Residual registration error after a fitted rigid transformation

Structure coordinate data before fitting

For every observation, store substrate, face, layer, fiducial ID, nominal coordinate, measured X and Y, repeat number, repositioning state, camera setup, and timestamp. Use consistent units and face orientation. Preserve raw pixels only with calibration metadata.

Classify repeats made without movement, after repositioning, and on independently processed parts. They estimate different variation. Do not pool them merely to increase sample count.

Fit translation and rotation with designated points

Choose fit fiducials distributed across the field. Estimate the rotation matrix and translation that minimize the declared residual criterion. Document weights, excluded readings, and algorithm. A rigid fit does not include scale, shear, or local warpage.

Compute predictions for all points from the same fitted transform. Residual is measured coordinate minus prediction. Preserve signed vector components before calculating magnitude.

Estimate repeatability at stable fiducials

At one stable coordinate, calculate mean signed residual and sample standard deviation from independent repeats under the stated repeat condition. A confidence interval on the mean narrows with sample count only when independence is credible.

For mean residual 8 µm, standard deviation 6 µm, nine independent repeats, and illustrative t multiplier 2.306, the interval is approximately 8 ± 4.6 µm. That arithmetic does not establish production registration capability.

Test prediction with held-out fiducials

Do not include every fiducial in the fit. Predict reserved locations and calculate their residuals. A small fitted error can coexist with large held-out error when the model absorbs datum noise or the field contains scale or local distortion.

Place held-out points near field extremes and functionally critical relationships. Rotate their role across development trials if more coverage is needed, but keep each evaluation’s held-out status fixed before calculation.

Plot residual vectors across the field

Uniform vectors suggest remaining translation. Vectors changing direction around the centre suggest rotation. Opposite signs at opposing edges suggest scale. A quadrant or local cluster suggests distortion, damaged marks, or measurement effects.

Compare residuals with screen mesh, frame, substrate support, layer sequence, and camera field. A confidence interval at one fiducial cannot detect these spatial signatures.

Keep coordinate uncertainty contributors visible

List calibration scale, lens distortion, edge threshold, focus, illumination, fiducial quality, fixture seating, temperature, and algorithm. Estimate contributions where evidence exists and mark others unknown. Do not subtract measurement uncertainty from observed residual to improve the result.

Use repeated stable artifacts to monitor camera behaviour and repositioning trials to monitor fixture transfer. Process variation requires independent parts. Report each level instead of one pooled number.

Report fit, repeatability, and prediction separately

The table prevents a precise mean from being mistaken for broad field performance.

Test registration prediction outside the fitting set

Hold back fiducials at the field edge, near critical product features and at locations that challenge the assumed rigid model. After estimating translation and rotation from designated fit points, predict the held-out coordinates without refitting. Report each signed residual with its local measurement uncertainty and compare patterns across repeated prints or setups. If held-out residuals grow radially, scale or substrate movement may be present; if they change after screen reinstallation, datum repeatability may dominate. Do not add transformation terms merely to reduce training residuals unless the physical question and data coverage justify them. The calculation record should preserve fit-point selection, excluded observations, coordinate convention, software revision and raw measurements so another reviewer can reproduce both the fit and the independent prediction.

Release the calculation with raw coordinates

Control coordinate frames, face orientation, fit and check roles, camera calibration, edge rule, repeat condition, algorithm, weights, confidence level, t multiplier, residual maps, exclusions, and owner. Retain raw coordinates for independent recalculation.

Revalidate after fiducial, artwork, screen, substrate, fixture, camera, calibration, algorithm, layer, or environment changes. RFQ review requires drawing registration and feature coordinates. No capability index or tolerance promise is made without a suitable production study.

Check residual distributions for outliers, serial correlation, and coordinate-dependent spread before applying a confidence formula. A narrow interval derived from many autocorrelated frames can overstate information. When the normal-model assumption is doubtful, report the raw distribution and use a justified alternative rather than hiding the shape.

Keep confidence on the mean distinct from a prediction interval for a future observation. The mean interval estimates systematic residual at the sampled condition; it does not contain most individual production points. Specify the engineering question before choosing an interval.

For vector residuals, calculate X and Y intervals separately and preserve covariance when it matters. Converting every vector to positive magnitude changes its distribution and removes direction. Direction is often the clue distinguishing translation, rotation, scale, and local distortion.

An additional processed substrate contributes process information; another camera frame of the same stationary mark mainly contributes measurement information. Count these units correctly when stating sample size. The calculation record should identify the independent experimental unit, not merely the number of rows in a file. Archive the exact coordinate set and analysis code or formula revision. Independent recalculation should reproduce fit parameters, residuals, interval bounds, and excluded-point decisions without manual reconstruction. If software changes numeric precision or fitting defaults, treat the result as a method change and compare both outputs on the archived coordinate set.