Two different errors, often measured as one

Backlash is a gap. When a motor reverses, its gear teeth leave one flank and travel across empty space before they touch the opposite flank of the mating gear. During that travel the motor turns and the limb doesn’t. Gear Technology defines it simply as “the clearance, or play, between the teeth of gears”, and in a robot the consequence is just as simple: the load isn’t where the controller thinks it is.

Torsional stiffness is what happens after contact. Teeth bend, a flexspline stretches, bearings deflect, and the output winds up by an angle proportional to the torque on it. No gap involved. Push harder, twist further.

Manufacturers blur the two on purpose, and not dishonestly. “Lost motion” – the figure most precision reducers quote – is the total rotation at the output when you load it to about ±3% of rated torque with the input locked, according to a 2020 review of compact robot gearboxes in Frontiers in Robotics and AI. It captures the gap plus the first sliver of spring. That’s a useful number. It says nothing about what happens at 50% of rated torque, which is where a humanoid’s knee lives during a squat.

What the reducers actually specify

The reducer choice sets the slack. Strain-wave gears have no clearance in the tooth mesh by design; Harmonic Drive’s CSG-2A engineering data says so directly, and adds that a small backlash of “a few seconds of arc” can appear only when an Oldham coupling is used on the input. Cycloidal RV reducers get there with many teeth in contact at once: Nabtesco lists backlash and lost motion both under 1 arc-minute across the RV-N range. Precision industrial planetaries from Wittenstein and Neugart quote lost motion of about 4–6 arc-minutes in the Frontiers review’s comparison.

Then there are the actuators humanoid startups actually buy by the hundred. Low-ratio planetary modules built for legged robots trade precision for cost, back-drivability and impact tolerance, and it shows. CubeMars specifies 9 arc-minutes on its AKE80-8, an 8:1 quasi-direct-drive unit with 30 N·m peak torque. EYOU’s RP series of humanoid planetary actuators, ratios 19.5:1 to 25:1, lists up to 15.

The spring nobody advertises

Here’s the catch with strain-wave gears. They win on backlash and lose on stiffness, because the flexspline that makes them work is a thin, deliberately elastic steel cup. Harmonic Drive publishes stiffness as three slopes – K1 at low torque, K2 in the middle, K3 above the T2 point – and K1 is the softest. That’s the region around zero torque, exactly where a joint sits when it’s holding a light pose or reversing direction.

Run the catalog numbers and something neat falls out. Load a CSG gear to its T1 point and the output winds up by about 1.7–2.0 arc-minutes, whatever the size. At T2 it’s 4–6 arc-minutes, and almost exactly 5.3 for sizes 20 to 32, because Harmonic Drive scales stiffness with the torque rating. So a gear sold as “zero backlash” twists four to five times its lost-motion spec before it’s anywhere near its rated torque.

Cycloidal reducers are much stiffer for their size. Nabtesco rates the RV-25N at 61 N·m per arc-minute – about 210,000 N·m/rad – though it’s also a 245 N·m reducer, too big and heavy for most humanoid joints. That’s the real trade in the reducer war: RV gears are stiff and heavy, strain-wave gears are light and springy, planetaries are cheap and sloppy.

Why a walking robot cares

Arc-minutes sound tiny. They aren’t, once you put a lever on them. One arc-minute at a 0.6 m shoulder-to-hand reach moves the hand 0.17 mm. Fifteen arc-minutes moves it 2.6 mm, and that’s one joint – shoulder, elbow and wrist errors stack, so a cheap arm can put a fingertip half a centimetre from where the controller placed it. Fine for carrying a tote. Not fine for threading a connector.

Legs have a different problem. A stance leg constantly swaps between pushing and being pushed, and every swap crosses the dead band. Gear Technology lists the symptoms: delays and oscillations at the start or stop of a move, noise and vibration at every change of direction. For a balance controller, a joint that briefly goes slack is a joint that briefly isn’t there.

Stiffness sets a ceiling on how fast the controller can react. A joint spring and the limb’s inertia make an oscillator with a natural frequency of 1/2π × √(K/J). Put a 0.1 kg·m² load – roughly a 2 kg forearm and hand at 0.2 m – on a size-20 strain-wave gear and you get about 57 Hz in the soft K1 region and 86 Hz on the K3 slope. Push the control bandwidth toward that and the limb starts ringing.

Fixing it in software: the second encoder

That second sensor is increasingly standard. A dual-encoder actuator reads the motor shaft for commutation and the output shaft for the true joint angle. EYOU’s RP series pairs a 17-bit motor encoder with a 19-bit output encoder behind the planetary stage; Robstride offers dual-encoder versions of its QDD motors specifically for backlash compensation. The controller closes the position loop on the output encoder, so the slack stops being invisible.

The same pair of readings does a second job. The gap between the motor angle (divided by the gear ratio) and the output angle, run through a stiffness model, is a torque estimate – which is why joint makers such as Honpine pitch dual encoders as a cheap stand-in for a joint torque sensor. The catch is that the model inherits every nonlinearity in Table 1. Estimate torque in the soft K1 region with a single average stiffness and the answer can be off by the spread between K1 and K3 – more than a factor of two on the size-20 gear.

What to check on a joint spec sheet

• Which number is quoted: backlash, lost motion or hysteresis loss. They’re measured at different loads and aren’t interchangeable.
• Torsional stiffness at low torque, not just the headline value. For strain-wave gears, K1 is the one that matters when a joint reverses.
• Whether there’s an output encoder. Without one, backlash and wind-up are invisible to the controller.
• How the figure holds up with wear. Planetary backlash grows as teeth wear; ask for end-of-life values, not only new-out-of-the-box ones.

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