BLOGS

Every robot joint receives two loads, not one. The first is torque about its own axis, and it gets a number on the datasheet, a curve in the catalog and a line in every sizing spreadsheet. The second is bending moment across that axis, produced by a payload hanging off the end of a link, and it usually gets nothing at all. Yet in humanoid ankles it routinely exceeds the joint's rated torque, and it arrives at the one component in the assembly that was never designed to resist it: the gear.

The result is a class of joint that passes every torque calculation, holds position perfectly on the bench, and fails in the field within a season. Not because the reducer was undersized for torque, but because nobody drew the second load path.

Two Loads, One Datasheet

Start with the geometry, because the geometry is the whole argument. A joint transmits torque through its axis. A link attached to that joint, carrying a mass at some distance, applies a force that is offset from the axis. That offset converts force into moment:

M = F × L

where F is the force at the attachment point and L is the perpendicular distance from the joint axis to the line of action. Torque and moment are both expressed in newton-metres, which is exactly why they get conflated. They are not the same quantity. Torque is reacted by the gear teeth. Moment is reacted by whatever structure holds the output flange square to the housing — and if nothing does, it is reacted by the gear teeth too, in a direction they cannot survive.

For a humanoid, the numbers are not marginal. Take a 70 kg machine, a body weight of roughly 690 N, and an ankle whose centre of pressure sits 80 to 100 mm forward of the joint axis during a normal step:

Read the ankle rows again. A joint rated at 40 N·m of continuous torque — a perfectly reasonable number for a humanoid ankle — sees around 200 N·m of bending moment on a hard landing. The moment is five times the torque rating, and it is not an abuse case. It is the design case.

This is the first place joint specifications mislead. A module can be honestly rated for 40 N·m continuous, 120 N·m peak, and still be structurally marginal in an ankle, because neither number describes its ability to resist a moment. The missing specification is not an oversight in marketing. It is missing because the moment is not supposed to reach the gear at all.

What a Moment Actually Does Inside a Joint

Follow the load from the foot upward. The ground reaction force enters the foot plate, travels through the ankle housing, and reaches the joint as a force applied some distance from the axis. Inside the joint, that force arrives at the output flange.

In a strain-wave joint, the output flange is bolted to the flexspline — a thin-walled alloy cup with teeth cut into its open end. It is radially compliant by design; that compliance is the mechanism. It is torsionally stiff, which is why it works as an output. It is not, in any useful sense, a structural member for bending. Load it with a moment and the cup ovalises asymmetrically, the tooth engagement zone shifts away from the major axis, and contact pressure concentrates on a fraction of the teeth that were supposed to share it.

The failure sequence is consistent enough to be predictable:

1. Tooth contact migrates. The engaged region, normally symmetric about the major axis, shifts toward the loaded side. Local contact stress rises faster than the applied moment because the sharing area shrinks at the same time.
2. Ratcheting. Beyond a threshold, the teeth skip rather than mesh. In component-set catalogs this threshold is given as roughly eight to nine times rated torque — a general figure that shifts with frame size — and it is a torque figure applied to the output. A moment converts into an equivalent tooth load far more efficiently than designers expect.
3. Flange distortion and lost concentricity. The flexspline stops running concentric with the circular spline. Backlash appears where there was none, and it appears progressively, which is worse than a manufacturing clearance because it keeps growing.
4. Fatigue at the cup root. Cracks initiate at tooth roots or at the stress concentration near the flange, exactly where a thin cup under reversed bending wants to crack.
5. Buckling. At sixteen to seventeen times rated torque, with the input held, the cup can deform plastically. This is not a wear-out mode. It is a single event.

In field terms, the symptom is a joint that develops backlash in a few months, gets noisy, then loses accuracy — and the reducer gets blamed for a load it should never have seen. The gear did not fail because it was weak. It failed because it was the only thing in the assembly stiff enough to resist the moment, so it did.

Why Span Beats Size

The fix is structural and almost embarrassingly simple: put two bearings between the output flange and the housing, spaced apart along the axis, so the moment is reacted as a force couple rather than by the gear.

The statics are worth writing down because they explain every packaging decision that follows. If the moment M is reacted by two bearings separated by a span s, the reaction force at each bearing is approximately:

F = M / s

For the ankle case, with M = 206 N·m:

Two conclusions fall out immediately. First, the reducer cannot be part of this path: a 40 mm span handing 5 kN to a flexspline flange is not a design, it is a countdown. Second, span is cheaper than size. A pair of bearings separated by 80 mm carries the same moment with half the reaction force of a pair at 40 mm, and reaction force is what drives bearing life.

Stiffness scales even more favourably. For a pair of bearings each with axial stiffness k, separated by span s, with rigid flanges and equal stiffness in both bearings, the tilting stiffness of the pair rises roughly with the square of the span:

K ≈ k · s² / 2

Double the span and you get four times the tilting stiffness, at the cost of axial length and mass. This is why the output end of a well-designed joint looks like a short, wide cylinder with a large bearing at each end, and why the gear sits in the middle of the housing doing only the job it was designed for.

The trade is real and it is not free. Every millimetre of span pushes the joint's mass outward from the body, which raises the reflected inertia of the link it drives. On a distal joint that penalty is paid twice: once in the actuator's own mass, and again in the torque the proximal joints need to swing it. Span is cheap compared with bearing size, but it is not free compared with nothing.

Crossed Roller or a Pair of Angular Contacts

Once the moment has its own path, the question becomes which bearings form it. Two architectures dominate, and the choice is a genuine trade rather than a hierarchy.

The crossed-roller ring wins on stiffness per millimetre, and that is usually the deciding factor in a joint where the output bearing is also the structural interface between two links. Its weakness is not mechanical but procedural: the ring is only as stiff as the surface it is bolted to. A housing face that is flat to 0.02 mm instead of 0.01 mm produces uneven preload, and uneven preload means the moment is carried by a subset of the rollers while the rest idle. The bearing then fails early, and the post-mortem says "bearing defect" when the real defect was the fixture that machined the housing.

The paired angular contact arrangement is more forgiving to manufacture and better at speed, and in a joint that spins fast — a wrist on a high-throughput arm, a spindle-adjacent axis — that advantage can outweigh the width penalty. It also lets you tune preload after assembly, which matters when the housing and the bearing are made of materials with different expansion coefficients.

Preload grade is a system decision, not a bolt-on. Light preload keeps drag torque low, which preserves backdrivability and keeps a force-controlled joint honest. Heavy preload maximises stiffness under moment but adds friction that the torque loop must overcome, and friction that varies with temperature is worse than friction that is merely high.

The Stiffness Chain and the Deflection Nobody Budgets

Here is where moment load stops being a durability question and becomes an accuracy question.

A joint's angular deflection under moment is θ = M / K, where K is the tilting stiffness of the whole load path — bearing, housing, flange, and the bolted joint between them. That deflection is not a clearance. It does not reverse cleanly, it is load-dependent, and it sits downstream of the encoder.

Take a joint with a tilting stiffness of 2 × 10⁵ N·m/rad — the right order of magnitude for a mid-size output bearing, with the full chain including the bolted flange lower than that — carrying the 235 N·m produced by a 20 kg payload on a 1.2 m lever:

• θ = 235 / (2 × 10⁵) = 1.18 × 10⁻³ rad ≈ 0.067° ≈ 4 arcmin
• At a 1.2 m lever, that is 1.4 mm of tool deflection

Now compare that with the number usually quoted for the joint: backlash, measured in arcminutes and typically an order of magnitude smaller. The specification sheet says the joint is precise to a few arcminutes. Under load, it is not. The bearing is deflecting more than the gear is backlashing, and no amount of gear quality fixes it.

This is the part that surprises control engineers. A dual-encoder joint measures motor-side and output-side angle, and that pair is genuinely powerful: it sees gear torsion, it sees lost motion, it can compensate for both. But it cannot see deflection that happens outside its own measurement boundary. If the output encoder is on the flange and the link is bolted outboard of the bearing, the tilt between flange and link is invisible to the loop — unless the encoder is deliberately mounted across the bearing, on the link side, which some high-accuracy joints do. The controller will otherwise happily command a position the structure cannot hold, and the error will look like an unmodelled disturbance.

For a fuller treatment of what a dual-encoder pair can and cannot see, see the loaded error budget of a robot joint. The tilt term belongs in that budget, and it is usually the largest single line in it at the distal joints.

The stiffness chain also has a soft link that is almost always the bolted interface. A flange held by eight M5 bolts at 8 N·m develops a clamped joint whose stiffness depends on face contact area, bolt preloading, and whether the faces were flat when they were machined. Lose preload — through relaxation, through thermal cycling, through a washer that was not hardened — and the joint loses stiffness long before anything visibly loosens. The bearing is fine. The gear is fine. The interface between them is breathing.

The 10/3 Exponent: How Moment Load Eats Life

Moment load does not only bend the structure; it consumes bearing life at a rate that is easy to underestimate. The standard rolling-bearing life relation is:

L₁₀ = (C / P)^p × 10⁶ revolutions

where C is the dynamic load rating, P is the equivalent load, and p is 3 for ball bearings and 10/3 for roller bearings. The exponent is where the trouble lives. Because P includes the reaction force generated by the moment, a modest overload produces a disproportionate life penalty:

A 20% underestimate in the moment — a slightly longer foot, a slightly higher landing impulse, a slightly heavier tool — halves the bearing's rated life. A 50% underestimate cuts it to a quarter. There is no graceful degradation curve here: the moment does not have to be wrong by much to be wrong by a lot.

Two practical consequences follow. First, the moment used for bearing sizing should be the worst credible case, not the nominal gait. Landing impulses, stumble recovery, and a person leaning on the machine all belong in the number. Second, the static safety factor deserves its own check, separate from life. Static capacity governs brinelling under shock; a joint that survives a million walking cycles can still be dented by one fall.

Life calculations and torque calculations answer different questions and should not share an input. The torque sizing question is "can the gear transmit this?" The bearing sizing question is "how long does the structure last while the load tries to bend it?" Mixing the two produces a joint that is correctly geared and structurally under-built.

Preload, Temperature and the Quiet Drift

Moment stiffness is not a constant. It drifts with temperature, and the drift is systematic rather than random.

A steel bearing ring in an aluminium housing is a thermal mismatch by construction. Aluminium expands at roughly 23 × 10⁻⁶ per kelvin, bearing steel at about 11.5 × 10⁻⁶. Across a 100 mm housing at a 40 K temperature rise, the differential is:

(23 − 11.5) × 10⁻⁶ × 40 × 0.1 m ≈ 46 µm

Forty-six microns is larger than the preload clearance in most precision bearing sets. Warm the joint and the housing grows away from the ring, preload falls, stiffness drops, and the deflection under moment increases — precisely when the joint is working hardest and dissipating the most heat. Cool it down and preload returns, sometimes with interest if the housing was over-constrained.

This is why preload cannot be specified as a single assembly value and forgotten. It has to be specified as a range that holds across the joint's operating temperature, and the housing material, the bolt pattern and the bearing grade all participate in that range. It is also why a joint that measures beautifully on a cold bench can feel vague after twenty minutes of walking.

The same logic explains a common field complaint: a joint that is stiff and precise when new, and noticeably softer after a few hundred hours. Nothing has worn out in the usual sense. Preload has relaxed, the bolted interface has settled, and the stiffness chain has lost a few percent at every link. The gear is still zero-backlash. The structure is no longer as square as it was.

A Six-Line Check Before the Drawing Is Signed

The whole argument compresses into a short procedure that can be run before a joint design is frozen, and it costs less than an hour.

1. Compute the moment at every joint for the worst credible case. Payload mass times lever for the arm; ground reaction force times centre-of-pressure offset for the leg; add the collision case. Do not use nominal gait.
2. Convert to bearing reaction. F = M / s for the span you have drawn. If the number exceeds the bearing's dynamic rating by a comfortable margin, you are not sizing bearings, you are guessing.
3. Confirm the gear sees none of it. For a component set, the permissible moment at the output flange is effectively zero — the catalog will tell you the output must be externally supported. If your design routes moment through the flexspline, stop and redraw.
4. Check the stiffness chain, not just the bearing. Bearing tilt plus housing deflection plus flange bending plus bolted-joint compliance, summed as a series. The softest link sets the total, and it is usually not the bearing.
5. Budget the deflection against the accuracy target. θ = M / K, then δ = θ × L at the tool or the foot. If the deflection is a large fraction of your accuracy budget, the joint is not accurate under load no matter what the repeatability figure says.
6. Set preload for the temperature range, and check static safety separately. Life uses the 10/3 exponent; shock uses the static rating. Both need the moment, and neither gets it from the torque curve.

A joint that passes all six is not necessarily a good joint. But a joint that skips line one is a joint whose failure mode has already been decided, and only the date is open.

What This Means for Joint Selection

The practical lesson for anyone specifying a joint module is that the output interface is a structural specification, not a mounting detail. The diameter of the output bearing, the span between its rows, the flatness of the housing face and the preload grade together determine how much of the machine's accuracy survives contact with a load. Torque density tells you what the joint can lift. Moment stiffness tells you whether it can hold what it lifted.

That is the reasoning behind the output-end architecture used across Ti5ROBOT's lightweight joint modules: a large-diameter output bearing carries the moment, the gear carries only torque, and the housing is machined as a single setup so that the bearing seat and the gear mounting face share an axis rather than being aligned to each other afterwards. The same principle scales into the humanoid actuator range, where the ankle and knee joints see the largest moments in the machine and have the least room to grow.

It also connects to two effects that are often discussed separately. The gear's own contribution to precision is bounded by the wave generator, as covered in what sets harmonic precision and life. The dynamic penalty of putting mass at the end of a link is covered in the reflected-inertia tax. The moment belongs in the same conversation, because it is the load that decides whether the other two ever get a chance to matter.

Draw the second load path. It is one line on the section view, and it is the difference between a joint that is specified correctly and a joint that is specified only for the load someone remembered to write down.