The Flexspline: Inside the Strain-Wave Gear in Humanoid Joints
Inside the strain-wave gear: the flexspline that bends twice per motor turn
Most humanoid rotary joints that need precision run through a strain-wave gear, and at its heart sits a thin steel cup that is squeezed into an ellipse and released twice on every turn of the motor. Two missing teeth set the gear ratio, the tooth profile sets the torque, and metal fatigue in that cup sets the life. Here is how the part works and what the catalogue numbers mean.
Sources: Harmonic Drive CSG/CSF catalogue; Wikipedia, Strain wave gearing; 36Kr, 2026.
Three parts and two missing teeth
A strain-wave gear, often called a harmonic drive after the company that commercialised it, has three parts. The circular spline is a rigid ring with internal teeth. The flexspline is a thin-walled steel cup with external teeth around its open end. The wave generator is an elliptical cam pressed into a thin ball bearing whose races can flex. Push the wave generator into the flexspline and the cup’s rim takes on the elliptical shape, so its teeth mesh with the circular spline only in two zones at the ends of the long axis. C.W. Musser patented the idea in 1957 at United Shoe Machinery; the first successful use followed in 1960.
The trick is in the tooth count. In a standard unit the flexspline has two teeth fewer than the circular spline. As the motor turns the wave generator, the two mesh zones travel round the ring, and after one full turn of the cam the flexspline has fallen back by those two teeth. With the circular spline fixed, the flexspline is the output and turns in the opposite direction to the input. The ratio is simply the flexspline tooth count divided by the tooth difference: 200 teeth against 202 gives 100:1.
Figure 1 · Front view of a strain-wave gear
Illustration: humanoid.guide, after the component description in the Harmonic Drive CSG/CSF catalogue. Tooth counts chosen for legibility; catalogue units at 100:1 carry about 200 flexspline teeth.
Because the gear reduces speed through the deformation of one part rather than through a train of meshing wheels, a single stage covers ratios that would need two or three planetary stages. The same envelope can hold anything from 30:1 to 320:1, depending only on the number of teeth cut. That is why the strain-wave gear is the usual choice for humanoid arm joints, where ratios of 50:1 to 160:1, zero backlash and a short axial length all matter.
Figure 2 · Animated · The wave goes round fast, the output creeps the other way
Illustration: humanoid.guide. Speeds drawn for 30:1, the lowest common catalogue ratio, so the output motion is visible; the wave of deformation turns with the cam while the flexspline material turns slowly backwards. CSS animation; stops when reduced motion is requested.
Why the ratio sets the bending
The flexspline has to bend far enough for its teeth to clear the circular spline at the short axis of the ellipse and to sink into it at the long axis. In the standard geometry the radial deflection at the long axis is half the module times the tooth difference, w₀ = 0.5 · m · ΔZ, so with a two-tooth difference the rim moves outward by about one module. Since the pitch diameter is the module times the tooth count, the deflection as a share of the diameter is one divided by twice the ratio.
This is the reason strain-wave gears rarely go below 30:1. A 30:1 flexspline has only 60 coarse teeth and must bend by about 1.7% of its diameter on every half turn of the cam. At 100:1 the 200 finer teeth need only 0.5%, and at 160:1 about 0.3%. Lower ratios mean higher alternating bending stress in a wall that is only about 1 to 1.5% of the rim diameter thick, according to the design guidance quoted by Zhu and co-authors.
Chart 1 · How far the flexspline must bend at each ratio
Calculation: humanoid.guide, for a two-tooth difference with radial deflection equal to one module (w₀ = 0.5 · m · ΔZ, Zhu et al., Gear Technology 2013). Ratio = Zf ÷ 2, pitch diameter = m · Zf, so deflection ÷ diameter = 1 ÷ (2 × ratio). Bars drawn to scale.
Table 1 · Tooth count and bending at the standard catalogue ratios
| Ratio | Flexspline teeth | Circular spline teeth | Deflection ÷ pitch diameter | Example: 50 mm pitch diameter |
|---|---|---|---|---|
| 30:1 | 60 | 62 | 1.67% | m 0.83 mm |
| 50:1 | 100 | 102 | 1.00% | m 0.50 mm |
| 80:1 | 160 | 162 | 0.63% | m 0.31 mm |
| 100:1 | 200 | 202 | 0.50% | m 0.25 mm |
| 120:1 | 240 | 242 | 0.42% | m 0.21 mm |
| 160:1 | 320 | 322 | 0.31% | m 0.16 mm |
Calculation: humanoid.guide, two-tooth difference, circular spline fixed. The module m equals the radial deflection at the major axis under the standard-deflection assumption. Ratios 50, 80, 100, 120 and 160 are the CSG series options (Harmonic Drive catalogue); 30:1 is offered in small sizes.
Tooth profile: from involute to S
Earlier strain-wave gears used involute teeth, which meet the circular spline over a narrow arc. Harmonic Drive’s current gears use a tooth form it calls the S tooth. The company says the engaged angle is about double that of the conventional form, which in turn doubles torsional stiffness, doubles the momentary and repeated peak torque ratings and doubles life at the same rated load. Its CSG/CSF catalogue states that up to 30% of all teeth are in mesh at the same time. Efficiency is not improved by the profile and stays in the range of 60 to 90%, depending on load, speed and lubrication.
Tooth profile is also where newer suppliers compete. Within the S-tooth family, the CSG series is the “high-torque” version: the catalogue claims 30% more torque than CSF and a rated life of 10,000 hours instead of 7,000, a 43% gain. Chinese makers such as Leader Harmonious Drive say they built their own path around tooth profile, materials and manufacturing, though they publish few details.
Cup, silk hat or pancake
The flexspline also comes in three shapes, and the choice shows up directly in a joint’s layout. The cup closes its far end with a thin diaphragm that bolts to a small output hub; it is the most compact and the most common. The silk hat turns that end outward into a brim, leaving the middle open for a large hollow bore through which cables or a second shaft can pass. The pancake drops the diaphragm altogether and becomes a short toothed ring; a second circular spline with the same tooth count as the flexspline takes the output, which makes the unit very short along the axis.
Figure 3 · Three flexspline shapes, side section
Illustration: humanoid.guide, after the flexspline types listed on Harmonic Drive’s component-set pages. Schematic, not to scale.
Table 2 · Flexspline shapes in the Harmonic Drive component-set range
| Shape | Example series | What the maker highlights | Where it fits in a humanoid |
|---|---|---|---|
| Cup | CSF-2A, CSG-2A, CSF/CSG-2A-LW, CSD-2A | Compact; CSG adds 30% torque over CSF; LW versions 30% lighter; CSD ultra-flat | General rotary joints where a small centre bore is enough |
| Silk hat | SHF-2A, SHG-2A | Large hollow through-hole; rated torque ranges 9.0–1,800 N·m (SHF) and 23–3,400 N·m (SHG) | Joints that route cables or a second shaft through the centre |
| Pancake | FB-2, FR-2 | Cup without a bottom, very short; FR has a double wave-generator bearing for higher torque | Axially tight joints such as wrists |
Source: Harmonic Drive LLC, component sets product pages; Harmonic Drive CSG/CSF catalogue. Application column: humanoid.guide.
How flexsplines fail
In most designs the flexspline is the part that limits life. Zhu and co-authors write that the failure of harmonic gear drives is mainly caused by fatigue fracture of the flexspline, which sees an asymmetric alternating stress. Their design method applies a stress concentration factor of 1.8 to 2.0 at the tooth root and a safety factor of 1.3 to 1.5, using a quenched and nitrided alloy steel in their example.
Material quality matters as much as geometry. A 2024 failure analysis in the Chinese journal Special Steel traced a flexspline fracture in service to coarse prior austenite grains, with grain size at the crack about 1.4 times that elsewhere, rather than to inclusions, and recommended grain-refining heat treatment. For humanoid builders the practical point is that two flexsplines with identical drawings can have very different lives, which is why buyers ask for fatigue test data and heat-treatment records, not just a torque rating. Falls add a further risk: the motor’s inertia appears at the output multiplied by the square of the ratio, which raises impact torques, as covered in our article on overload protection.
Who makes them for humanoids
Japan’s Harmonic Drive Systems long held a near monopoly on precision strain-wave gears. In China, Leader Harmonious Drive (Leaderdrive) has become the main domestic alternative. According to 36Kr, it supplies Unitree, Agibot and UBTech, has Tesla certification, once supplied about 70% of Unitree’s upper-limb reducers in 2025, and held 27.5% of the Chinese harmonic reducer market in 2025. Its average selling price fell from about 1,900 yuan in 2017 to about 1,100 yuan in 2024. The company reported capacity of about one million units at the World Robot Conference in August 2026, with 1.5 million planned by the end of 2026 and humanoid orders above 40% of its order book.
Demand per robot is large. Citing Guojin Securities, 36Kr puts the count at about 14 harmonic reducers per Tesla Optimus today, rising to more than 20. Not every joint uses one: many leg joints rely on planetary gears or roller screws instead, where shock loads and back-drivability matter more than zero backlash.
What to ask a strain-wave supplier
- Tooth profile and generation. Ask which profile is used and whether rated, repeated peak and momentary peak torques are given for the exact ratio you need; ratings differ by ratio within one size.
- Flexspline material and heat treatment. Steel grade, grain size control and surface treatment decide fatigue life as much as the drawing does.
- Rated life and its basis. Hours at rated torque and input speed, and whether the figure refers to the flexspline, the wave-generator bearing or both.
- Shape and bore. Cup, silk hat or pancake, and the hollow bore available for cable routing.
- Torsional stiffness at low torque. The wind-up region near zero torque is where strain-wave joints lose position accuracy.
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
What is a flexspline?
The thin-walled, externally toothed steel part of a strain-wave (harmonic) gear. The wave generator forces it into an ellipse so that its teeth mesh with the rigid circular spline at two opposite zones. In the usual layout it is also the output.
How does a strain-wave gear get its ratio?
The flexspline has two fewer teeth than the circular spline. Each turn of the wave generator moves the flexspline back by two teeth, so with the circular spline fixed the ratio is the flexspline tooth count divided by two: 200 teeth give 100:1, turning opposite to the input.
Why do strain-wave gears rarely go below 30:1?
At a fixed diameter, a lower ratio means fewer, larger teeth, and the flexspline must bend by about one tooth module. At 30:1 that is about 1.7% of the pitch diameter, against 0.5% at 100:1, so bending stress in the thin wall rises quickly.
What is the S tooth profile?
Harmonic Drive’s tooth form for its current gears. The company says it roughly doubles the engaged tooth angle versus the earlier involute form, which doubles torsional stiffness, peak torque ratings and life, with up to 30% of all teeth in mesh at once.
How do flexsplines fail?
Mainly by fatigue fracture of the flexspline, usually starting at the tooth root, where the alternating bending stress is concentrated. Material defects such as coarse grains have been identified as the cause in published failure analyses.
Sources
- Harmonic Drive, CSG/CSF gear units catalogue (two-tooth difference, S tooth up to 30% teeth engaged, CSG 30% more torque and 10,000 h vs 7,000 h life, ratio options)
- Harmonic Drive, S tooth profile (engagement doubled, stiffness, peak torque and life doubled, 60–90% efficiency)
- Harmonic Drive, Component sets (cup, silk hat and pancake series, SHF/SHG torque ranges, LW versions)
- Wikipedia, Strain wave gearing (operating principle, 30:1 to 320:1, Musser 1957 patent, opposite rotation)
- Zhu et al., Minimal tooth number of flexspline in harmonic gear drive with external wave generator, Gear Technology, Oct 2013 (w₀ = 0.5 · m · ΔZ, wall thickness, fatigue as main failure, stress factors, 10–30% teeth in contact)
- Mu et al., Fracture analysis of toughness fracture failure on flexspline of harmonic reducers during the service, Teshugang 45(1), 2024 (coarse austenite grains as root cause)
- 36Kr, Leader Harmonious Drive and humanoid demand, 2026 (≈14 harmonic reducers per Optimus, capacity, prices, market share, Unitree share)
- humanoid.guide, Leaderdrive harmonic reducers surge as humanoid demand lifts shares (supplier context)