How to Calculate Crossed Roller Bearing Life for Robot Joints

How to calculate crossed roller bearing life for a robot joint comes down to one equation and an honest load case. The equation is the ISO 281 rated-life formula, and the load case is where most calculations go wrong, because a robot joint does not push a bearing in one clean direction. It loads the bearing radially, axially, and in overturning moment at the same time, and it reverses constantly. Get the equation right and the load case wrong and the answer is precise fiction.

This article works through the calculation with a real bearing from the Yuanhe RAU range, using the published ratings from the crossed roller bearings catalog: RAU 5008, with a 50mm bore, a basic dynamic radial load rating C of 5.10 kN, and a static rating C0 of 7.19 kN. Every number in the worked example follows from those published values plus the formulas below.

01What L10 Life Actually Means

L10 is the basic rating life defined in ISO 281: the number of revolutions that 90 percent of an identical group of bearings will complete before the first sign of material fatigue appears. It is a statistical statement about a population, not a promise about one bearing. Half of a group can be expected to run several times longer than the L10 value, and a small share will fail earlier.

Two consequences follow for a robot builder. First, the L10 result is a design tool for comparing bearings and load cases, not a warranty hour count. Second, the rating assumes conditions that a robot joint only partly meets, which is why the oscillation caveat at the end of this article matters as much as the formula.

02The Life Equation

The basic rating life is calculated as follows, per ISO 281:

The exponent is where crossed roller bearings differ from ball bearings in practice. Line contact spreads the load, which raises C for a given envelope, but it also makes the life curve steeper: the p = 10/3 exponent means a 20 percent increase in load cuts the calculated life by roughly half. Load assumptions deserve scrutiny before the arithmetic.

03Build the Load Case From Radial, Axial, and Moment Loads

A crossed roller bearing carries radial loads, axial loads in both directions, and overturning moment loads in a single ring set, with cylindrical rollers arranged at 90-degree alternating angles in V-shaped raceways. The load case for a robot joint therefore has three columns. The radial component comes from link weight and tool forces at the bearing centerline. The axial component comes from payload direction changes and any thrust in the mechanism. The overturning moment comes from the tool tip offset, and in most robot joints it is the largest column of the three.

To fold the moment into the calculation, a first estimate treats it as an axial force couple acting at the roller pitch diameter dp:

This estimate is a sizing tool, not the final word. Raceway geometry and roller distribution shift the exact relationship, which is why a full rating check on a moment-dominated joint is confirmed with the bearing maker. But the estimate shows the scale of what the moment does, and scale is what catches undersized bearings early.

04Worked Example With RAU 5008 Under a Radial Load

Take a joint where the radial component dominates and the axial and moment contributions are small. The equivalent load P equals the radial load Fr. With Fr = 1.5 kN against C = 5.10 kN, the calculation ledger reads as follows.

The same bearing at 2.5 kN tells the story of the steep exponent: the load ratio drops to 2.04 and the calculated life falls to about 1.08 x 10^7 revolutions. A 67 percent higher load removes more than 80 percent of the calculated life. When a joint specification is close to the edge, every tenth of a kilonewton in the load case matters.

05From Revolutions to Working Hours

Robots are rated in hours or years, so the revolution count has to be converted. The conversion is:

The phrase to underline is average speed. A robot axis does not hold one speed, so n should be the time-weighted mean over the duty cycle, and the load P should be the equivalent load over that same cycle. For a variable load spectrum, ISO 281 gives the standard way to combine partial loads and times into one equivalent load P, weighting each load by its share of time and by P raised to the p power. Averaging the loads without the exponent quietly understates the real damage.

06Oscillation and Short Strokes Change the Answer

ISO 281 assumes continuous rotation, and a robot joint does not rotate continuously. A wrist axis sweeps a limited arc and reverses. The first correction is bookkeeping: count the cumulative angle. A joint sweeping 90 degrees each way through 12 cycles per minute turns 180 degrees per cycle, which is 0.5 revolution, so it accumulates 6 equivalent revolutions per minute, or 5,760 per 16-hour day.

Applied to the 1.5 kN case, 5.91 x 10^7 revolutions lasts about 10,000 working days, far beyond the life of the robot. Applied to the 2.5 kN case, 1.08 x 10^7 revolutions lasts about 1,870 days, roughly 5 years of single-shift service. Oscillation moves the answer from absurd to relevant.

The second correction is physical. Under short-stroke oscillation the same raceway arc carries every cycle, the grease film is squeezed and restored in one spot, and slow micro-slip at the roller contacts can frett the raceway long before classical fatigue would appear. Accuracy loss, not spalling, is usually what ends a robot bearing’s useful life. The standard practice is to treat the L10 result as an upper bound, apply a service factor, and check the failure mechanisms that oscillation accelerates. Our guide to crossed roller bearing failure modes and prevention lists what those mechanisms look like in service.

07The Static Check and the Safety Margin

Rated life handles repeated load. Peak load needs a separate static check, because a single overload can dent a raceway even when the average load is mild. The static safety factor is S0 = C0 / P0, where P0 is the peak equivalent load and C0 is the static rating, 7.19 kN for the RAU 5008.

In the moment example above, a 60 Nm overturning moment corresponds to about 1.05 kN of raceway force, far below C0, so the static margin is comfortable. A moment of several hundred newton meters on the same bearing would move the check to the center of the conversation. For a robot joint, include the worst-case stall or collision load in the static check, not the nominal cycle load. If the peak sits close to C0, the bearing envelope needs to grow or the load path needs to change.

08Use the Result and Confirm With the Maker

The calculated life is a screen, not a verdict. Compare the result against the target service life of the robot, and when the margin is thin, question the load case before the bearing. The steep p = 10/3 exponent means the load assumptions deserve the same care as the arithmetic, and the inspection grade of the bearing, P5, P4, or P2, shapes how uniformly that load is shared around the raceway. The grades are explained in our article on crossed roller bearing accuracy grades and what P5, P4, and P2 mean.

For a moment-dominated joint, send the load spectrum to the bearing maker for the final rating check. The RAU catalog values are the published starting point; the raceway-level calculation for a combined load case is the maker’s job, and the RAU series crossed roller bearings span bores from 20mm to 350mm in standard grades P5, P4, and P2, with every bearing shipped with dimensional inspection data measured at four quadrant positions.

09Frequently Asked Questions

10Key Points Before You Calculate

• L10 = (C / P) ^ (10/3) x 10^6 revolutions, with P as the equivalent load over the full duty cycle.
• Fold the overturning moment into the axial column as M / dp before comparing loads.
• Convert revolutions to hours with the time-weighted average speed, and weight variable loads with the p exponent.
• Check the worst-case peak against C0, and apply a service factor for oscillating short-stroke duty.