Crosstalk correction: true forces from three contaminated channels

What is calculated?

Ideally each measuring point of a multi-component element would respond only to ‘its’ force (Keil eq. 10.83: identity matrix). Manufacturing tolerances and gauge positioning errors, however, cause crosstalk: the reading F_x* also contains shares of F_y and F_z. Keil describes this with the crosstalk matrix of eqs. (10.79)–(10.82), whose coefficients are determined during calibration by applying one force at a time and reading all channels. In service the system must then be solved for the true forces. Keil's excavator tooth (eqs. 10.84–10.86) is stored as the default: the transverse force F_y crosstalks by 36.5 % into the F_x channel – without correction F_x would be far too large.

Equations

F_x* = q_xx·F_x + q_xy·F_y + q_xz·F_z (Keil Gl. 10.79)

F_y* = q_yx·F_x + q_yy·F_y + q_yz·F_z (Keil Gl. 10.80)

F_z* = q_zx·F_x + q_zy·F_y + q_zz·F_z (Keil Gl. 10.81)

D = det(Q) (Keil Gl. 10.82); F_i = D_i/D (Cramersche Regel, D_i mit Spalte i durch F* ersetzt)

Fehler ohne Korrektur = (F_i* − F_i)/|F_i|

Assumptions and typical mistake

Linear crosstalk (coefficients independent of load level) and three components; moments can be treated the same way if regarded as channels. The coefficients apply only to the calibrated range and load point; hysteresis and temperature effects are not included.

Swapping rows and columns of the matrix (q_xy is the influence of F_y on channel X, not the other way round); entering coefficients in percent instead of as a factor (36.5 % = 0.365); entering readings in mV/V instead of kN although the diagonal is set to 1.

What you enter – and where the values come from

Reading channel X: F_x* [kN]

The reading of measuring channel X, i.e. the measuring point installed for the force F_x – already converted to force (amplifier scaled to kN). The asterisk means: contaminated by crosstalk, not yet the true force. Source: the reading of the amplifier or data acquisition for this channel under load.

Reading channel Y: F_y* [kN]

The reading of measuring channel Y, i.e. the measuring point installed for the force F_y – already converted to force (amplifier scaled to kN). The asterisk means: contaminated by crosstalk, not yet the true force. Source: the reading of the amplifier or data acquisition for this channel under load.

Reading channel Z: F_z* [kN]

The reading of measuring channel Z, i.e. the measuring point installed for the force F_z – already converted to force (amplifier scaled to kN). The asterisk means: contaminated by crosstalk, not yet the true force. Source: the reading of the amplifier or data acquisition for this channel under load.

q_xx (channel X ← F_x)

The main sensitivity of channel X: reading of the channel divided by the applied force F_x when only F_x acts. If the display was adjusted to the force during calibration this value is 1. Source: calibration with F_x acting alone.

q_xy (channel X ← F_y)

The crosstalk coefficient: reading of channel X divided by the applied force F_y when only F_y acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_y falsely appears as F_x. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_y acting alone (Keil Fig. 10.42).

q_xz (channel X ← F_z)

The crosstalk coefficient: reading of channel X divided by the applied force F_z when only F_z acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_z falsely appears as F_x. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_z acting alone (Keil Fig. 10.42).

q_yx (channel Y ← F_x)

The crosstalk coefficient: reading of channel Y divided by the applied force F_x when only F_x acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_x falsely appears as F_y. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_x acting alone (Keil Fig. 10.42).

q_yy (channel Y ← F_y)

The main sensitivity of channel Y: reading of the channel divided by the applied force F_y when only F_y acts. If the display was adjusted to the force during calibration this value is 1. Source: calibration with F_y acting alone.

q_yz (channel Y ← F_z)

The crosstalk coefficient: reading of channel Y divided by the applied force F_z when only F_z acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_z falsely appears as F_y. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_z acting alone (Keil Fig. 10.42).

q_zx (channel Z ← F_x)

The crosstalk coefficient: reading of channel Z divided by the applied force F_x when only F_x acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_x falsely appears as F_z. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_x acting alone (Keil Fig. 10.42).

q_zy (channel Z ← F_y)

The crosstalk coefficient: reading of channel Z divided by the applied force F_y when only F_y acts (Keil eqs. 10.79–10.81). A value of 0.1 means: 10 % of the force F_y falsely appears as F_z. Zero for ideal decoupling. Enter as a factor, not in percent. Source: calibration curve with F_y acting alone (Keil Fig. 10.42).

q_zz (channel Z ← F_z)

The main sensitivity of channel Z: reading of the channel divided by the applied force F_z when only F_z acts. If the display was adjusted to the force during calibration this value is 1. Source: calibration with F_z acting alone.

What this calculator is for

Multi-component force-measuring elements made from one part (excavator tooth, wheel hub, robot link, tool holder), wind-tunnel balances, multi-component transducers with incomplete mechanical decoupling.

What comes next

1. Calibrate: apply only F_x, only F_y, only F_z in turn and read all three channels each time; coefficient q_ij = reading of channel i divided by applied force j.
2. Enter the nine coefficients here (main diagonal usually 1 after display calibration).
3. In service enter the three readings; the results are the true forces, the error figures show how much the raw readings were contaminated.

How to read the results

• The three true forces are the result; the error figures show how far off the uncorrected readings were. If all errors are below about 1 %, the correction can be skipped in everyday use.
• The correction is only as good as the calibration: recheck the coefficients regularly, especially after modifications or re-bonding.
• Keil's default values illustrate the principle: for a pure force F_x = 10 kN the channels read 10 / 0.9 / 0.46 kN – the calculator turns that back into 10 / 0 / 0.

What each value means

True force F_x [kN]

The force actually acting in the x-direction after the shares of the other forces have been removed from the reading – the value to work with.

True force F_y [kN]

The force actually acting in the y-direction after correction.

True force F_z [kN]

The force actually acting in the z-direction after correction.

Error without correction, channel X [%]

The percentage by which the raw reading of channel X would have deviated from the true force without correction. Shows whether the correction matters for this channel (Keil's excavator tooth: strongly, due to 36.5 % crosstalk from F_y).

Error without correction, channel Y [%]

Error of the uncorrected reading of channel Y in percent of the true force.

Error without correction, channel Z [%]

Error of the uncorrected reading of channel Z in percent of the true force.

Determinant of the crosstalk matrix D

The determinant of the crosstalk matrix (Keil eq. 10.82). 1 for complete decoupling; the further away, the more strongly the channels couple and the more sensitive the correction is to reading errors. Keil's excavator tooth: ≈ 0.96.

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.4 simultaneous measurement of several load components, eqs. (10.79)–(10.86), Figs. 10.41–10.42 (excavator tooth).

The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.

Frequently asked questions

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• Basic strain-gauge bridge equation: signal from four strains

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