Torque–Speed Curves in Humanoid Joints: Why Every Actuator Hits a Voltage Wall
Torque–speed curves in humanoid joints: why every actuator hits a voltage wall
A joint actuator cannot deliver its peak torque at its top speed. Current sets the torque ceiling, the battery voltage sets the speed ceiling, and back-EMF couples the two into one envelope. Where a humanoid’s motions sit inside that envelope decides how fast it can walk, kick or catch itself.
Sources: CubeMars AK80-9 and RobStride 04 product data; Human-Level Actuation for Humanoids (arXiv 2511.06796); KAIST Hound study, arXiv 2312.17507.
Two limits, one envelope
Spec sheets usually list a joint’s peak torque and its maximum speed as two separate numbers. They are not available at the same time. Torque in a permanent-magnet motor is proportional to current, so the drive’s current limit (and the magnets’ tolerance for it) puts a flat ceiling on torque. Speed is limited by something else: a spinning motor generates a back-EMF that rises with speed, and the drive can only push current into the winding while its supply voltage exceeds that back-EMF.
In the simple model used by KAIST’s Hound team, the supply voltage has to cover two things at once: the resistive drop that makes torque, (R/Kt)·τ, and the back-EMF, ω/Kv. Their sum cannot exceed the bus voltage. Plotted with speed on one axis and torque on the other, that rule is a sloping line. The current limit is a flat line above it. Together they enclose the region a joint can actually reach, and a third, lower line marks the torque the motor can hold without overheating.
Figure 1 · The operating envelope of an electric joint actuator (motoring quadrant)
Illustration: humanoid.guide, after the motor operating region in the KAIST Hound study (arXiv 2312.17507) and MathWorks’ field-weakening description. Schematic, not to scale; real envelopes bend because of inductance, saturation and friction.
The voltage budget: back-EMF eats the headroom
The sloping line is easiest to read as a budget. At standstill the back-EMF is zero, and the whole supply voltage can drive current through the winding resistance, so torque is limited only by how much current the drive allows. As the joint speeds up, back-EMF claims a growing share of the same voltage, less is left to push current, and available torque falls. At the no-load speed the back-EMF equals the supply and no torque is left at all.
The two slopes are linked by one physical constant. In SI units the torque constant (N·m per ampere) and the back-EMF constant (volts per rad/s) are the same number for a given motor. CubeMars’ AK80-9 data shows it: a 100 rpm/V motor works out to 0.0955 V·s/rad, and the listed torque constant is 0.095 N·m/A. A winding with more torque per amp therefore always reaches its voltage wall at a lower speed. Rewinding a motor moves the envelope along the speed axis; it does not enlarge it.
Figure 2 · Animated · Moving along the voltage limit: the same supply voltage, split two ways
Illustration: humanoid.guide, from V = (R/Kt)·τ + ω/Kv (KAIST Hound study, arXiv 2312.17507). CSS animation; stops at the midpoint when reduced motion is requested.
What the spec sheets say
Off-the-shelf actuator modules used in research humanoids and quadrupeds publish enough to see the trade. Rated speed at rated torque is always below no-load speed, and the bigger, higher-torque modules are the slower ones at the same 48 V. The motor speed constant can be backed out from the no-load figure: output speed times gear ratio, divided by supply voltage.
Table 1 · Published torque and speed figures for joint actuator modules
| Actuator | Supply | Ratio | Peak torque | Rated point | No-load output speed | Implied motor Kv |
|---|---|---|---|---|---|---|
| CubeMars AK80-9 V3.0 | 48 V | 9:1 | 22 N·m | 9 N·m at 390 rpm | 570 rpm · 59.7 rad/s | ≈107 rpm/V (listed 100) |
| RobStride 02 | 48 V | 7.75:1 | 17 N·m | 6 N·m (tested at 100 rpm) | 410 rpm · 42.9 rad/s | ≈66 rpm/V |
| Unitree GO-M8010-6 | 24 V | 6.33:1 | 23.7 N·m | Not published | 30 rad/s (max speed) | ≈76 rpm/V |
| RobStride 04 | 48 V | 9:1 | 120 N·m | 40 N·m at 167 rpm | 200 rpm · 20.9 rad/s | ≈38 rpm/V |
Sources: CubeMars AK80-9 product page; OpenELAB RobStride 02 and 04 listings (speeds ±10%); Unitree GO-M8010-6 product page. Implied Kv = no-load output speed × ratio ÷ supply voltage, humanoid.guide arithmetic. rad/s converted from rpm.
Chart 1 · No-load output speed at rated supply, against the human ankle push-off band
Sources: as Table 1; ankle band from Human-Level Actuation for Humanoids (arXiv 2511.06796). No-load speed is the zero-torque end of the envelope; usable speed under load is lower.
Every module in the chart clears the ankle push-off band with no load. That is the wrong comparison, though, because push-off is when the ankle also produces most of its positive work. The RobStride 04 shows how much the margin shrinks under load: 200 rpm (20.9 rad/s) with nothing attached, 167 rpm (17.5 rad/s) at its 40 N·m rated torque, and less again towards its 120 N·m peak. Put the same module behind a further reduction to raise torque and the speed divides by that ratio. A 2:1 linkage would leave it below 9 rad/s at rated torque, at the bottom edge of the band.
The battery moves the wall
Because no-load speed is proportional to supply voltage, the envelope is not fixed for a battery-powered robot. Unitree lists a maximum pack voltage of 67.2 V for H1 and H1-2. That is consistent with a 16-cell lithium-ion string at 4.2 V per cell; Unitree does not publish the cell count. If so, the pack sits nearer 51–54 V close to empty, and every joint’s voltage-limited speed falls by roughly a fifth with it, while peak torque at low speed is unchanged. A motion that works on a full battery can saturate at the end of a shift.
The other direction matters too. When a joint brakes, the motor works as a generator and pushes current back into the bus. Hound’s team charged its 20-series pack to 81 V rather than its 84 V maximum to leave headroom for that reverse current, on a motor controller rated for 100 V. For comparison, ANYmal and the TYTAN quadruped in a 2025 sim-to-real study run 48 V buses, and Unitree quotes its GO-M8010-6 speed at 24 V.
Why learned controllers need the envelope
Simulators usually model a joint with a rectangular limit: a maximum torque and a maximum speed, available in any combination. A reinforcement-learning policy will find and use the corner of that rectangle, high torque at high speed, that the real motor cannot deliver. The KAIST team behind Hound clipped Hound’s simulated torques to the trapezoid-like motor operating region during training. With the constraint the 45 kg quadruped reached 6.5 m/s on a treadmill; without it the best was 4.5 m/s, and the unconstrained policy fell at 5 m/s on hardware.
That 2025 study (arXiv 2509.06342) makes the same point from the other side. Its PACE method clips simulated torques to an idealised permanent-magnet envelope set by the maximum torque and the back-EMF-limited torque, and lists per-joint limits such as 25 N·m and 94 rad/s for the TYTAN hip motor. Simulated ANYmal still ran about 4 m/s while the hardware peaked near 3 m/s, which the authors attribute to a 32 A current limit the model did not include. They name bus-voltage and current limits as the next things to fit. The paper reports deployment on humanoids including Fourier’s GR-1, but gives envelope figures only for the quadrupeds.
Field weakening: buying speed with current
Drives have one electrical way past the voltage wall. Field-oriented control normally puts all current on the q-axis, which makes torque. Driving the d-axis current negative opposes the rotor magnets’ field and lowers the back-EMF, so the motor can run above its base speed. MathWorks describes the result as a roughly constant-power region: speed rises, maximum torque falls. Kollmorgen’s AKD2G drive raises d-axis current automatically once the motor voltage nears the bus voltage.
The costs are why humanoid joints use it carefully. The d-axis current makes no torque but still heats the copper, so it competes with torque for the same thermal budget. Kollmorgen warns that if the power stage disables above base speed, the back-EMF can exceed the bus voltage, and that d-axis current must be capped to avoid demagnetising the magnets. A University of Twente research project is studying field weakening for explosive robot motions such as jumps, pairing it with winding-temperature prediction because the extra current is short-term borrowing against heat.
What to check on a joint spec sheet
- Speed at rated torque, not only no-load speed. The AK80-9 loses about a third of its speed between zero load and rated torque.
- The voltage behind every speed figure. The GO-M8010-6 speed is quoted at 24 V, the RobStride and CubeMars figures at 48 V. Speeds are only comparable at the same supply.
- A published torque–speed curve. Two or three points are not an envelope. Ask for the curve at minimum and maximum pack voltage.
- Whether field weakening is used, and what it costs. Extra speed from the drive is current that is not making torque.
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 torque–speed curve?
The boundary of the torque a motor or joint actuator can produce at each speed. It is flat at low speed, where current sets the limit, and falls towards zero at the no-load speed, where back-EMF uses up the supply voltage.
Why can a joint not reach peak torque at top speed?
Producing torque needs current, and pushing current into the winding needs voltage left over after the back-EMF. Back-EMF rises with speed, so the faster the joint turns, the less voltage remains for current and the lower the available torque.
Does battery voltage change a humanoid’s joint speed?
Yes. No-load speed scales with supply voltage, so the voltage-limited part of the envelope shrinks as the pack discharges. Low-speed peak torque, which is set by current, does not change.
What is field weakening?
A drive technique that injects negative d-axis current to oppose the magnets’ field and lower the back-EMF, letting the motor exceed its base speed at reduced torque. It costs extra copper heat, and drives must guard against overvoltage and demagnetisation.
Sources
- Actuator-Constrained Reinforcement Learning for High-Speed Quadrupedal Locomotion, arXiv 2312.17507 (V = (R/Kt)τ + ω/Kv model, motor operating region, 81 V pack, 6.5 vs 4.5 m/s)
- Towards bridging the gap: Systematic sim-to-real transfer for diverse legged robots, arXiv 2509.06342 (PACE, back-EMF-limited torque envelope, TYTAN limits, 48 V buses, ANYmal 4 vs 3 m/s)
- Human-Level Actuation for Humanoids, arXiv 2511.06796 (ankle push-off at 8–12 rad/s)
- CubeMars AK80-9 V3.0 KV100 product page (48 V, 9/22 N·m, 390/570 rpm, 0.095 N·m/A)
- OpenELAB, RobStride 02 listing (48 V, 7.75:1, 17 N·m peak, 410 rpm no-load)
- OpenELAB, RobStride 04 guide (48 V, 9:1, 120/40 N·m, 200/167 rpm)
- Unitree GO-M8010-6 joint motor (24 V, 6.33:1, 23.7 N·m, 30 rad/s)
- Unitree H1 / H1-2 product page (67.2 V maximum battery voltage, M107 joint motor)
- MathWorks, Field-Weakening Control (base speed, negative d-axis current, constant-power region)
- Kollmorgen AKD2G user manual, Field Weakening (d-axis control near bus voltage, overvoltage and demagnetisation warnings)
- University of Twente RaM, Field-weakening and fatigue management for explosive robotics (field weakening for jumps, thermal management)