From Bearing and Range to a Grid Reference: the Target Location Error Budget
A sensor reports an angle and a range. Someone else has to act on a position. This paper assembles the error budget between the two and shows which term dominates at which range.
A surveillance system does not deliver angles. It delivers a place on the ground that somebody will drive to, illuminate or report up a chain. Between the sensor’s measurement and that place sits a chain of uncertainties, and the useful question is not how accurate any one component is but how large the resulting circle on the map turns out to be.
This paper builds that budget for the common case of a ranging sensor mounted on a positioner, and works a numerical example.
What contributes
Split the sources by what they affect.
The sensor measures range and angle. A laser rangefinder contributes a range uncertainty that is largely independent of distance — a few metres of timing resolution, near enough constant whether the target is at 2 km or 20. A radar contributes both a range uncertainty and its own angular uncertainty from beamwidth and processing.
The positioner contributes angle only. Its accuracy, its repeatability and any backlash add directly to the bearing and elevation reported with the measurement.
The structure contributes angle too. Mast or tripod deflection under wind tilts the whole assembly. The encoder cannot see it, so it is indistinguishable from a positioner error and adds to the same term.
Alignment contributes a fixed offset. Any residual misalignment between the sensor’s boresight and the positioner’s axes — and between the positioner’s zero and true north — appears in every measurement until it is surveyed out.
Combining them
Independent errors combine in quadrature, not by addition. For the angular terms:
Down-range uncertainty comes from the rangefinder and stays roughly constant. Cross-range uncertainty is the angular error multiplied by the distance, so the ellipse stretches sideways as the target gets further away.
The total angular uncertainty in each axis is the root sum of squares of the contributions:
σtotal = √(σ²sensor + σ²positioner + σ²structure + σ²alignment)
That angle becomes a distance by multiplying by range. For small angles the tangent may be dropped, provided the angle is expressed in radians:
cross-range error = R × σtotal (rad)
And the three orthogonal components combine the same way into a single figure:
TLE = √(σ²range + (R·σaz)² + (R·σel)²)
A worked example
Take a PES10 rotator carrying a ranging sensor on a 10 m mast. Use the published positioner accuracy of 0.042° for the worst case, a laser rangefinder with 5 m range uncertainty, an angular uncertainty of 0.05° from the sensor itself, and allow 0.05° for structural deflection in moderate wind and 0.02° for residual alignment.
Per axis:
σtotal = √(0.05² + 0.042² + 0.05² + 0.02²) = √(0.0025 + 0.00176 + 0.0025 + 0.0004) ≈ 0.084°
In radians that is 0.084 × π/180 ≈ 0.00147 rad. Now evaluate at three ranges:
| Range | Cross-range (per axis) | Range term | Total TLE |
|---|---|---|---|
| 500 m | 0.7 m | 5 m | 5.1 m |
| 2 km | 2.9 m | 5 m | 6.5 m |
| 5 km | 7.4 m | 5 m | 10.4 m |
| 10 km | 14.7 m | 5 m | 21.4 m |
The pattern is the point of the exercise. At 500 m the rangefinder dominates and the positioner is almost irrelevant. By 5 km the angular terms have overtaken it, and by 10 km they set the answer almost alone.
What this means for procurement
Specify at the range you care about. A requirement for “10 m target location error” is meaningless without a range attached. The example above meets it at 5 km and fails at 10 km with no change to the hardware.
Attack the dominant term. At short range, buying a better positioner changes nothing measurable — the money belongs in the rangefinder. At long range the reverse is true, and it is worth checking which of the angular terms is largest before assuming it is the positioner. Structural deflection frequently is.
Survey the alignment, then survey it again. Alignment is the cheapest term to reduce and the easiest to lose. It costs nothing but procedure, and it walks every time the head is removed for maintenance.
Ask what the structure contributes. A positioner accurate to 0.028° on a tripod that deflects 0.2° in wind is a system accurate to 0.2°. This is why our mast and tripod pages publish a wind rating against a stated payload area rather than a bare load figure — the deflection, not the strength, is what limits pointing.
Choosing components against the budget
Once the budget is written the component choice becomes arithmetic rather than preference. Our published positioner accuracies span 0.12° on the PED08 to 0.028° on the PED12, a factor of four; at 5 km that difference is 8 m of cross-range error, and at 500 m it is 0.8 m and invisible under the range term.
The three questions worth settling before a model is chosen: what range, what sensor, what structure. The positioner is usually the last decision, not the first.
Related reading: the terms used above are defined in accuracy, repeatability and resolution, and the case where the platform itself is moving is covered in sizing stabilisation.
Technical Discussion and Quotation
Tell us the platform and the constraint. Our engineering team answers with a configuration that fits it, usually within two working days.
