed ratio.
Robot joints almost always use the same arrangement: the motor turns the sun, the ring is bolted to the housing, and the carrier is the output. The ratio is then 1 + Zring / Zsun, where Z is the tooth count. The planets’ own tooth count drops out of the formula; they only have to fit between the other two. Ben Katz’s Mini Cheetah actuator uses stock 0.5-module gears with a 20-tooth sun and a 100-tooth ring, so its ratio is 1 + 100/20 = 6:1. Sun and carrier turn in the same direction, which is one reason planetary stages are easy to drop into a motor housing.
Figure 1 · Animated · The Mini Cheetah’s 6:1 stage, sun in, carrier out, ring fixed
Illustration: humanoid.guide, using the tooth counts of the Mini Cheetah stage (Katz 2018). Gears drawn as touching pitch circles without teeth; speeds follow the fixed-ring kinematics (carrier = sun ÷ 6, planet = −1.5 × carrier). CSS animation; stops when reduced motion is requested.
The tooth-count rules that cap one stage
Two geometric rules decide which gear sets are possible. First, the planets must bridge the gap between sun and ring, so with standard gears Zring = Zsun + 2 × Zplanet. Second, for planets spaced evenly around the sun, (Zsun + Zring) divided by the number of planets must be a whole number. The Mini Cheetah set passes both: 20 + 2 × 40 = 100, and (20 + 100) / 3 = 40.
To raise the ratio you need a bigger ring relative to the sun. The sun cannot shrink much: it carries the full motor torque on few teeth, and Katz lists a sun-gear torque limit of about 11 N·m for his design. A 2025 optimisation study of single-stage gearboxes for a T-Motor U12 sets 20 teeth as the minimum. Every design it found uses exactly that 20-tooth sun, so higher ratios come only from larger planets and ring. At 10.2:1 the ring already needs 184 teeth, and with three planets the planets grow until they almost touch each other. That is why single planetary stages in robot joints stay roughly between 4:1 and 10:1.
Table 1 · Optimised single-stage sets for one motor: tooth counts, ratio and modelled efficiency
| Ratio | Sun | Planet | Ring | Planets | Module | Mass | Model efficiency |
|---|---|---|---|---|---|---|---|
| 5:1 | 20 | 30 | 80 | 4 | 0.6 mm | 1.375 kg | 90.9% |
| 6:1 | 20 | 40 | 100 | 3 | 0.5 mm | 1.508 kg | 92.3% |
| 7.2:1 | 20 | 52 | 124 | 3 | 0.5 mm | 1.657 kg | 93.3% |
| 8.1:1 | 20 | 61 | 142 | 3 | 0.5 mm | 1.823 kg | 93.7% |
| 9:1 | 20 | 70 | 160 | 3 | 0.5 mm | 1.959 kg | 94.1% |
| 10.2:1 | 20 | 82 | 184 | 3 | 0.5 mm | 2.224 kg | 94.4% |
Source: Comparison between External and Internal Single Stage Planetary gearbox actuators for legged robots, arXiv 2506.16356 (2025), external-layout designs for a T-Motor U12. Mass is the whole actuator. Efficiencies are model predictions, not measurements. Every row satisfies ratio = 1 + ring/sun and ring = sun + 2 × planet.
Inside the motor or beside it
Quasi-direct-drive actuators use large-diameter, short motors, which leaves an empty bore in the middle of the stator. The Mini Cheetah put its gearbox in that bore. The sun is press-fit to the rotor, the ring is press-fit into the housing, and the carrier’s hardened planet pins stick out of the front face as the output. The whole actuator weighs 480 g, and Katz measured about 0.005 rad (roughly 17 arcmin) of backlash at the output, with no anti-backlash measures.
The 2025 arXiv study calls this the internal layout and compares it with the external one, where the gearbox sits in front of the motor, as in the MIT Cheetah 3. For its test motor the internal layout came out lighter (1.386 kg against 1.508 kg at 6:1) at the same modelled efficiency. The catch is size: the ring has to fit inside the stator, so the internal layout produced no workable design above about 7:1. Cheetah 3, with its external single stage, runs at 7.67:1 and 230 N·m.
Figure 2 · Two ways to package a single planetary stage with the motor
Illustration: humanoid.guide, schematic side views, not to scale. Sources: Katz 2018; Bledt et al., MIT Cheetah 3 (IROS 2018); arXiv 2506.16356 for the ≈7:1 limit of the internal layout with a T-Motor U12.
Going past 10:1: two stages, compound planets and Wolfrom trains
When a joint needs more torque from a small motor, designers stack stages. Two simple stages multiply their ratios: the Unitree G1 knee, according to a China-Amass teardown, uses a two-stage planetary for 22.5:1 and 120 N·m in an actuator of about 1 kg, with crossed-roller output bearings and magnetic encoders on both the rotor and output sides. Industrial precision gearboxes such as Apex Dynamics’ AD series offer single stages from 4:1 to 10:1 and two-stage units from 16:1 to 100:1.
Compound planetaries use stepped planets with two tooth counts, meshing with two different rings or suns, to get a large ratio in one plane of gears. A Wolfrom train is the extreme case: one ring is fixed, a second ring with slightly different teeth is the output, and the small difference gives ratios in the hundreds. The COMPAct study (arXiv 2510.07197) built a 7.2:1 single stage and a 14:1 compound stage and modelled a 16.36:1 Wolfrom and a 22.05:1 double stage. At Vrije Universiteit Brussel, Crispel and colleagues reported a Wolfrom-based gearbox at 222:1 with peak efficiency above 80%, 80 N·m repeatable peak torque, 650 g and under 1 N·m back-drive torque. Wolfrom trains normally lose a lot of power to internal circulating power; the VUB team reworked the configuration and gear meshing to cut that loss.
Table 2 · Published planetary reducers in legged and humanoid joints
| Actuator | Layout | Ratio | Published figures |
|---|---|---|---|
| MIT Mini Cheetah | Single stage, inside stator bore, 3 planets | 6:1 | 480 g actuator; ~0.005 rad backlash; ~90–95% efficiency; ~$300 BOM |
| MIT Cheetah 3 | Single stage, external | 7.67:1 | 230 N·m max; 21 rad/s; ±10% static torque error, mainly gear friction |
| Unitree G1 knee | Two-stage planetary | 22.5:1 | 120 N·m; ~1 kg; crossed-roller bearings; dual encoders (teardown) |
| COMPAct single stage | Single stage, MN8014 motor | 7.2:1 | 60–80% measured efficiency; 0.59° backlash; 242.7 N·m/rad |
| COMPAct compound | Compound planetary | 14:1 | 60% measured efficiency; 2.6° backlash; 201.6 N·m/rad |
| VUB Wolfrom prototype | Wolfrom (3K) train | 222:1 | >80% peak efficiency; 80 N·m peak; 650 g; <1 N·m back-drive |
Sources: Katz 2018; Bledt et al. 2018; China-Amass G1 teardown; COMPAct, arXiv 2510.07197; Crispel et al., IEEE/ASME AIM 2024. Stiffness figures are transmission stiffness as reported by COMPAct.
Backlash and efficiency: catalogue versus lab
Catalogue numbers for precision planetaries look excellent. Apex Dynamics lists its AD series at 97% or better for one stage and 94% or better for two, with backlash classes of 1, 3 or 5 arcmin for single stages and 3, 5 or 7 arcmin for two stages. Those are ground, preloaded industrial units, rated at their nominal load and speed.
Robot-built stages measure lower. Katz reports 90–95% over most of the Mini Cheetah’s range and models friction as a fixed 0.09 N·m plus 0.04 N·m per N·m of load. COMPAct measured 60–80% on its single stage and 60% on its compound stage, with backlash of 0.59° and 2.6° (about 35 and 156 arcmin). That gap matters because quasi-direct-drive joints estimate torque from motor current. Every unmodelled newton-metre of gear friction becomes torque error: Cheetah 3’s authors put their static torque error at about ±10%, mainly from the gears.
Chart 1 · Planetary efficiency: catalogue, model and measured
Sources: Apex Dynamics AD series data (via Walker Industrial); arXiv 2506.16356 (model, external 9:1 design); Katz 2018; COMPAct, arXiv 2510.07197; Crispel et al., AIM 2024. Yellow segments show a reported range. Catalogue values are minimums at rated conditions; lab values come from different test methods and are not directly comparable.
What to ask about a planetary joint
- How many stages, and which layout? A single stage inside the stator is short and back-drivable but capped near 7:1; two stages reach 20:1 and beyond at the cost of extra friction and backlash.
- Measured efficiency at robot loads. Ask for efficiency across torque and speed, not one catalogue number at rated load. Low-load efficiency is what a standing or walking robot sees most of the time.
- Backlash at the output, in arcmin or radians. Hobbed robot gears can be ten times looser than ground industrial units; check whether the figure is no-load or under a stated torque.
- Sun-gear and peak-torque limits. The small sun carries the full motor torque on a few teeth. Ask what limits peak torque and what happens to it in a fall.
- Back-drive torque. One number that combines ratio and friction and tells you how well current-based torque control will work.
Go deeper: The Humanoid Actuation Report
249 pages on motors, reducers, screws, bearings, encoders and drives, the vendors behind every major platform, and the cost curves that decide adoption to 2036.
The Humanoid Actuation Report All joints & actuators articles All humanoid.guide reportsKeep reading · Joints & actuators
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FAQ
How do you calculate a planetary gear ratio?
With the ring fixed, the sun as input and the carrier as output, the ratio is 1 + ring teeth ÷ sun teeth. The MIT Mini Cheetah’s 20-tooth sun and 100-tooth ring give 6:1. The planet tooth count does not appear in the formula.
Why don’t humanoid joints use a single planetary stage for 50:1?
The ring would need about 50 times as many teeth as the sun. The sun cannot shrink below roughly 20 teeth and still carry motor torque, so the ring and planets become impractically large. Single stages in robot joints stay near 4:1 to 10:1, and higher ratios use two stages, compound planets or strain-wave gears.
Which humanoid robots use planetary gearboxes?
Published examples include the Unitree G1 knee (two-stage, 22.5:1, from a teardown) and the quasi-direct-drive modules derived from the MIT Mini Cheetah design. Many humanoid makers do not publish their reducer types.
Is a planetary gearbox back-drivable?
A single low-ratio stage usually is, which is why quasi-direct-drive legs use it. Back-drivability falls as ratio and friction rise, although research Wolfrom designs such as VUB’s 222:1 prototype report under 1 N·m of back-drive torque.
Sources
- B. Katz, A Low Cost Modular Actuator for Dynamic Robots, MIT thesis, 2018 (6:1 stage in the stator bore, 3 planets, 0.5-module stock gears, ~0.005 rad backlash, 90–95% efficiency, friction model, 11 N·m sun limit, 480 g, ~$300 BOM)
- Comparison between External and Internal Single Stage Planetary gearbox actuators for legged robots, arXiv 2506.16356, 2025 (tooth counts, masses and modelled efficiency; internal layout limited to about 7:1)
- COMPAct: Computational Optimization and Automated Modular design of Planetary Actuators, arXiv 2510.07197, 2025 (measured efficiency, backlash and stiffness of single and compound stages; Wolfrom and double-stage models)
- S. Crispel et al., A high efficient, high gear ratio Planetary Gear Transmission for Human-Centred Robotics, IEEE/ASME AIM 2024 (222:1 Wolfrom-based gearbox, >80% peak efficiency, 80 N·m, 650 g, <1 N·m back-drive)
- VUB research portal, Wolfrom planetary gear trains for lightweight human-centered robotics (Wolfrom alpha-I concept)
- G. Bledt et al., MIT Cheetah 3: Design and Control of a Robust, Dynamic Quadruped Robot, IROS 2018 (single-stage 7.67:1, 230 N·m, 21 rad/s, ±10% torque error)
- China-Amass, Unitree G1 knee joint actuator teardown (two-stage 22.5:1, 120 N·m, ~1 kg, crossed-roller bearings, dual encoders)
- Apex Dynamics AD series high-precision planetary gearboxes (Walker Industrial) (ratios, ≥97% / ≥94% efficiency, backlash classes)
- C. Glinsky, Drivetrain Hub gear notebooks, Chapter 4: Planetary Gears (planetary geometry and assembly conditions)