Industry Applications #06
Tolerance Allocation: Where Does Accuracy Leak Away?
An optical measurement station: every part on the drawings is marked ±0.02 mm, and the machine shop delivers every part within specification. Once assembled and measured, the position error is 0.12 mm.
Nobody did anything wrong, yet the result is wrong. Part of it comes from tolerance stack-up, and that is only the first of four accounts.
1. Two ways to stack, a factor of 2.2 apart
Five parts in a chain, each with a tolerance of ±0.02 mm. What is the total error?
Worst case
Assume all parts deviate to the same extreme at the same time:
T = Σ|tᵢ| = 5 × 0.02 = ±0.10 mm
Root sum square (RSS)
Assume the actual size of each part is normally distributed within its tolerance band, and the parts are independent of each other:
T = √(Σtᵢ²) = 0.02 × √5 = ±0.045 mm
Which one to use?
- Use worst case: for few parts (3 or fewer), for safety or interference questions (will it jam, will it collide), or for small batches without statistical meaning
- Use RSS: for many parts, volume production, and parts from stable processes
- Calculate both: the most common approach in practice. Set the design target with RSS, and use worst case to check "even if everything happens to deviate to the limit, will anything go wrong"
Note that RSS has an easily overlooked precondition: independence. If the five parts come from the same machine, the same tool and the same batch of material, their deviations are correlated and lean the same way, and RSS then badly underestimates. Be especially careful when a batch of parts is located from the same fixture datum.
2. Tolerance is not just machining tolerance
The calculation above covers only the dimensional chain. The accuracy budget of an opto-mechanical system has to include at least three other accounts.
2.1 Thermal expansion
The coefficient of linear expansion of the material times the length times the temperature rise gives the displacement. For a length of 300 mm and a rise of 5 K:
| Material | Coefficient of linear expansion α | Elongation |
|---|---|---|
| Aluminum alloy | 23 × 10⁻⁶ /K | 34.5 µm |
| Carbon steel | 11 × 10⁻⁶ /K | 16.5 µm |
| Invar | 1.2 × 10⁻⁶ /K | 1.8 µm |
An aluminum machine, a 10 µm accuracy requirement and a factory whose temperature changes by 5 K: this combination does not work even on paper.
This item comes up most often in complaints of "accurate in the morning, drifting by the afternoon". The temperature rise does not come only from the environment either: the servo motor itself is a heat source, and after a few hours of continuous running, structural parts near the motor are noticeably warmer than the room.
There are three common countermeasures, each with different costs and conditions: separating the measurement datum from heat sources, switching to low-expansion materials (Invar, carbon fiber, glass ceramics), or temperature compensation (measuring the structure's temperature and calculating a correction).
2.2 Deformation under load
Static drawing tolerances do not include dynamic deformation. Reaction forces when the stage accelerates bend the column, and the longer the overhang and the higher the acceleration, the more obvious the deformation. This item usually has to be estimated by finite element analysis (FEA), or overhung structures avoided at the design stage.
A practical test: if measuring the same position "at rest" and "just after stopping" gives different results, it is not a tolerance problem but a stiffness or settling time problem, which is covered in servo selection and motion profile planning.
2.3 Optical depth of field
This item is specific to opto-mechanical systems, and the one most often missed: the mechanism's tolerance stack in the Z direction must not exceed the lens's depth of field, or the image will go out of focus.
For a typical inspection station configuration (3.45 µm pixels, F/5.6, a permissible circle of confusion of two pixels, magnification 0.0845, i.e. a field of view of about 100 mm and a GSD of about 41 µm), the depth of field is about 11.7 mm. That is the total front-to-back range; as a ± value it is only ±5.85 mm. It sounds generous, but note that this budget has to cover:
- Fixture positioning repeatability
- The thickness tolerance of the part itself
- Flatness of the conveyor or stage
- Thermal and load deformation of the structure
If the total range of the four items (twice the ± values) adds up to more than the depth of field, image quality comes and goes, and the symptom is intermittent, which is harder to trace than a steady error. The full depth of field calculation is in how to configure a vision inspection station.
3. The accuracy budget table: merging four accounts into one
In practice we draw up a table listing every error source, estimate a number for each, and combine them by RSS:
| Error source | Estimate | Controlled by |
|---|---|---|
| Screw lead error (grade C5, thread length 500–630 mm) | 30 µm | Purchasing specification |
| Mechanical dimension chain (RSS) | 45 µm | Mechanical design |
| Thermal drift (steel, effective length 300 mm, 3 K rise) | 10 µm | Environment and layout |
| Encoder and servo positioning | 5 µm | Control tuning |
| Image measurement (half the GSD) | 20 µm | Optical configuration |
| RSS total | 59 µm | — |
The most useful thing about this table is not the 59 µm, but that it shows you which item dominates. In this example the mechanical dimension chain (45 µm) accounts for about 60 % of the sum of squared errors: upgrading the screw from C5 to C3 (30 µm down to 16 µm) brings the total only from 59 µm to 53 µm, while halving the dimension chain brings it down to 44 µm. Spend the budget on the largest item first.
That is the nature of RSS: the largest item dominates the result. To improve it, the most effective place to start is the largest item.
4. Where accuracy usually leaks away
Summarizing from real projects, the three most common leaks:
- Nobody keeps the overall account: the mechanical engineer manages mechanical tolerances, the control engineer servo accuracy, and the optical engineer the lens; each of them gets their part right, but nobody adds it all up
- The measurement datum and the machining datum differ: the datum surface on the drawing and the surface the part rests against during measurement are not the same, adding a conversion error nobody has calculated
- Only the static case was calculated, not dynamics or heat: at acceptance the machine is cold, parts are measured one at a time, slowly, and everything is fine; once on the line, running continuously at full cycle time, the numbers start to drift
None of the three is a technical challenge; they are process problems: what is missing is a budget table that puts everyone's numbers together.
Closing: tolerance is a budget, and someone has to manage the total
Think of tolerance as a budget and things become much clearer: the total is the accuracy the customer requires, each stage spends part of it, and when it is overspent someone has to decide where to save it back, whether by raising the machining grade, changing materials, changing the structure, or going back to the customer about the specification.
The worst case is not overspending but nobody keeping the books. If you only start looking for causes once assembly is finished and 0.12 mm is measured, few options are left.
