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A well-defined resistance-versus-position curve does not always provide a precise position from a measured resistance. A shallow segment amplifies reading error; a plateau represents many positions; a reversal may give two separated positions with the same output. The receiving system must preserve these distinctions. The engineering task is to identify the complete position set compatible with the observation and the calibration evidence, not simply to return whichever numerical root a solver finds first.
Key design decisions
- Define the electrical observation at the same loading, temperature and contact boundary as the calibration.
- Invert the response interval over the complete valid position domain, preserving separate branches when necessary.
- Distinguish a deterministic compatibility bound from a statistical calibration interval; do not attach a confidence percentage without its model.
1. Decide what the inverse result should mean
Let the forward calibration f(x) describe resistance at position x under specified conditions. The inverse question starts with a new resistance observation and asks which positions could have produced it. Keep this separate from choosing the forward table's breakpoints: the inverse may be ambiguous even when that table represents the intended curve accurately.
State whether the output is card travel, shaft angle or an installed quantity obtained through another mechanical relationship. Do not silently convert card travel into tank volume or valve position. Name the valid domain and the meaning of an out-of-domain result. A last-value hold or endpoint clamp is a controller policy, not an additional calibration observation.
2. Use reciprocal slope only where the local inverse exists
On a smooth segment with nonzero slope, a small resistance change produces an approximate position change equal to the resistance change divided by the local slope. Units reveal the operation: ohms divided by ohms per millimeter gives millimeters. A steeper output segment is less sensitive to a given resistance error in this inverse sense.
For an assumed reading bound of plus or minus 0.4 ohm, a slope of 8 ohms per millimeter gives approximately plus or minus 0.05 millimeter. A slope of 0.2 ohm per millimeter gives plus or minus 2 millimeters. The approximation must remain inside a segment where the slope is represented adequately. At zero slope, dividing by a small artificial number does not recover position information.
Δx ≈ ΔR/f′(x); Xcompatible = {x ∈ D : Rlow ≤ f(x) ≤ Rhigh}
- f(x) is the declared forward resistance curve; f′ has units of ohms per position unit.
- D is the valid physical position domain.
- [Rlow,Rhigh] is a defined resistance interval after the explicitly included observation/model bounds.
- Xcompatible is a set and may be an interval, several intervals, the entire plateau or empty.
The reciprocal-slope expression is local and requires nonzero slope. The set expression is deterministic interval inversion of the stated curve, not automatically a probabilistic confidence statement.
3. Invert the interval before rounding a curved result
Assume a demonstration curve R(x)=100+x², with resistance in ohms and x in millimeters over the nonnegative domain from zero to ten millimeters. A reading of 125 ohms gives a nominal position of 5 millimeters. If the total stated resistance compatibility interval is 116 to 134 ohms, the compatible positions are 4 to approximately 5.831 millimeters.
The interval is not symmetric around 5. Its lower distance is 1 millimeter and its upper distance is approximately 0.831 millimeter. A local slope calculation would use 10 ohms per millimeter and produce plus or minus 0.9 millimeter, which is only an approximation. Exact interval inversion is straightforward here and makes the nonlinear asymmetry visible. These assumed numbers are not a production calibration or a sensor accuracy claim.
4. Preserve plateaus and separated branches
If a defined curve is exactly 200 ohms from 4 to 6 millimeters, an exact reading of 200 ohms cannot distinguish any position in that region using resistance alone. Adding more decimal places to the lookup table does not change that fact. A compatible reading interval may also extend into the neighboring sloped regions, so include them rather than returning only the plateau midpoint.
If the earlier quadratic example were instead valid from minus ten to plus ten millimeters, 125 ohms would have roots at minus five and plus five millimeters. The 116-to-134-ohm interval would correspond to two separated ranges: approximately minus 5.831 to minus 4, and 4 to 5.831 millimeters. A previous position estimate may support branch selection only through a separately justified motion/history rule; it is not contained in the instantaneous resistance.
5. Keep observation and calibration uncertainty in the same problem
A narrow repeatability band from repeated readings does not include every uncertainty in the forward curve. The curve may have estimated coefficients, fixture-position uncertainty, interpolation residuals or specimen-to-specimen variation. State which of these apply to the intended use before deriving the inverse interval.
For a statistically fitted calibration, uncertainty in the estimated response and in a new observation both matter. Inverting a fitted mean curve alone is not the same as obtaining a calibration interval for the new reading. If using deterministic worst-case bounds instead, define their coverage and combination rule without labeling the result 95 percent. The interval arithmetic examples here intentionally use stated compatibility bounds, not a fitted probability model.
6. Return an explicit result status when inversion is inadequate
The receiving interface should distinguish an ordinary estimate from an ambiguous or unsupported value. This lets the equipment designer decide how to respond without confusing a numerical convention with measured position.
| Mathematical result | Engineering interpretation | Information needed next |
|---|---|---|
| One narrow connected interval | Position is locally identifiable within the stated bounds | Check against the allowed position error |
| One wide interval in a shallow region | Reading contains limited position information | Revisit electrical uncertainty or the required curve sensitivity |
| A plateau or multiple intervals | One reading does not select one position | Independent position, direction or justified state information |
| Empty compatible set inside the valid domain | Reading and model are inconsistent under the stated bounds | Check connection, operating state and calibration identity |
| A root exists only outside the calibrated domain | Numerical extrapolation is not supported calibration | An explicit extended-domain study or invalid-result policy |
7. Verify the inverse with cases that challenge its logic
Test interval endpoints, table joins, exact plateaus, decreasing segments and observations just outside the valid response range. Compare returned position sets against the forward curve: every returned boundary should map to the appropriate response boundary or physical domain limit. This round-trip check detects swapped inequality signs on decreasing branches.
Include numerical precision and lookup quantization in the implementation review. A rounded pair of distinct resistance breakpoints can become equal and accidentally create a plateau; reversed point ordering can create a false branch. Preserve the validated forward data and inversion algorithm revision together so a firmware change cannot silently reinterpret an unchanged card specification.
8. Validate inferred position against an independent coordinate
Compare the inverse result with independently observed physical position at selected challenging regions, not only at the calibration points. Include forward and reverse travel when contact history or mechanism backlash can change the relation. Do not merge those branches into one average curve unless that operation still meets the position requirement.
Record when the physical reference falls outside the predicted compatible set and investigate the missing contribution. A persistent offset after remounting points toward coordinate transfer; a direction-dependent difference suggests state dependence; isolated electrical jumps may not belong to a smooth curve at all. The final inverse specification should make these failure states visible rather than smoothing them into an apparently precise position.
Review a resistor-card inverse-position requirement
Provide the forward curve and reading uncertainty so usable position information can be assessed across the full travel.
- Validated forward curve, interpolation rule, physical domain, coordinate origin and units.
- Electrical boundary, loading, temperature, wiper condition and the basis of the reading/model intervals.
- Required inferred-position accuracy, allowed ambiguous states and any independent branch-selection information.
- Raw independent position comparisons, direction history and lookup/firmware representation.
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