Measurement principles and uncertainty

Why this matters

A drone knows everything about itself through sensors. If a sensor reads slightly wrong, the controller decides on wrong numbers, and the result is a drone that will not hold still, an altitude that drifts, or survey data that cannot be used. This course teaches how each type of sensor measures, how it can be wrong, and how to calibrate it.

The whole course uses one hypothetical case: a lab building a fixed-wing VTOL drone to survey farm plots, which must calibrate its whole sensor set before mission flights. All numbers are made up for practice unless a source is given. The Python code for every module can be downloaded from /downloads/uat-205/.

The measurement chain

A single measurement passes through several stages. The sensor converts a physical quantity, such as temperature or pressure, into an electrical signal. Signal conditioning amplifies and filters it, the ADC converts it to a number, and software applies calibration to convert it back into the desired unit. Every stage can add error, so a complete measurement result always has both a value and an uncertainty (Fraden, 2016).

Characteristics to read from the datasheet before choosing a sensor:

• Range: the lowest to highest measurable value; the top of the range is full scale (FS).
• Resolution: the smallest change the system can distinguish, which is not the same as accuracy.
• Sensitivity: output change per unit input, such as mV/°C.
• Nonlinearity: deviation from the straight line used to represent the sensor, often given as %FS.
• Bandwidth and response time: how quickly the sensor follows changes.

The calibration curve

Calibration compares sensor readings with reference values of known uncertainty, then finds the relationship used to correct readings. The simplest method is two-point, using the two end points. Least squares uses every point to find the line with the smallest sum of squared errors, as described in the NIST statistics handbook.

Once the equation is found, the uncertainty of the result is needed. The GUM (JCGM 100:2008), clause 5.1.2, gives the law of propagation of uncertainty: if $y=f(x_1,x_2,\dots )$ and the inputs are uncorrelated, the combined uncertainty is $u_c^2=\sum c_i^2{u}^2(x_i)$, where $c_i=\partial f/\partial x_i$ is the sensitivity coefficient, telling how strongly that input affects the result.

import math
import statistics as st

T_REF = [0, 10, 20, 30, 40, 50]                   # °C from the reference thermometer
V = [0.502, 0.598, 0.701, 0.797, 0.905, 1.001]     # V from the sensor
FS = 50.0                                          # °C full scale
U_V, U_REF = 0.002, 0.10                           # V, °C standard uncertainties

two_point = ((V[-1] - V[0]) / (T_REF[-1] - T_REF[0]), V[0])
mt, mv = st.mean(T_REF), st.mean(V)
gain = sum((t - mt) * (v - mv) for t, v in zip(T_REF, V)) / sum((t - mt) ** 2 for t in T_REF)
least = (gain, mv - gain * mt)

for name, (a, b) in (("two-point", two_point), ("least squares", least)):
    res = [(v - b) / a - t for t, v in zip(T_REF, V)]
    print(f"{name:<14} {a * 1000:.3f} mV/°C, offset {b:.4f} V, residuals "
          + " ".join(f"{r:+.2f}" for r in res) + f" °C, worst {max(map(abs, res)) / FS:.2%} FS")

a, b = least
res = [(v - b) / a - t for t, v in zip(T_REF, V)]
u_fit = math.sqrt(sum(r * r for r in res) / (len(res) - 2))
c_v = 1 / a                                        # °C per V (sensitivity coefficient)
u_c = math.sqrt((c_v * U_V) ** 2 + U_REF ** 2 + u_fit ** 2)
print(f"c_V = {c_v:.1f} °C/V, u from voltage {c_v * U_V:.2f} °C, u_fit {u_fit:.2f} °C")
print(f"combined u_c = {u_c:.2f} °C, expanded U = {2 * u_c:.2f} °C (k = 2)")

two-point      9.980 mV/°C, offset 0.5020 V, residuals +0.00 -0.38 -0.06 -0.44 +0.38 +0.00 °C, worst 0.88% FS
least squares  10.034 mV/°C, offset 0.4998 V, residuals +0.22 -0.21 +0.05 -0.38 +0.38 -0.05 °C, worst 0.76% FS
c_V = 99.7 °C/V, u from voltage 0.20 °C, u_fit 0.31 °C
combined u_c = 0.38 °C, expanded U = 0.77 °C (k = 2)

ADC and sampling

An $N$-bit ADC with reference voltage $V_{\text{ref}}$ divides the range into $2^N$ levels; one level is one LSB. Analog Devices tutorial MT-001 shows that ideal quantization noise has an rms value of $\text{LSB}/\sqrt{12}$, and the ideal SNR for a full-scale sine wave is $6.02N+1.76$ dB.

The other key issue is the sampling rate. Following Shannon (1949), you must sample faster than twice the highest frequency in the signal. Otherwise high frequencies disguise themselves as low frequencies that are not really there, which is called aliasing, like car wheels in a video that seem to turn slowly or backwards. On a drone, propeller vibration sampled too slowly can become a slow oscillation that the controller tries to correct. Signal conditioning therefore needs an anti-aliasing filter before the ADC.

import math

N, VREF = 12, 3.3
lsb = VREF / 2 ** N
print(f"LSB {lsb * 1000:.3f} mV, quantization noise {lsb / math.sqrt(12) * 1000:.3f} mV rms, "
      f"ideal SNR {6.02 * N + 1.76:.1f} dB")

def alias(f, fs):
    return abs(f - round(f / fs) * fs)

for f, fs in ((180, 200), (1170, 1000), (450, 1000)):
    print(f"{f:>5} Hz sampled at {fs:>4} Hz appears at {alias(f, fs):>3.0f} Hz")

diff = max(abs(math.sin(2 * math.pi * 180 * n / 200) + math.sin(2 * math.pi * 20 * n / 200)) for n in range(200))
print("180 Hz samples match an inverted 20 Hz wave:", diff < 1e-9)

LSB 0.806 mV, quantization noise 0.233 mV rms, ideal SNR 74.0 dB
  180 Hz sampled at  200 Hz appears at  20 Hz
 1170 Hz sampled at 1000 Hz appears at 170 Hz
  450 Hz sampled at 1000 Hz appears at 450 Hz
180 Hz samples match an inverted 20 Hz wave: True

Module lab

Common mistakes

Summary

• The measurement chain has several stages, and a result needs both a value and an uncertainty.
• Least squares uses every calibration point and reveals nonlinearity through the residuals.
• Combined uncertainty follows the GUM law of propagation with sensitivity coefficients.
• An ADC adds quantization noise of LSB/√12, and signals must be filtered before sampling to prevent aliasing.

Check your understanding

1. What is the LSB of a 10-bit ADC with a 5 V reference?
2. If $T=(V-b)/a$ with $a=0.01$ V/°C, what is the sensitivity coefficient of $V$?
3. Three uncertainty sources of 0.3, 0.4 and 0 °C combine to what?
4. A 90 Hz signal sampled at 100 Hz appears at what frequency?
5. Why filter before the ADC rather than in software afterwards?