# Calibration of a 3-Component External Balance for MAVs Wind Tunnel Research
## 3 Force Measurements with Balance  
When the balance is loaded with a load vector or state of load \([F_{L}]\), the resulting output signal is measured to generate the matrix \([F_{R}]\). These matrices can be related by the following expression (1) [19]:  
\[[F_{R}]=[K][F_{L}] \tag{1}\]  
where \([K]\) is the calibration matrix and the subscripts \"\(R\)\" and \"\(L\)\" are indicating reading and applied load, respectively. Measured forces will be given by the recorded signal in the load cells, but when a pure load is applied, all load cells record a signal, mainly due to misalignments and deformations of the balance, producing an error called the interaction load effect. The balance equation gives applied loads as a function of the measured signal using the following formula:  
\[[F_{L}]=[E][F_{R}] \tag{2}\]  
where \([E]\) is the evaluation matrix, calculated as \([E]=[K]^{-1}\).  
This equation is used during the wind tunnel tests so that applied loads are unknown, output signals are measured, and matrix A is previously determined using the calibration process. Usually, in practice, forces acting on balance are given by the following matrix expression:  
\[[F]=[A][U]+[R] \tag{3}\]  
where \([F]\) is the force matrix, \([A]\) is the unknown matrix to be determined by calibration, \([U]\) is the output signals matrix containing the output of the three load cells, and finally, \([R]\) is the residual matrix of the fitting mathematical model used.  
The forces acting on the external 3-component balance can be expressed as a function of the output voltages of the 3 load cells by the following second-order polynomial expression:  
\[F_{i}=\sum_{j=1}^{N}A_{ij}U_{j}+\sum_{j=k}^{N}\sum_{k=1}^{N}A_{ijk}U_{j}U_{k} +R_{i} \tag{4}\]  
where \(F_{i}\) is the force or moment of the balance at position \"\(i\)\" within the matrix of forces and moments, \(U_{j}\) are the output voltages of the load cells (\(j=1,2,3\)) and the term \(A_{ij}\) represents the linear terms of the mathematical model. Moreover, \(A_{ijk}\) corresponds to the second-order contributions, and \(R_{i}\) is the residual or fit error. The upper index of the summation is \(N=3\), since there are only 3 output signals from the load cells.  
The generic development of a force or moment component, resulting from a combination of the 3 output voltage signals recorded in the acquisition system, is detailed below. Therefore, Expression (4) is broken down into first-order linear terms and up to second-order terms (quadratic and cross terms).  
Figure 5: 3D view (**left**) and manufactured assembly (**right**) corresponding to the junction between the external balance and the wind tunnel model.  
To obtain the matrix A an estimator of the force matrix \(\big{[}\hat{\mathbf{F}}\big{]}\) is defined as follows:  
\[\big{[}\hat{F}\big{]}=[A][U] \tag{11}\]  
So that the residual matrix is given as follows:  
\[[R]=[F]-\big{[}\hat{F}\big{]} \tag{12}\]  
The matrix \([A]\) is obtained by multivariate linear regression (linear in parameters and nonlinear in variables), resolving it by the method of least squares in matrix form. The problem solution is to find the matrix with the minimum square error, and multiplying both sides of the equation by \([U]^{\mathsf{T}}\), we obtain the following equation:  
\[\big{[}\hat{F}\big{]}[U]^{\mathsf{T}}=[A][U][U]^{\mathsf{T}} \tag{13}\]  
and finally, the matrix \([A]\) is as follows:  
\[[A]=\big{[}\hat{F}\big{]}[U]^{\mathsf{T}}\Big{(}[U][U]^{\mathsf{T}}\Big{)}^{-1} \tag{14}\]  
where \(T\) and \(-1\) indicate transpose matrix and inverse, respectively.