BLOGS

Every joint datasheet now leads with a resolution figure — 19 bits, 0.001°, 0.022°. All of those numbers are measured with the joint unloaded, stationary and at room temperature. The moment a humanoid joint carries a load, the mechanism's own compliance writes an error two orders of magnitude larger. This article rebuilds the error budget from first principles and shows which terms actually decide where the hand ends up.

Something shifted in the robot-joint conversation this month. Encoder resolution moved from a footnote to a headline. Chinese engineering accounts spent late October publishing primer after primer on magnetic versus optical versus inductive feedback for humanoid joints, and one teardown-style post led with a number that reads like a specification: 0.022 degrees of resolution from a 14-bit magnetic encoder module, sealed against oil, water and dust.

It is a good number, honestly reported. 0.022° is 79 arc-seconds, and it comes from simple division: 360° over 16,384 counts. But put that module in a real joint and the joint will not position to 0.022°. It will position to whatever the reducer, the bearing, the housing and the load decide — and in a loaded joint that figure is usually measured in arc-minutes, not arc-seconds.

The gap between resolution and accuracy is not a trick of marketing. It is the difference between two questions: how finely can the sensor divide a revolution, and how far does the output actually move when the controller asks for one more count. Only the second one matters to a robot picking up a part.

Bits, Counts, and the Arc-Second

Start with the arithmetic, because most of the confusion lives there. An encoder's "bits" describe how many distinct codes it can emit per revolution. Convert that to angle and the numbers fall out immediately.

*Table: resolution arithmetic. One arc-second is 1/3,600 of a degree; one arc-minute is 60 arc-seconds.*

Two reference points make the table readable. A harmonic reducer's lost motion — the dead band you feel when you reverse direction — is typically quoted at about one arc-minute on good industrial units, and two to three arc-minutes on cheaper ones. That is 60 to 180 arc-seconds. A 19-bit encoder resolves 2.5 arc-seconds. The sensor is therefore 24 to 72 times finer than the mechanism it is watching.

That ratio is the whole story in one line. Resolution tells you the size of the ruler's markings. Accuracy tells you how far the ruler itself has slipped.

The Five Terms That Eat the Resolution

A joint's output error is not one number but a sum of independent contributions, each measured at the output flange. Five of them dominate in a humanoid joint, and only one of the five comes from the encoder.

*Table: order-of-magnitude error budget at the joint output. Terms are not independent — hysteresis and compliance interact — but the ranking holds across joint sizes.*

Read the table again and the design priority becomes obvious. The encoder term is the smallest row. Everything that decides real accuracy lives downstream of the sensor.

Term One: A Micron of Eccentricity Costs Eight Arc-Seconds

Optical and magnetic encoders both work by comparing a pattern on a rotating disc or magnet against a fixed reading head. If the pattern's centre does not coincide with the axis of rotation, the apparent angle wobbles once per revolution, and the wobble is not a rounding error.

For a small offset *e* at a reading radius *r*, the angular error is approximately *e/r* radians. At a 25 mm reading radius, one micron of eccentricity is 4×10⁻⁵ radians — 8.3 arc-seconds. Two microns, entirely normal for a pressed-in magnet ring on a production line, is 16.6 arc-seconds: that alone has consumed a 17-bit encoder's entire resolution, and it is 6.7 times the 19-bit count.

This is why an uncalibrated 19-bit magnetic encoder is not a 19-bit device in practice. Magnetic encoder ICs of the 12-to-14-bit class typically publish integral nonlinearity in the range of ±0.2 to ±0.4 degrees before factory calibration, and modules that reach single-digit arc-seconds do it by measuring the eccentricity once, storing a compensation table, and applying it in the drive.

The lesson is structural rather than vendor-specific: encoder accuracy is an assembly property. A sensor that is calibrated on a bench and then bolted to a joint with a different tolerance stack is a different sensor.

Term Two: The Reducer's Dead Band Is Not a Defect

Reverse a harmonic joint by a few arc-seconds and nothing happens at the output. The flexspline teeth are still taking up the gap they left on the way in. This is lost motion, and it is a property of the transmission, not of the control loop.

The numbers are well established across the industry: harmonic drives typically specify around one arc-minute of lost motion on precision grades, with hysteresis loss on the same order, and two to three arc-minutes on standard grades. Planetary gearboxes start at one to three arc-minutes for low-backlash grades, sit at roughly three to eight for precision general-purpose units, and go wider still for standard industrial models. Cycloidal and RV reducers reach about one arc-minute or better, at the cost of mass and a different stiffness profile.

For a controller this dead band has two consequences. First, any position command smaller than the lost motion produces no motion at all — which is why very fine resolution in the feedback path does not translate into very fine motion at the output. Second, the dead band is direction-dependent and load-dependent, so it cannot be calibrated away with a single offset. It can be compensated with a model, but the model has to know which way the joint is travelling and how much torque is in the teeth.

Wear moves the number. We covered the mechanisms in how a robot joint actually wears out — the short version is that lost motion grows with cycle count, and a joint specified at one arc-minute when new may be at two or three by the time it has finished its useful life.

Term Three: Where the Accuracy Actually Goes

Now the term that dwarfs everything else, and the one that almost never appears on a datasheet: torsional wind-up.

A geared joint is a spring. The flexspline, the output bearing, the flange, the shaft and the housing all deflect under torque, and the deflection is roughly proportional to load. Published torsional stiffness for a wrist-sized harmonic unit lands near 10⁴ newton-metres per radian at rated torque; larger units are stiffer, but the ratio of stiffness to rated torque stays in the same family.

Run the arithmetic with that stiffness. A 50 N·m load deflects the joint by 0.005 radian — 0.29°, or 17 arc-minutes. That is 1,030 arc-seconds. A 100 N·m load doubles it. Compared with the 2.5 arc-second resolution of a 19-bit encoder, the elastic term is roughly four hundred times larger.

*Table: elastic wind-up at the output, computed as torque divided by torsional stiffness. Real joints add hysteresis on top, so the unloaded return path is offset from the loaded one.*

This is not an argument against stiff joints; it is an argument about what a positioning specification means. An unloaded repeatability figure of a few arc-seconds is real and useful — it describes how consistently the joint returns to the same place with no load. But a task that involves pushing, lifting, or holding a tool is governed by the loaded curve, and the difference between a 10⁴ and a 5×10⁴ N·m/rad joint is a factor of five in where the tool actually sits.

If a joint datasheet gives you resolution and repeatability but not torsional stiffness, you cannot predict loaded accuracy from it. Stiffness, lost motion and rated torque are the three numbers that close the loop — resolution alone leaves the budget unfinished.

Term Four: Heat Moves the Zero

Thermal error is quieter than compliance but it arrives every time the robot works hard. The mechanism is differential expansion. If the encoder scale, the housing and the gear all heat by the same amount and share the same coefficient of expansion, the joint grows uniformly and the angle reading stays correct.

Real joints are not uniform. An aluminium housing at 23 ppm/K, a steel flexspline at around 11 ppm/K and a magnet ring bonded to a hub at yet another coefficient expand differently. The error is the mismatch in strain, so the radius cancels out: a five ppm/K mismatch under a 20 K rise is 10⁻⁴ radians of scale error — about 20 arc-seconds, whatever the joint's size. Longer duty cycles and larger temperature swings scale it linearly.

None of this is visible in a 25 °C bench test. It becomes visible in a joint that has been holding a 60% duty cycle for twenty minutes, which is exactly when a humanoid arm is doing productive work. We looked at the thermal budget itself in the thermal case for integrated joint design; the accuracy consequence is a separate line item that rarely makes it into the same discussion.

There is a second-order effect worth knowing. Many magnetic sensing schemes are temperature-compensated for the magnet's field strength but not for the mechanical growth of the mount, so the residual error appears as a slow angular drift that looks like controller integral wind-up and is often tuned out as such — masking the real cause.

Term Five: The Structure Is Part of the Sensor

The last term is the one no joint vendor controls: the arm. A 500 mm forearm with a realistic section deflects under its own weight and the payload, and that deflection appears as angular error at the wrist. Gravity sag of 50 to 500 arc-seconds is normal across the reach envelope of a humanoid arm, and it changes sign as the arm crosses the horizontal.

This matters for how you interpret the other four terms. The joint may be repeating to ten arc-seconds while the hand returns to a different place depending on pose, because the structure is doing the moving. A calibration performed in one pose does not transfer to another. This is the practical reason absolute accuracy in robotics is expensive: you are not compensating a sensor, you are compensating a kinematic model with elastic terms.

Motor-Side, Joint-Side, or Both

The placement of the encoder changes what the controller can know, and the trade is often misunderstood.

A motor-side encoder sits before the reducer. Its resolution at the output is divided by the gear ratio: a 19-bit sensor behind a 100:1 reducer appears to offer 0.025 arc-seconds at the output, a figure so far below the mechanism's own error that it is meaningless. What the motor-side encoder is genuinely good for is commutation and velocity control — it sees the motor directly, with no gear compliance in between, which is where a current loop wants its feedback.

A joint-side encoder sits after the reducer, on the output flange. It resolves the output directly, which makes it the honest choice for positioning: it includes the gear's error rather than hiding it. The cost is that it sits in the load path and must survive the torque, the temperature and the assembly tolerances of the output stage, and its eccentricity error is not attenuated by the ratio.

Dual encoders use both, and the interesting quantity is the difference between them. Multiply the motor angle by the inverse ratio, subtract the joint angle, and what remains is the transmission error — the reducer's lost motion and wind-up, measured live, in the loop. That difference signal is the basis for two capabilities: backlash compensation that knows which direction the teeth are loaded, and a software torque estimate, since the same deflection divided by stiffness gives the torque the joint is transmitting. We described the architecture in our analysis of dual encoders and the accuracy ceiling; the accuracy argument here is narrower and simpler: only a dual-encoder joint can tell you its own error while it is working.

The technique has limits worth stating plainly. The torque estimate is only as good as the stiffness model, the model changes with temperature, and hysteresis means the same deflection corresponds to two different torques depending on direction. It is a good sensor for control and a poor one for metrology.

What to Specify Instead of a Resolution Figure

If 2.5 arc-seconds of resolution does not predict where the hand goes, what does? Six numbers, all measurable, cover most of the ground:

• Unloaded repeatability, both directions. Ten approaches to the same point from each side, measured at the output. This is the honest ceiling on fine positioning.
• Lost motion, measured not quoted. Command a small reversal and measure the dead band at the flange, at operating temperature, after the joint has run for an hour.
• Torsional stiffness over the working range. A load-deflection curve, not a single number — the curve is not linear near zero torque because of the hysteresis loop.
• Loaded positioning error at rated payload. The only figure a customer can compare directly with their task tolerance.
• Thermal drift over a duty cycle. Hold a representative load for thirty minutes and log the zero.
• Dual-encoder tracking error, if both sensors exist. The difference signal is a free self-test: it should be repeatable and direction-consistent, and it should not grow with temperature.

A joint that publishes these six is telling you more than one that publishes twenty-three bits.

The Ti5ROBOT View

We build integrated joints, so this arithmetic is not academic for us: it is the reason the specification sheet for a module such as the Lightweight Joint Module series is written around stiffness, lost motion and thermal behaviour rather than a headline bit count, and the reason the Humanoid Robot Actuator line carries feedback on both sides of the transmission.

The design consequence is a chain of decisions that have to be made together. A stiffer output stage buys loaded accuracy but adds mass at the worst possible radius. A finer encoder buys resolution that the reducer's dead band will absorb unless the assembly is calibrated for eccentricity. A dual-encoder architecture buys self-knowledge, at the cost of a second sensor, a second cable path and a calibration step in production. There is no configuration that wins on all three; there is only a configuration that matches the task.

For a humanoid, that match usually means accepting that the wrist will not be as accurate as its encoder and making the joints that touch the world stiff, cool and well characterised — then using the difference signal to know, in real time, how far off they are.

The Number That Survives Contact

Resolution is a property of a sensor. Accuracy is a property of a joint, and loaded accuracy is a property of a joint plus its structure plus its thermal history. The 0.022° figure that circulated this month is a correct description of a 14-bit magnetic encoder doing exactly what it was designed to do. It is not a description of a robot.

The useful question is not how finely the joint can count, but how far it moves when it is asked, when it is loaded, and when it is warm. Answer that, and the bit count becomes what it always was: one term in a budget, and not the largest one.

Source: Chinese engineering coverage of humanoid joint encoder selection, October 2026 (magnetic, optical and inductive feedback; 14-bit magnetic module at 0.022° resolution); lost-motion and hystereseis figures stated as typical grades from published reducer catalogues rather than a single vendor; torsional stiffness treated as an order-of-magnitude value for wrist-sized harmonic units; encoder eccentricity, thermal scale error and wind-up computed from standard angular-error arithmetic and reproduced in the tables; earlier Ti5ROBOT articles on joint wear, thermal limits and dual-encoder architecture.