I'm looking for the theory of an experiment that is giving me a hard time to perform.

I have an instrument composed of a rigid horizontal square plate rigidly supported under each corner by a load-cell.

Considering the origin of the system at the center of the platform, the problem statement is "find the point of application (PosX, PosY, relative to the origin at the center of the platform) of the ground reaction force of an object placed on this platform, given the vertical force measured in each of the load-cells at the corners (Z1, Z2, Z3 and Z4)".

I know that the total force vertical component so as the moments around X and Y axis, with the corners numbered clockwise from "upper left" corner as seen from above:

$Fz = Z1 + Z2 + Z3 + Z4$

$Mx = -Z1 - Z2 + Z3 + Z4$

$My = -Z1 + Z2 + Z3 - Z1$

and then:

$PosX = Mx/Fz$

$PosY = My/Fz$

Unfortunately, though, the system is hyperstatic, and then some papers and texts from manufacturers mention a cross-talk between sensors, thus requiring some sort of calibration matrix that should multiply the measured values to obtain the actual values:

$actual \begin{vmatrix}Fz \\ Mx \\ My\end{vmatrix} =\begin{vmatrix} C_1 & C_2 & C_3 \\ C_4 & C_5 & C_6 \\ C_7 & C_8 & C_9 \end{vmatrix} \times measured\begin{vmatrix}Fz \\ Mx \\ My\end{vmatrix}$

So the question proper is: "What's the theoretical basis for this calibration matrix (considering a square plate with hyperstatic support conditions of one support per corner) and how should one be derived from an array of experimental measurements in the form $(Z1, Z2, Z3, Z4) = f(Fz, PosX, PosY)$ ?

I'm not sure this question applies here, but it seems to me that it does, since it has a theoretical nature (although derived from a very practical prolem). If it is not the case, then I would appreciate very much any suggestion regarding where I could find help.

Thanks for reading!


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